Modeling the Biomechanics of the Lamina Cribrosa Microstructure in the Human Eye

Abstract

Glaucoma is among the leading causes of blindness worldwide that is characterized by irreversible damage to the retinal ganglion cell axons in the lamina cribrosa (LC) region of the optic nerve head (ONH), most often associated with elevated intraocular pressure (IOP). The LC is a porous, connective tissue structure that provides mechanical support to the axons as they exit the eye and the biomechanics of the LC microstructure likely play a crucial role in protecting the axons passing through it. There is a limited knowledge of the IOP-driven biomechanics of the LC microstructure, primarily due to its small size and the difficulty with imaging the LC both in vitro and in vivo. We present finite element (FE) models of three human eye posterior poles that include the LC microstructure and interspersed neural tissues (NT) composed of retinal axons that are constructed directly from segmented, binary images of the LC. These models were used to estimate the stresses and strains in the LC and NT for an acute IOP elevation from 0 to 45 mmHg and compared with identical models except that the LC was represented as a homogenized continuum material with either homogeneous isotropic neo-Hookean properties or heterogeneous properties derived from local connective tissue volume fraction (CTVF) and predominant LC beam orientation. Stresses and strains in the LC and NT microstructure were investigated, and results were compared against those from the models wherein the LC was represented as a homogenized continuum. The regionalized volumetric average stresses and strains showed that the microstructural model yielded similar patterns to our prior approach using an LC continuum representation with mapped LC CTVF/anisotropy, but the microstructural modeling approach allows analysis of the stresses and strains in the LC and NT separately. As expected, the LC beams carried most of the IOP load in the microstructural models but exhibited less strain, while the encapsulated NT exhibited lower stresses and much higher strains. Results also revealed that the continuum models underestimate the maximum strains in the LC beams and NT by a factor of 2–3. Microstructural modeling should provide greater insight into the biomechanical factors driving damage to the axons (NT) and LC connective tissue remodeling that occur in glaucoma. The methods presented are ideal for modeling any structure with a complex microstructure composed of different materials, such as trabecular bone, lung, and tissue engineering scaffolds such as decellularized LC. Matlab code for mesh generation from a segmented image stack of the microstructure is included as Supplemental Material.

Graphical Abstract

1. Introduction

Glaucoma, a leading cause of blindness worldwide, is the result of damage to the retinal ganglion cell axons as they pass out of the eye to the brain through the optic nerve head (ONH). The axons are supported by the lamina cribrosa (LC), a porous, connective tissue structure that spans the scleral canal that provides structural support as they leave the eye [1, 2]. Although experimental evidence has shown that glaucomatous damage initiates in the laminar region [3], the mechanisms of damage are not well understood [4]. There is a strong association between elevated intraocular pressure (IOP) and glaucoma onset and progression, and IOP lowering is the only effective treatment for the disease [5]. Elevated IOP has been hypothesized as a significant factor in inducing mechanical deformation or strain within the laminar region [2, 3, 6, 7] and it may cause a complex biological cascade, impairing axoplasmic transport of neurotrophic factors, inducing tissue hypoxia and activating glial cells [4, 8], which ultimately results in neuronal cell death. Mechanical strains in the ONH are dependent on eye-specific tissue morphology and material properties [7, 9]. This may contribute the wide variance in susceptibility to glaucoma, with some eyes developing or progressing at epidemiologically defined “normal” levels of IOP, while others are particularly resistant to IOP-related injury.

The LC connective tissue microarchitecture shares many morphologic similarities to porous, open-celled structures such as trabecular bone and engineered foams. The inhomogeneous and anisotropic characteristics of these materials can profoundly influence their structural load-bearing and directional stiffness characteristics. Prior studies have shown that LC pore and beam size vary regionally through the optic nerve head [10–13], and it has been hypothesized that this inhomogeneity is related to the distinctive patterns of glaucomatous damage and visual impairment[11, 14, 15]. The fabric tensor concept has been used to describe the structural anisotropy of LC pores and beams by quantifying the density and anisotropy of the LC microstructure [15, 16]. The mean intercept length (MIL) method can characterize local fabric in terms of the both the predominant orientation of the LC beams and relative strength of their anisotropy. The MIL method has been used to quantify and compare materials and tissues with complex microstructures in other fields [17, 18]. It can also be used to map LC material properties in Finite Element (FE) models of the posterior eye and ONH in which the LC is represented as an anisotropic heterogeneous bulk material [15, 19]. It is also known that the peripapillary sclera and pia have anisotropic collagen structures, where the collagen fibrils are preferentially aligned circumferentially around the ONH and optic nerve, respectively [20, 21].

While these continuum-modeling approaches can estimate the stress and strain in homogenized laminar tissues, a major disadvantage is that they cannot separate the mechanics of the LC beams from the contained neural tissues (NT). Several studies have used a two-step multiscale approach to simulate the stresses and strains in the LC beams and NT. In the first step, a macro-scale FE model of the ONH, often part of a larger model of the posterior eye, is used to calculate macro-scale deformations using a homogenized continuum model of the tissues in the laminar region. In the second step, the macro-scale deformation results are used as a boundary condition for a microscale model of the LC in a specific sub-region. The multiple scales of this two-step modeling approach are not fully coupled, where the micro-scale results are impacted by the macro-scale model, but the micro-scale model has no impact on the macro-scale results. We first used this multiscale modeling approach to construct a macroscale model of the posterior pole of the eye to investigate the role of acute IOP elevation in the regional distribution of stress and strain in the LC microstructure [22]. More recently, Sigal and colleagues also employed macroscale, beam-aware LC FE models to calculate ONH deformations at an IOP of 30 mmHg. Similar to our approach, the resulting deformations were then used as displacement boundary conditions for a small region of the LC microscale model [23]. The mechanical insult to the NT within the pores of the LC was calculated as a function of the LC microstructure and mechanical properties. Another study by the same group using the same FE modeling approach to determine how the architecture of the LC microstructure, specifically the size and shape of the LC pores, influences the IOP-induced deformation of the NT within the LC pores [24]. Grytz and coworkers proposed a fully coupled, two-scale analysis of the idealized LC to translate the IOP load at the macroscale to the mechanical insult of the axons within the microstructure of the LC. Their simulation suggests that the collagen structures of the LC and the surrounding peripapillary sclera effectively provide mechanical support to the axons by protecting them from high tensile stresses even at elevated IOP levels [25].

The primary advantage of the multiscale modeling approach is that it can calculate the stresses and strains in the LC and NT microstructures separately, although prior approaches have several limitations [22–25]. Boundary conditions play a pivotal role in FE simulations and assigning suitable boundary conditions at the micro- and/or macro-scale is not a trivial task. In the aforementioned studies that used a two-step multiscale approach, the macro-scale displacement boundary conditions were only applied to the peripheral surface nodes of the cut edges of the subregion of the LC micro-scale models. However, the LC deformation at the macro-scale is also present at the anterior and posterior LC surfaces, as well as on the LC microstructure surfaces inside the pores within the subregion, so this is an imperfect boundary condition. Fully coupled multi-scale modeling approaches often assume that characteristic lengths of the microstructure are multiple magnitudes smaller than characteristic lengths at the macro-scale. This assumption may not be valid for the LC, where LC beams diameters have been reported to be only ~ 5 times smaller than the LC thickness [13]. Hence, there is a need to develop a better approach to capture realistic mechanical conditions for the complex LC microstructure and ensure accurate stress and strain estimates therein. Another major disadvantage of the two-step multiscale approach is that the LC microstructure models ignore the important load-bearing role of other components of the eye, especially the prelaminar NT and retrolaminar optic nerve that are continuous with the NT in the LC. In addition, modeling the LC microstructure in small chunks cannot estimate the contours of the stresses and strains across the entire LC region of the ONH. Our more recent studies benefited from a more comprehensive macro-scale model that included the retina, sclera, optic nerve, and pia, in addition to locally variable connective tissue volume fraction (CTVF) and LC beam orientation using MIL data using a LC microstructure-based material formulation [26, 27]. While this is a marked improvement in modeling the macro-scale parent model, the problems inherent to modeling the mechanical environment of the LC microstructure remain.

Given the limitations of the multiscale LC modeling approaches outlined above, Midgett et al., conducted an experimental study in human cadaver eyes using an inflation test from 10 to 45 mmHg to quantify the deformation response of the human LC with respect to both age and anatomical region [20]. They evaluated the strain tensors, as well as the in-plane principal and maximum shear strains, using digital volume correlation (DVC) for both the central and peripheral regions of the LC and the nasal, temporal, inferior, and superior quadrants surrounding the central retinal vascular trunk. In this study, older age was associated with lower strains, the maximum shear strain was larger in peripherally, and tensile strains were lowest in the nasal quadrant. In a second study, the inflation response of the LC and adjacent peripapillary sclera were estimated in post-mortem human eyes with no history of glaucoma [28]. The displacements and deformations induced within the human LC microstructure by an acute IOP increase from 10 to 50 mmHg were also experimentally measured [29]. These experimental inflation studies all relied on second harmonic image generation to image the collagenous LC beams after all retrolaminar tissues had been dissected away, which ignores the important contributions of the pia and retrolaminar optic nerve to LC biomechanics [30–34]. Also, limitations in microscope field of view required that small regions of the LC be imaged in series and then stitched together to form the full LC microstructure, resulting in mismatches at the image region boundaries due to regional changes in deformation and strain that accumulate during the extended time necessary for imaging.

Recent advances in clinical imaging have allowed for in vivo characterization of the deformation of the human ONH using optical coherence tomography, a 3D imaging technique that yields image volumes that can be processed with digital volume correlation [35], or digital image correlation [36]. These latter imaging approaches can estimate bulk strains in the ONH tissues as a whole, but lack the resolution necessary to measure deformations or estimate strains in the LC or NT separately, and can only provide data in the portions of the LC that aren’t obscured by overlying vasculature, Bruch’s membrane, or sclera. While these studies have improved our knowledge of laminar and ONH biomechanics, they are limited by either imaging the LC microstructure from the posterior side in experimental inflation after having removed the important load-bearing retrolaminar tissues, or imaging the ONH in vivo without the resolution to discern the LC microstructure. Hence, there is the need for development of a comprehensive posterior eye model that includes the other components of the eye, wherein the LC microstructure can be captured as a whole with suitable boundary conditions.

This is the first study that incorporates the complete and patient-specific LC microstructure into a macro-scale FE model of the human eye, where the micro-scale stresses and strains can be computed across the entire LC and NT microstructure within a model in which the other important load-bearing tissues are represented. These models were used to estimate the stresses and strains in the LC and NT under acute IOP elevation and compared with identical models in which the LC was represented as a continuum material with either homogeneous, isotropic neo-Hookean properties or our prior approach using heterogeneous anisotropic properties derived from local LC connective tissue volume fraction (CTVF) and the predominant orientation of the LC beams [37], after adjusting the LC material properties to ensure identical LC deformations across approaches within eyes. The goal of the current work was to determine if modeling the LC and NT microstructure directly resulted in significantly different stresses and strains compared to prior studies that employed a homogenization or multi-scale subregion modeling approach. The findings of this study have implications for not only understanding the biomechanics of the retinal axons that transmit visual information to the brain, and the load bearing LC that protect them from damage, but also for investigating the mechanisms of glaucomatous damage.

2. Materials and Methods

2.1. 3D Eye-specific FE Model of the Human Optic Nerve Head

The macro-architecture of the model is first defined by 3D delineation of anatomic tissue surfaces within a high-resolution, histologic, fluorescent 3D reconstruction of the posterior eye and ONH obtained from three human donors of European descent with normality confirmed by ophthalmic clinical record review, as described previously [38]. Briefly, human donor eyes were procured from the Alabama Eye Bank within 6 hours postmortem, then immersion fixed in aldehyde fixative while the internal pressure was held at 10 mmHg. The ONH and surrounding peripapillary sclera were trephined, embedded in light-blocking paraffin, then sectioned on a microtome at 1.5 μm thickness, with a fluorescent image (1.5×1.5μm resolution) of the exposed block face acquired after each section is taken. The resulting image stack was aligned using laser displacement measurements of the block position after each section was taken, and combined into an image volume [38]. Custom delineation software (MultiView, courtesy of Dr. Claude F. Burgoyne) was used to section the 3D fluorescent ONH reconstruction volume at forty, equally spaced, radial, sagittal planes centered on the ONH [39]. Within each radial image section, the anatomic surfaces were delineated using 2D Bezier curves to define the morphology of the neural canal, LC, peripapillary sclera, and pia [40, 41]. 3D surfaces are fit to the families of Bezier curves defining each anatomic surface and the resulting eye-specific geometries of the ONH and peripapillary sclera are then fit into a larger generic posterior scleral shell with anatomic shape and thickness [15, 26, 27]. Finally, a parameterized, anatomic surface defining the anterior surface of the prelaminar NT, retina, and choroid is added [26]. These surfaces are then utilized to build the high-fidelity, eye-specific, 3D FE macro-scale mesh of the human posterior eye that includes the sclera, LC, pre- and retro-laminar NT, retina, and pia as presented in Figs. 1a & b. A coarser mesh was generated for the anterior areas of the retina and sclera, as well as the posterior area of the optic nerve, and the mesh was more refined where more accurate estimates of stresses and strains were needed, including the peripapillary sclera, scleral flange, LC, and immediate retrolaminar optic nerve and pia.

Fig. 1.

The 3D human eye-specific FE model viewed from (a) the posterior and (b) anterior. Cross sections through (c) temporal-nasal and (d) superior-inferior axes showing the detailed structure of the ONH are presented. The LC in the model consists of the collagenous LC microstructure and the interspersed NT as viewed from the (e) posterior and (f) anterior. Cross sections through (g) temporal-nasal and (h) superior-inferior axes of the laminar volume showing the LC and NT microstructures.

To create the mesh of the LC microstructure, the LC was segmented from the original 12-bit grayscale images of the laminar region using a custom automated algorithm that leverages a priori knowledge of the 3D trabeculated structure of the LC as described previously [42]. The volume tetrahedral mesh, composed of 10-noded elements, was generated using a custom Matlab (Mathworks, Natick, Massachusetts, US) script [27] as described fully in section 2.2 below. The connectivity of the elements across tissue/geometric boundaries was maintained at their nodal interfaces using an automated mapping algorithm, which allowed us to avoid defining contacts or tying constrains between the model components, thereby eliminating any possible contact complications [27]. Three different 3D eye-specific FE models of the human ONH with the same element edge length of ~14–20 μm in the LC were constructed. The complete, volume-meshed LC and NT micro-structure was translated and rotated into the same coordinate system as that of the macro-scale model using a customized Matlab script and incorporated into our eye-specific 3D human ONH model by cutting the volume-meshed LC beams based on the LC volume in the parent FE model. A customized Matlab script was used to identify and retain the LC beams that lie inside the LC volume of the ONH, while the remainder of the microstructure residing outside the LC boundary were deleted. All elements outside the beams themselves or in the pores between the LC beams were considered NT. The final quadratic tetrahedral (C3D10) mesh of the LC beams and NT were incorporated into the human ONH model as shown in Fig. 1. Detailed information on the number of elements/nodes, element length, Jacobian, volume aspect ratio, volume skewness, and simulation time for three different eye models of donors 118, 119, and 129 are summarized in Table 1.

Table 1.

The number of elements/nodes, element edge length, Jacobian, volume aspect ratio, volume skewness, and simulation time for the three donor eye models.

Eye model	Number of elements	Number of nodes	Element length (μm) Min/Max	Jacobian	Volume aspect ratio	Volume skewness	Simulation time (hr)
Model 118	1,152,302	1,551,302	14/20	0.999	1.055	0.003	53.80
Model 119	704,621	959,334	14/20	0.999	1.062	0.004	28.21
Model 129	978,411	1,318,984	14/20	0.999	1.061	0.005	32.23

2.2. Creation of the LC Volume Mesh from Binary Images of the LC

The binary images of the LC (Fig. 2a) are stacked in a volume representing the entire LC microstructure. The images were stacked to construct a triangular isosurface image volume using a constrained Delaunay tetrahedralization approach as shown in Fig. 2b. Since the isosurface was not closed, the surface edges located on the bounding box of the volume were calculated and a piecewise linear complex was added to form a closed surface using a free Iso2Mesh library [43, 44]. The triangular surface meshes were directly generated from the input isosurface volume using a modified surface extraction program built on a free library, CGAL [45]. A mesh repairing process based on a surface simplification approach was applied to the resulting surface to remove topological complications, such as isolated vertices (vertices that are not contained by any surface element), duplicated triangles (multiple triangles that share the same vertices), and nonmanifold vertices [46]. These shortcomings would lead to a cascade of difficulties with tetrahedral mesh generation. A customized Matlab script based on the free JMeshLib library [46], was used to find the non-manifold vertices and repair them by replication.

Fig. 2.

The reconstruction of the eye-specific 3D LC microstructure from the segmented binary image stack. (a) The stack of segmented LC binary images were used to construct (b) an isosurface image volume of the LC microstructure. The isosurface image volume was then volume-meshed using our meshing algorithm, as shown in the (c) isometric and (d) posterior views.

The input volume may contain multiple sub-regions or tissue types, in this case LC beams and NT, so correctly tagging the resulting 3D tetrahedral elements with the associated region is critical for the FE analyses. The determination of an interior point for each sub-region in a volumetric image could be a solution; however, it is not a trivial task because the extracted isosurfaces are not necessarily convex. A distance-field method was used to robustly determine an internal point for each sub-domain. First, all the edge voxels of each sub-domain were identified then set to 1 inside the boundary and 0 elsewhere. Thereafter, a Gaussian smoothing kernel (smoothing factor α = 0.005) was applied for 10 iterations to the resulting array using a custom Matlab script. This produced a 3D field where the values are inversely related to the distances from the domain boundaries. The application of this method assures that the resulting points are located strictly inside the specified sub-regions and are roughly N (depending on the geometry of the model) voxels away from the boundaries. Here, an N=3 or 4 was found to work robustly for LC with only minimal additional computation. Since our model contained pore regions, these regions were simply tagged with a label and the same algorithm was applied to determine the interior points. These regions were eventually removed from the final tetrahedral mesh using their regional labels.

It was important to ensure that the surfaces constructed from the binary images did not have openings or holes, especially at their peripheral boundaries. Some software packages, such as Meshmixer and Mimics, can close a nonmanifold surface. Here, we developed an approach that assumed that the edges of the open surfaces are all located inside the volume-bounding box, and thus piecewise linear complexes on the bounding box are needed to enclose the surfaces. All open edges of the surfaces were computed and then organized into a list of disjointed loops to generate water-tight surfaces. If a loop crossed the intersections of two or three bounding box faces, it was further decomposed into sub-loops that conform to a single face. A regional exclusion algorithm was then applied to flatten the bounding faces by the polygons enclosed by the edge loops. The Matlab code for mesh generation from a segmented image stack of the microstructure is included as Supplemental Material. Our previously proposed mesh generation algorithm [27] was then used to create quadratic tetrahedral meshes (C3D10) from the closed, watertight, triangular surface mesh. The final meshed LC microstructure is shown in Figs. 2c & d.

The final generated mesh exhibit different densities by controlling the surface and volume-element density parameters, which enables us to accurately capture surface features. A boundary overlay comparison was performed between the 2D binary image of the LC and its corresponding mesh section to assess the accuracy of the volume-meshed LC relative to the ground truth images. In addition, a customized Matlab script was developed to cut the binary images and construct a volume mesh to compare the accuracy of our 3D meshing algorithm to the LC binary image volume.

The microstructure model with an element edge length of ~ 5 μm for the LC beams results in models with millions of elements that cannot be run on our server due to lack of memory (40-core Intel® Xeon® CPU E7–4870@2.40 GHz computer with 512GB RAM). Hence, developing models of no more than two million elements was critical. To accomplish that, the LC FE model of the 3D eye-specific human ONH was re-meshed to both keep its nodal connectivity at the components’ interfaces, and be dense enough to capture the microstructural geometry of the LC and NT for biomechanical simulation purposes. Therefore, the isolated LC in the 3D eye-specific ONH models for donors 118/119 was meshed at three different element edge lengths: ~ 10, 15, and 20 μm that maintain the precise microstructural geometry of the LC beams and also run within a reasonable simulation time. These meshes of the LC only resulted the element/node counts of 1,294,585/1,881,515, 561,867/823,055, and 365,828/509,059 for the LC with the element edge lengths of ~ 10, 15, and 20 μm, respectively. As stated previously, an element edge length of ~5 μm led to meshes with >2 million elements, and so this case could not be simulated. To compare the mesh geometries, the 2D area and 3D volume of LC beams meshed at 5 μm edge length were compared to LC models with element edge lengths of ~ 10, 15, and 20 μm using a custom Matlab script. To compare their simulation results, the LC FE models with 10, 15, and 20 μm element edge lengths were assigned homogeneous isotropic neo-Hookean material properties with a shear modulus of 0.226 MPa (the same as that was used for donor 118). The pressure of 20 mmHg was then applied on the anterior surface of the LC while the peripheral nodes of the LC were constrained in the X, Y, and Z directions. The resultant regional volumetric stress and strain in the LC and NT microstructures were compared for the three LC element edge lengths.

2.3. Material Properties and Boundary Conditions

Two different material properties were used to represent the LC as a homogenized material in the continuum models, mapped CTVF/heterogeneous anisotropic and isotropic hyperelastic neo-Hookean, which were compared to a LC microstructural model with isotropic hyperelastic neo-Hookean properties for the LC and NT. The microstructural model of the LC explicitly includes the beam structure so the size, CTVF, and orientation of the beams are inherent to the microstructural model itself.

A mesh-free approach was employed to define the heterogeneous mechanical properties of the sclera and pia in all of our models, as well as for the LC in the continuum model with mapped CTVF/anisotropy, as described in detail in our recent publications [26, 27]. Briefly, we took advantage of the assumption that material parameters can be measured or approximated at discrete locations, which are arbitrary and independent of the FE mesh. This is a reasonable assumption given that material properties in biologic structures tend to change in a smooth fashion within tissues, only changing rapidly oat discrete tissue boundaries, where the material property descriptions also change, such as the transition between the LC and the adjacent sclera that are represented as separate solids in the model. We implemented our mesh-free approach in the open-source FE package CalculiX [47], which allows us to approximate the local LC CTVF (Fig. 3a), as well as the direction and strength of the predominant LC beam orientation via a 3D structure tensor at the Gauss points of the FE mesh (Fig. 3b). The local variation LC CTVF and degree of anisotropy and is illustrated in Fig. 4 for the three human donor eyes modeled in this study. The regional CTVF exhibits considerable regional variation within and between eyes. There are distinct regions in which the CTVF is markedly higher, particularly in areas encompassing the central vascular trunk, where the connective tissue structure is robust.

Fig. 3.

(a) A segmented 3D LC reconstruction of a human eye, showing large variation in local CTVF and predominant LC beam orientations; local CTVF and the fabric tensor were calculated using MIL in overlapping conical volumes with discrete center locations that serve as control points for the mesh-free approach used in the homogenized continuum models of the LC region. (b) Contour plot of the CTVF in the FE model of the LC. Red dots represent the control points at which the microstructure was evaluated. (c) First principal directions of the fabric tensor, where the color of the principal directions represents the local degree of anisotropy; The mesh-free approach leads to a smooth approximation of both the material parameter and the fabric tensor across the entire LC region; (d) Distribution of the preferred collagen fibril orientation in the material description of the peripapillary sclera and pia. Adapted from Grytz, et al. with permission [26].

Fig. 4.

The distribution of the predominant LC beam orientation (arrows) plotted over the regional magnitude of the LC CTVF (colors) in the continuum models with mapped CTVF and anisotropy for human donor eyes (a) 118, (b) 119, and (c) 129.

We also used our mesh-free approach to assign material anisotropy to the sclera and pia, as the collagen fibrils are preferentially aligned circumferentially around the scleral canal and retrobulbar optic nerve in the peripapillary sclera and pia, respectively. The circumferential orientation was defined at the scleral canal as the tangential direction to an ellipse fit to the anterior insertion points of the LC into the scleral canal wall and pia (Fig. 3d). The material parameters of the eye-specific FE models with three different material modeling approaches for the LC of eye models from donors 118, 119, and 129 are listed in Tables 2, 3, & 4, respectively.

Table 2.

The material parameters of the eye-specific FE model of donor 118 based on the age (40 year-old female) and ethnicity (European) as determined from our prior studies [64]. The bulk modulus was set to κ=100μ for all tissues.

Tissue	Shear modulus (μ) in MPa	Elastic modulus of collagen fibers (Efib) in MPa	Crimp angle of collagen fibers (θ0)	Crimp parameter (R0/r0)	Average CTVF
Continuum model with mapped CTVF/anisotropy
Retina	0.01	-	-	-	-
Sclera	0.226	54.48	6.29	4.54	-
Lamina cribrosa	0.226	54.48	6.29	4.54	-
Pia	0.226	54.48	6.29	4.54	-
Optic nerve	0.01	-	-	-	-
Continuum model with isotropic neo-Hookean properties
Retina	0.01	-	-	-	-
Sclera	0.226	54.48	6.29	4.54	-
Lamina cribrosa	0.172	-	-	-	0.38
Pia	0.226	54.48	6.29	4.54	-
Optic nerve	0.01	-	-	-	-
Microstructural model with neo-Hookean properties
Retina	0.01	-	-	-	-
Neural tissue	0.01	-	-	-	-
Sclera	0.226	54.48	6.29	4.54	-
Lamina cribrosa	0.452	-	-	-	-
Pia	0.226	54.48	6.29	4.54	-
Optic nerve	0.01	-	-	-	-

Table 3.

The material parameters of the eye-specific FE model of donor 119 based on the age (40 year-old male) and ethnicity (European) as determined from our prior studies [64]. The bulk modulus was set to κ=100μ for all tissues.

Tissue	Shear modulus (μ) in MPa	Elastic modulus of collagen fibers (Efib) in MPa	Crimp angle of collagen fibers (θ0)	Crimp parameter (R0/r0)	Average CTVF
Continuum model with mapped CTVF/anisotropy
Retina	0.01	-	-	-	-
Sclera	0.298	35.72	5.31	4.54	-
Lamina cribrosa	0.298	35.72	5.31	4.54	-
Pia	0.298	35.72	5.31	4.54	-
Optic nerve	0.01	-	-	-	-
Continuum model with isotropic neo-Hookean properties
Retina	0.01	-	-	-	-
Sclera	0.298	35.72	5.31	4.54	-
Lamina cribrosa	0.142	-	-	-	0.47
Pia	0.298	3572	5.31	4.54	-
Optic nerve	0.01	-	-	-	-
Microstructural model with neo-Hookean properties
Retina	0.01	-	-	-	-
Neural tissue	0.01	-	-	-	-
Sclera	0.298	35.72	5.31	4.54	-
Lamina cribrosa	0.410	-	-	-	-
Pia	0.298	35.72	5.31	4.54	-
Optic nerve	0.01	-	-	-	-

Table 4.

The material parameters of the eye-specific FE model of donor 129 based on the age (40 year-old female) and ethnicity (European) as determined from our prior studies [64]. The bulk modulus was set to κ=100μ for all tissues.

Tissue	Shear modulus (μ) in MPa	Elastic modulus of collagen fibers (Efib) in MPa	Crimp angle of collagen fibers (θ0)	Crimp parameter (R0/r0)	Average CTVF
Continuum model with mapped CTVF/anisotropy
Retina	0.01	-	-	-	-
Sclera	0.216	58.14	6.44	4.54	-
Lamina cribrosa	0.216	58.14	6.44	4.54	-
Pia	0.216	58.14	6.44	4.54	-
Optic nerve	0.01	-	-	-	-
Continuum model with isotropic neo-Hookean properties
Retina	0.01	-	-	-	-
Sclera	0.216	58.14	6.44	4.54	-
Lamina cribrosa	0.160	-	-	-	0.27
Pia	0.216	58.14	6.44	4.54	-
Optic nerve	0.01	-	-	-	-
Microstructural model with neo-Hookean properties
Retina	0.01	-	-	-	-
Neural tissue	0.01	-	-	-	-
Sclera	0.216	58.14	6.44	4.54	-
Lamina cribrosa	0.304	-	-	-	-
Pia	0.216	58.14	6.44	4.54	-
Optic nerve	0.01	-	-	-	-

Regarding the applied boundary conditions, the nodes of the sclera and retina along the cut face of the globe at the equator were selected and assigned to allow for radial displacement only in the plane of the equator. The inner surface of the retina was also selected to apply the pre-stressing load of 10 mmHg with no deformation [48] followed by an IOP elevation to 45 mmHg. Contact definition was not necessary between the tissue components, as our fully automatic meshing program maps the surface of the components onto each other and maintains the connectivity of the nodes at the component interfaces. A customized Matlab script was used to detect the boundary condition nodes and surfaces, equivalence the nodes at the component interfaces, define the materials’ sections, define the element sets, and write the final input file. A 40-core Intel® Xeon® CPU E7–4870@2.40 GHz computer with 512GB RAM was used to run the simulations in the open-source FE package CalculiX.

To compare results across models, stresses and strains were averaged regionally, either as volume averages or weighted percentiles. To calculate the weighted percentile, where the element volumes were used as weights, the stress or strain results in the entire LC and NT elements were first ranked from the lowest to the highest. Thereafter, the accumulated elemental volume was calculated as V_n-1+V_n and the accumulated volume was then divided by V_Total, resulting in a volume percentile. The stress or strain associated with the 95^th volume percentile was reported. When comparing the model results across different approaches, we attempted to isolate the changes to the mechanical effects of laminar homogenization and/or microstructural modeling approaches. This necessitated that one of the resultant outcome variables – deformation, stress, or strain – be held constant across all models. Since ocular biomechanics studies most often focus on stress and strain outcomes, LC deformation was held constant across all models within eyes by adjusting the LC material properties, so that the LC stress and strain distributions and magnitudes could be directly compared across the three modeling approaches.

3. Results

3.1. Connective Tissue Volume Fraction Comparison

The regional maps of LC CTVF and predominant LC beam orientation are shown in Fig. 4 for the three human donor eyes. Eyes were intentionally chosen such that they represent a wide range of CTVF values, with donor 119 exhibiting a relatively dense LC structure and donor 129 exhibiting a relatively sparse LC microstructure. One would expect that regional strain patterns would be inversely proportional to LC CTVF, which is apparent in comparisons of Fig. 4 to Figs. 8–10. The CTVF in donor 119 was found to be higher superiorly and nasally (periphery and midperiphery) as well as temporal periphery, with the most porous regions occurring variably in the inferior midperiphery. However, when it comes to donor 129, the highest CTVF was observed at the superior periphery and midperiphery as well as the nasal; periphery, whereas the lowest CTVF was found at the temporal periphery and inferior midperiphery as presented in Fig. 4.

Fig. 8.

Polar plot of the regionalized volumetric average (a) von Mises stress, (b) 1^st principal stress, (c) 3^rd principal stress, (d) maximum shear stress, (e) 1^st principal strain, (f) 3^rd principal strain, (g) maximum shear strain, and (h) displacement in the LC of donor 118 for three different material descriptions: the continuum model with mapped LC CTVF/anisotropy, continuum model with homogeneous isotropic neo-Hookean LC properties, and the microstructural LC model with neo-Hookean properties.

Fig. 10.

Polar plot of the regionalized volumetric average (a) von Mises stress, (b) 1^st principal stress, (c) 3^rd principal stress, (d) maximum shear stress, (e) 1^st principal strain, (f) 3^rd principal strain, (g) maximum shear strain, and (h) displacement in the LC of donor 129 for three different material descriptions: the continuum model with mapped LC CTVF/anisotropy, continuum model with homogeneous isotropic neo-Hookean LC properties, and the microstructural LC model with neo-Hookean properties.

3.2. Mesh Comparison

The boundaries of the generated LC volume mesh (~ 5 μm element edge length) were compared against the binary LC images at the same section as displayed in Fig. 5a-5c using a custom Matlab script. The boundary overlay comparison yielded an error of 0.75 and 0.96% in the area and volume, respectively, for this high element density. A more detailed analysis for three different regions of the LC volume is illustrated in Fig. 5d-5e, wherein comparisons yielded errors in image to mesh area of 1.52%, 1.35%, and 1.16% for regions 1, 2, and 3, respectively. The boundary overlay comparison between the binary images of the LC and our mesh shows that the meshing algorithm can precisely capture the detailed microstructure, orientation, and size of the LC beams across the entire LC.

Fig. 5.

The boundaries of the LC beams in (a) a binary image and (b) a corresponding mesh section were extracted and the (c) overlay comparison was performed. To quantify the accuracy of the FE mesh compared to the binary images, (d) three different regions were selected and digitally sectioned; (e) The boundaries of the binary images and mesh sections of the LC are overlaid for comparison in these three regions.

3.3. Finite Element Edge-Length Comparison

An element length of ~ 5 μm yielded models that were too large to be simulated efficiently on our computers, so we re-meshed to element edge lengths of ~ 10, 15, and 20 μm, and compared the mesh and LC binary image boundaries for these meshes with reduced element count. The boundary overlay comparison between the original mesh (~ 5 μm) (Fig. 5) and meshes with element edge lengths of ~ 10, 15, and 20 μm is shown in Figs. 6a-6c, including area/volume error estimates of 3.7/4.1%, 5.3/5.7%, and 6.1/6.5%. To assess the impacts of area and/or volume errors between the original mesh (~ 5 μm) of the LC and that of the re-meshed models with lower element densities on the FE simulation results, models of the LC and NT microstructure only were subjected to a pressure of 20 mmHg applied on their anterior surfaces. The resultant stresses and strains were afterward compared for three different element edge lengths of ~ 10, 15, and 20 μm. The volumetric average von Mises, 1^st, 2^nd, and 3^rd principal stresses and strains are shown in Fig. 6d for 25 regions of the ONH. The results exhibited nearly identical patterns and magnitudes of the stresses and strains in the LC and NT FE models regardless of the element edge length, demonstrating that variation in element edge length up to 20 μm had little effect on simulations of mechanical behavior. Therefore, an element edge length of ~ 20 μm was selected for all subsequent FE simulations of the human ONH in this study, as this mesh density preserves simulation accuracy while also minimizing simulation time.

Fig. 6.

The boundaries of the LC beams in the binary images were compared to the volume-meshed FE model with the LC microstructure element edge length of (a) ~ 10, (b) ~ 15, and (c) ~ 20 μm, showing the errors in the area and volume. (d) The regionalized volumetric average von Mises, 95^th percentile 1^st principal stress and strain, and 5^th percentile 3^rd principal stress and strain in the LC and NT microstructures are shown for the three different element edge lengths for an applied pressure of 20 mmHg. Note that the peak stresses and strains changed less than 3.5% as element edge length increased from 10 to 20 μm, indicating that the mesh is converged.

3.4. Optic Nerve Head Finite Element Modeling

The contours of the von Mises, shear, 1^st, and 3^rd principal stresses and strains, and LC displacement in the nasal-temporal cross sections through the ONH are shown in Fig. 7 for the continuum model with inhomogeneous anisotropic material properties based on mapped LC CTVF and anisotropy, the continuum model with isotropic, homogeneous neo-Hookean properties, and microstructural model with neo-Hookean properties. Stresses and strains were compared across the models wherein sclera and LC material properties were assigned such that the models yielded identical displacement of the central, anterior LC surface, thereby isolating the differences in stress and strain patterns and values to material property description and LC homogenization. In the model of donor 118, the highest CTVF was observed in the inferior and temporal periphery while the lowest CTVF was located in the inferior and inferior-nasal midperiphery. The regionalized volumetric average stresses, strains, and displacement in the LC for all three material property cases in models of donors 118, 119, and 129, are depicted in Figs. 8, 9, & 10, respectively, with the microstructural model results split into three tissue components: LC beams and NT considered together, LC beams only, and NT only. The contours of the 1^st principal strain the continuum model with mapped CTVF/anisotropy and microstructural model with neo-Hookean properties of the eye #118 are illustrated in Fig. 11.

Fig. 7.

The contours of the (a) von Mises, (b) 1^st principal, (c) 3^rd principal, (d) maximum shear stresses, and (e) 1^st principal (f) 3^rd principal, (g) maximum shear strains, as well as (h) LC displacement of donor 119 are shown for the nasal-temporal section through the models with three different material descriptions: the continuum model with mapped LC CTVF/anisotropy, continuum model with homogeneous isotropic neo-Hookean LC properties, and the microstructural LC model with neo-Hookean properties.

Fig. 9.

Polar plot of the regionalized volumetric average (a) von Mises stress, (b) 1^st principal stress, (c) 3^rd principal stress, (d) maximum shear stress, (e) 1^st principal strain, (f) 3^rd principal strain, (g) maximum shear strain, and (h) displacement in the LC of donor 119 for three different material descriptions: the continuum model with mapped LC CTVF/anisotropy, continuum model with homogeneous isotropic neo-Hookean LC properties, and the microstructural LC model with neo-Hookean properties.

Fig. 11.

Contour plot of the 1^st principal strain in (a) the continuum model with mapped LC CTVF/anisotropy and (b) the microstructural LC model with neo-Hookean properties, plotted on identical scales for comparison. Note that the general pattern of strain is similar, but the LC microstructure mode #118l captures the higher strains present in the lamina cribrosa itself.

The regional volumetric average stresses and strains for all three material property cases for models of donors 118, 119, and 129 the LC are also plotted separately for the peripheral and mid-peripheral regions in the temporal-superior-nasal-inferior-temporal (TSNIT) pattern common to clinical ophthalmological data are shown in Fig. 12. The peripapillary sclera showed the highest von Mises, shear, and 1^st principal stresses (tensile) (Fig. 7), demonstrating the critical role the circumferential ring of collagen fibrils plays in protecting the vulnerable ONH from IOP. The LC beams in the microstructural model with neo-Hookean properties showed higher von Mises and shear stresses compared to the NT, showing that they are the primary load-bearing tissues in the LC. The NT exhibited much larger strains in all modes compared to the LC beams, showing their particular vulnerability to biomechanical insult. Stresses and strains showed eye-specific regional patterns, as expected, that were reasonably conserved and within similar ranges across the three models with different LC material property/tissue descriptions within eyes. In general, the LC in the microstructural models exhibited much higher stresses and lower strains than either the NT alone or the continuum materials where the LC and NT are homogenized (Figs. 8–10 & 12). In particular, von Mises stresses were much higher in the LC microstructure compared to either the NT alone or the continuum materials, while 1^st principal (tensile) stresses were similar or lower in the LC microstructure, indicating that the shear and compressive stresses play a large role in LC biomechanics. The results also showed a generally similar magnitude and regional pattern of the stresses and strains for the continuum model with mapped CTVF/anisotropy and microstructural model with isotropic neo-Hookean properties, although direct comparisons are difficult due to the homogenization in the continuum models versus models with the microstructure represented faithfully (Fig. 12). The volumetric average and 95^th weighted percentile stresses and strains for the LC volume in all models of donors 118, 119, and 129 are summarized in Tables 5 and 6.

Fig. 12.

Plots of the volumetric average von Mises stress, 1^st principal stress, and 1^st principal strain in the temporal, superior, nasal, and inferior regions of the LC in donors (a) 118, (b) 119, and (c) 129, showing the continuum model with mapped LC CTVF/anisotropy, continuum model with homogeneous isotropic neo-Hookean LC properties, and the microstructural LC model with neo-Hookean properties (LC+NT, LC only, and NT only). Solid square and open circles represent the peripheral and mid-peripheral regions, respectively, and data are shown in the temporal-superior-nasal-inferior-temporal (TSNIT) configuration typical of clinical ophthalmic data presentation.

Table 5.

Volumetric average stresses and strains in the entire full LC region of the three donor eye models, for all material property assignment approaches.

Donor #	118					119					129
Material models	Continuum CTVF/Anisotropy	Continuum neo-Hookean	Microstructural neo-Hookean			Continuum CTVF/Anisotropy	Continuum neo-Hookean	Microstructural neo-Hookean			Continuum CTVF/Anisotropy	Continuum neo-Hookean	Microstructural neo-Hookean
Components	LC	LC	LC + NT	LC Only	NT Only	LC	LC	LC + NT	LC Only	NT Only	LC	LC	LC + NT	LC Only	NT Only
von Mises stress (kPa)	18.92	14.22	16.3	22.13	10.46	16.22	18.92	17.33	33.24	1.41	12.65	19.68	14.83	27.85	1.81
1st Principal stress (kPa)	53.65	38.57	39.68	46.94	32.42	26.86	53.65	14.08	20.55	7.60	26.49	43.83	10.32	13.93	6.72
Shear stress (kPa)	31.94	24.05	21.47	25.79	17.15	15.24	31.94	8.07	15.43	6.88	16.65	27.89	7.27	12.92	5.08
3rd Principal stress (kPa)	−10.23	−9.53	−3.27	−4.65	−1.88	−3.63	−10.23	−2.06	−10.31	6.17	−6.81	−6.95	−4.22	−11.91	3.45
1st Principal strain (%)	3.64	3.43	3.49	2.83	4.14	2.27	3.64	2.47	1.79	3.14	3.61	2.39	2.95	1.99	3.92
Shear strain (%)	4.41	4.44	2.44	1.91	3.42	2.96	4.41	3.02	2.21	3.82	4.31	3.23	3.71	2.49	4.94
3rd Principal strain (%)	−5.18	−5.45	1.40	1	2.70	−3.66	−5.18	−3.57	−2.64	−4.50	−5.01	−4.07	−4.48	−3	−5.96

Table 6.

95^th weighted percentile stresses and strains in the LC region of the three donor eye models, for all material property assignment approaches.

Donor #	118					119					129
Material models	Continuum CTVF/Anisotropy	Continuum neo-Hookean	Microstructural neo-Hookean			Continuum CTVF/Anisotropy	Continuum neo-Hookean	Microstructural neo-Hookean			Continuum CTVF/Anisotropy	Continuum neo-Hookean	Microstructural neo-Hookean
Components	LC	LC	LC + NT	LC Only	NT Only	LC	LC	LC + NT	LC Only	NT Only	LC	LC	LC + NT	LC Only	NT Only
von Mises stress (kPa)	161.97	53.53	114.66	114.66	8.87	84.60	31.32	105.64	105.64	5.08	85.05	44.65	104.97	104.97	7.60
1st Principal stress (kPa)	138.7	85.22	102.03	102.03	81.26	111.15	57.60	130.24	130.24	53.27	91.39	68.92	89.12	89.12	−47.27
Max Shear stress (kPa)	89.81	29.38	64.63	64.63	4.93	48.14	16.71	58.30	58.30	2.84	47.10	24.93	59.69	59.69	4.29
3rd Principal stress (kPa)	−101.46	−56.54	−111.19	−111.53	77.80	29.12	−37.71	−84.47	−84.47	50.93	−47.01	−45.73	−111.53	−111.53	48.87
1st Principal strain (%)	7.01	4.74	18.84	10.53	18.84	4.68	4.30	10.10	7.95	10.10	7.51	5.91	18.83	12.58	18.83
Max Shear strain (%)	7.67	6.19	20.82	11.27	20.82	6.24	5.68	12.16	8.47	12.16	8.56	7.32	16.61	11.03	16.61
3rd Principal strain (%)	−9.64	−7.64	−22.8	−12.89	−22.80	−7.79	−7.11	−14.52	−9	−14.52	−10.58	−8.72	−18.49	−13.08	−18.49

4. Discussion

While it is largely accepted that the elevated IOP causes damage to the ONH in glaucoma, the mechanisms by which the IOP-induced forces and deformations contribute to NT degeneration and loss of vision are still not well understood. Given that glaucoma is a neurodegenerative disease, accurate characterization of the effects of IOP on the NT of the laminar region is critical to understanding one of the key exposures that drive the pathogenesis and progression of the disease [3–5, 8, 49]. Various experimental [20, 29, 35, 36] and numerical [22–25] studies have investigated the response of the LC to IOP elevation. While many of these studies have contributed to our knowledge of laminar biomechanics, they have oftentimes neglected the important roles of the complex boundary conditions imposed on the ONH by the other components of the eye, such as the retina, pia, and retrolaminar optic nerve, that distribute IOP and cerebrospinal fluid pressure loads in the LC [30, 50, 51]. In addition, the LC and NT have been simulated as a single continuum material [15, 24], a small region of the LC microstructure [22, 23], or a repeating idealized microstructural cell [25]. To the best of our knowledge, the models proposed herein are the first to include the complete, eye-specific, 3D microstructure of the LC within a realistic eye geometry.

4.1. 3D Eye-specific ONH model and Meshing

In this study, we developed and tested an eye-specific microstructural FE model of the human LC (Fig. 1) and ONH that is incorporated into an eye-specific FE model of the human posterior eye, including the retina, sclera, pia, and optic nerve (Fig. 2). Complex, heterogeneous, anisotropic, hyperelastic material properties that mimic the collagen architecture were assigned to the sclera and pia mater. This new approach allows for the estimation of deformation, stress and strain fields in the entire LC and NT microstructure at once, while simulating the important roles of the retina, sclera, pia and optic nerve that serve to distribute pressure loads through these tissues. The LC in the continuum model with mapped CTVF/anisotropy properties distributes anisotropic and heterogeneous material description throughout the homogenized LC continuum based on local assignment of stiffness and anisotropy (Fig. 4). In the microstructural model with the neo-Hookean properties, only the sclera and pia were assigned anisotropic and heterogeneous material properties, and the structural inhomogeneity in the LC microstructure was modeled explicitly (Fig. 3). The LC was meshed directly from the binary images using a custom algorithm (Fig. 2), which was shown to be highly accurate with a ~ 5 μm element edge length (Fig. 5). Due to the complex geometry of the LC and the high computational cost associated with these very small elements, a mesh density study was conducted to estimate the possible error due to reducing mesh density in the LC and NT microstructure to manageable levels (~ 10–20 μm element edge length) (Fig. 6). Results (volumetric average von Mises stress, 95^th percentile 1^st principal stresses and strains, as well as 5^th percentile 3^rd stresses and strains) show that an element edge length of ~ 20 μm can be used for our simulations, as it preserves both the microstructural geometry and predicted stress and strain patterns in the LC and NT with less than 3.5% change in the stresses and strains from 10 to 20 μm element edge lengths (Fig. 6). Three eye-specific FE models of the ONH were constructed from the 3D ONH reconstructions of human donor eyes; models were subjected to pre-stressing from 0 to 10 mmHg IOP with no deformation to estimate the stresses present in the tissues when they were immersion fixed at that pressure, and then loaded with an IOP elevation from 10 to 45 mmHg.

4.2. Comparison of the material models

The results in terms of the von Mises stress, shear and 1^st principal, and 3^rd principal stresses and strains, as well as the displacement, are presented in Fig. 7 for the nasal-temporal cross section of donor eye 119. To allow for quantitative assessment of the resultant stresses and strains, volumetric average stresses and strains were also calculated and plotted in 25 regions of similar volume for each donor eye (Figs. 8, 9, & 10). Two different material models were assigned to each continuum eye-specific model to assess the role of the LC microstructure in the LC mechanical response. While the LC was assigned different shear moduli based on donor age [52], the NT in all models was assigned the same shear modulus of 0.01 MPa, an assumption that matches prior computational studies (Tables 2, 3, & 4). The contours of the stresses, strains, and deformations in the ONH FE models in the nasal-temporal cross section showed a very similar pattern in the LC and peripapillary sclera across all models within donor 119 (Fig. 7). Results revealed that LC CTVF and anisotropy are heterogeneous both within and between eyes and the stresses and strains vary greatly from one region to another (Figs. 8, 9, & 10) in the models that include those morphological factors (Fig. 4). In general, stresses were higher, and strains were lower, in regions of higher CTVF within eyes. In addition, the average volumetric and 95^th weighted percentile stresses and strains in the two continuum models were considerably different (Tables 5 & 6), although the regional patterns were generally similar. Note that donor 119, at 79 years of age, was assigned a much higher LC shear modulus and lower collagen crimp angle than the other donors (118 at 40 years old and 129 at 34 years old) based on our prior work on age-related changes in scleral material properties [53]. Hence, the lower strains in donor 119 were consistent with age-related stiffening of the collagenous LC, sclera and pia. Regional patterns in the LC mechanical response are partially driven by the overall thickness and shape of the LC volume within eyes (identical for all models within eyes), with the remainder of the difference driven primarily by differences in the assignment of material properties (mapped or isotropic neo-Hookean) and/or structural heterogeneity (homogenized continuum or microstructural LC plus NT). The regional LC stress and strain patterns were more similar between the mapped CTVF/anisotropy models and the microstructural models within eyes (Figs. 8–10), which also was expected, since only the isotropic continuum models ignore the role of regional LC CTVF and beam orientation (Figs. 3–4).

The importance of LC structural heterogeneity in the ONH mechanical response warrants a more thorough examination of the differences between the continuum model with the mapped CTVF/anisotropy and the microstructural model within eyes. The stresses and strains in the microstructural FE models showed that the LC beams are the primary load-bearing component in the laminar region, while the NT bears much higher strains (Fig. 7). The contours of the 1^st principal strain in the LC of the continuum model with the mapped CTVF/anisotropy and the LC+NT of the microstructural model with neo-Hookean properties revealed the same patterns of strain, however, strains were ~ 3 times higher in the microstructural model (Fig. 11). While the continuum and microstructural models yielded the same pattern of strain, the continuum model cannot capture the realistic mechanical behavior of the LC microstructure under an applied IOP. A quantitative assessment of the regional stresses and strains is shown for all three eyes in Figs. 8, 9, & 10 and the circumferential patterns in the four anatomical regions (temporal, superior, nasal, and inferior, or TSNIT) was plotted in a manner similar to clinical data presentation in Fig. 12. It is important to note that the regional patterns of different components of the stress and strains are not plotted using the same scale within or across the eyes to maximize the pattern visibility, so the TSNIT curves in Fig. 12 are most useful for comparing stress and strain magnitudes. Recall that the 1^st and 3^rd principal stress/strain represent the tensile and compressive components, respectively [54]. The von Mises stress is a single scalar representing distortional stress that is useful for comparing overall stress levels. The regional patterns of the stresses and strains between the continuum model with the mapped CTVF/anisotropy and the microstructural model were generally similar, with the exception of the 1^st principal strain in donor 119 (Fig. 9). The regional patterns of the von Mises stress in the LC of all models was generally higher in the inferior and temporal regions (Fig. 12). The difference between the volumetric average and 95^th percentile von Mises stress at 25 regions between the continuum model with the mapped CTVF/anisotropy and the microstructural model was 6.8–17.2% and 16.9–33.6%, respectively (Tables 5 & 6). Tensile stress was generally higher in the inferior and temporal regions, although eye 129 did not show a distinct regional pattern (Fig. 12). Both the continuum models with the mapped CTVF/anisotropy and the microstructural models showed high compressive stresses at the inferior periphery and superior-temporal regions. The shear stress was highest in general in the superior, inferior, and temporal regions, with relatively higher magnitudes for the continuum models (Figs. 8–10).

Regarding the strains in the laminar tissues, the patterns generally showed the largest strain magnitudes in compression (3^rd principal strain) followed by tension (1^st principal strain), and finally shear (Tables 5 & 6). The shear strain was much smaller than other modes of strain and an order of magnitude smaller than the tensile strains on average [28], which is in good agreement with that observed by Sigal et al., [54]. The tensile strain showed a very similar pattern between the continuum model with the mapped CTVF/anisotropy and the microstructural model except for donor 119 (Fig. 12), with higher values in the temporal quadrant and lower levels in the superior and nasal quadrants. Although the regional pattern did not match between the models in donor 119 (there is no obvious explanation for this result), volumetric average tensile strain in all 25 regions of the LC in donor 119 showed less than 8.8% difference between the continuum and microstructural models (Table 5). The strains for all specimens showed substantial regional variation, but the patterns differed greatly between eyes, which was expected given the eye-specific patterns of LC CTVF and predominant beam orientation (Fig. 4). The magnitude of strains measured in the LC and peripapillary sclera were within the expected ranges based on prior experimental inflation studies, which reported average tensile strains of approximately 3% for inflation from 5 to 45 mmHg [28, 55], with local strains as large as 10% [56]. This is in good agreement with the results of our study as the average strain in the LC and peripapillary sclera was ~3% with the regional maximums as large as ~ 6% taken over a relatively large regional volume. Midgett and co-workers reported the smallest tensile strain in the nasal quadrant and the highest in the inferior or temporal quadrants [28] in their experiments, matching our findings. They also found that the shear strain was the smallest in the nasal and superior quadrant and highest in the inferior and temporal quadrants of the LC, which is also in agreement with our results (Figs. 8–10). The average maximum principal strain in our eye models varied between ~ 2.3–3.6% and 2.5–3.5% under an IOP elevation from 10 to 45 mmHg for the continuum and the microstructural models, respectively, which is within the ranges of 2.5–4.5% [29] and 1.5–4.1% reported previously [57]. Prior studies have shown that the maximum principal strain was 0.9–1.1% lower in the nasal quadrant of the LC compared to the other three quadrants [20], again matching our findings (Fig. 12). The maximum principal strain was not considerably different in the central and peripheral regions, which is in good agreement with that observed by Midgett et al., [20]. The maximum tensile strain was also higher in the peripheral LC regions compared to the inner peripapillary sclera, which is in good agreement with the observations of Midgett et al., [28] and Yanhui et al., [58]. Additionally, our results showed the LC beams exhibited larger tensile strain compared to the peripapillary sclera, with an equal shear strain, which is in agreement with previous reports [28]. The compressive strain (3^rd principal strain) was found to be higher at the temporal and inferior-temporal regions, specifically at the midperiphery, with a similar volumetric average compressive stress between the continuum and microstructural models excluding model 118, which showed a higher strain in the microstructural model (Tables 5 & 6). The 95^th weighted percentile revealed higher stresses and strains in the microstructural model, with the highest stresses and strains in the LC and NT, respectively (Table 6).

Shear strain has frequently been mentioned in the literature as being potentially very important in retinal ganglion cell loss in glaucoma [49, 59, 60]. It has been reported that the damage to the neuro-retinal rim in glaucoma occurs mainly in the inferotemporal and superotemporal regions [61, 62], preserving NT in the nasal region [61]. Quigley and Addicks [63] performed histological evaluations of optic nerve axons at the LC and found that axon loss occurred in all regions, but was greater in the superior and inferior regions of the LC, corresponding to thinner LC beams and larger pores in these regions [11]. Shear strains have been reported to be larger in the peripheral region than the central region [28], and the maximum shear strain was largest at the boundary with the significantly stiffer sclera [20, 57]. In our models, the shear strain was generally higher in the midperiphery compared to the periphery. The large volume of the regions prevents a close inspection of the shear strains in the LC at its insertion into the scleral canal wall, although this will be the focus of future studies. We also observed a larger shear strain in the peripheral regions compared to the central region, and the shear strain was the highest in the peripheral of donor eye 119 that had the stiffest peripapillary sclera due to advanced age [64]. In addition, our microstructural model showed a lower shear strain in the peripapillary sclera compared to the central LC regions (Fig. 7) which is in good agreement with the observation of Midgett et al., [28]. The peripapillary sclera has an anisotropic collagen fibril structure that is oriented circumferentially around the ONH, which led to a larger shear strain in the peripheral region of the LC [20]. The finding that the shear and tensile strains were largest in the peripheral region is consistent with the initial loss of peripheral vision in clinical glaucoma [20]. Histological studies by Minkler showed that the peripheral retinal ganglion cells occupy are located peripherally in the LC and optic nerve as well [65], so the larger shear and tensile strains predicted in these regions may contribute to glaucoma damage in the peripheral NT. Overall, the regional variation in the strain magnitude corresponds with the known regional variation in early axonal injury in glaucoma, which begins in the inferotemporal and superotemporal regions [11, 62] that show the highest strains in our models. Finally, our models predicted central LC posterior displacement of ~ 30–80 μm, which matches the displacement range of ~30–80 μm observed by Midgett et al., in experimental inflation studies [20, 28].

One of the key findings of this study is that the continuum model with mapped CTVF/anisotropy underestimated the strains in the LC and NT by a factor of ~ 2–3 (Fig. 12, Tables 5 & 6). The difference between the continuum and microstructural models mostly recognized when we calculated the 95^th weighted percentile stresses and strains in the entire LC and NT components (Table 6). This suggests that a microstructural model with a simple isotropic neo-Hookean hyperelastic material model can better estimate the complex mechanical response of the LC under an IOP elevation by capturing structural heterogeneity using accurate modeling of the collagenous microstructure. Furthermore, microstructural modeling revealed large differences in the magnitudes of stress and strain in the LC microstructure compared with the integrated NT microstructure in the same region, so these models can serve as a platform for future studies on the biomechanical drivers of LC remodeling and NT damage in glaucoma. Although only three eye-specific models were simulated in this study, the model from the oldest donor exhibited a much stiffer LC, with the lowest tensile, compressive, and shear strain magnitudes. Remodeling of the LC microstructure in glaucoma has been widely reported, including observations that glaucomatous eyes have thicker beams and smaller pores compared to healthy eyes [66]. Excessive strains in the LC beams could result in astrocytes and fibroblast activation and elicit biological responses such as extracellular matrix remodeling [1, 16, 67, 68] and this new modeling approach will allow us to investigate the biomechanical mechanisms underlying that phenomenon by incorporating suitable remodeling algorithms [16, 69]. In addition, damage from mechanical deformation may also include a vascular component, as capillaries pass through and around the LC beams, nourishing the axonal bundles passing through the ONH [70, 71]. Strain and remodeling of the LC beams could compromise perfusion and nourishment to the axon bundles [72]. Although investigating the mechanisms of LC remodeling was not an objective of this study, continuum models cannot address the remodeling of the beams due to their homogenization approach. The microstructural model presented in this study addresses this weakness. Boundary conditions play a pivotal role in FE simulations and assigning suitable boundary conditions at the micro- and/or macro-scale is not a trivial task. In the aforementioned studies that used a two-step multiscale approach, the macro-scale displacement boundary conditions were only applied to the peripheral surface nodes of the cut edges of the subregion of the LC micro-scale models [22, 23, 73]. However, the LC deformation at the macro-scale is also present at the anterior and posterior LC surfaces, as well as on the surfaces of the LC microstructure inside the pores within the subregion, so this is an imperfect boundary condition. While the NT strains reported by Voorhees and colleagues for subregions of human eyes [73, 74] are similar in magnitude and distribution to those reported herein for the full LC microstructural model, the LC strains they report are much lower than those presented in Table 6. Hence, this suggests that modeling the full LC microstructure with the surrounding tissues is important in simulating LC biomechanics.

The methods proposed herein are ideal for modeling any structure with a complex microstructure composed of different materials, such as trabecular bone, foams, tissue engineering scaffolds, decellularized LC [75], and lung. In bone tissue engineering using biodegradable scaffolds, the geometry of the porous scaffold microstructure is a key factor in controlling mechanical function during remodeling [76–78]. The bone regeneration process occurs over long periods with successive complex events, so detailed in vivo observation is difficult even if only the mechanical factors are assessed. Therefore, computational simulation could be a helpful for evaluating the change in mechanical function during the regeneration process [76, 79]. However, difficulties in generating an accurate FE mesh of the bone scaffold microstructure has hampered prior studies, irrespective of whether a voxel meshing method [76, 80–83] or commercial software [79, 84] was used. Voxel-based modeling and simulation has some advantages, such as simplicity in expressing the morphological change using remodeling algorithms, the stepped surface of voxels on curved surfaces is a significant disadvantage [76]. Accurately representing the microstructure in the model geometry is the key in microstructural FE meshing and modeling. The meshing algorithm proposed herein ensures accurate meshing for complicated microstructural geometries using simple binary images in any image format as input.

4.3. Limitations

This study is limited by the following considerations. First, there is no experimentally validated gold standard for the material properties of the LC we can use as a reference for the deformations, stresses, and strains presented in the analysis. Experimental strain and displacement data for the full 3D microstructure of the LC beams are currently unavailable for the posterior eye due to the difficulty in directly measuring these quantities, either in vivo or under controlled experimental conditions where the retrolaminar optic nerve and pia are intact. Second, while our 3D reconstruction methodology allows high-resolution, full-field reconstructions of the LC microstructure, sclera, and pia that are the basis of our models, there is some geometric artifact due to tissue shrinkage associated with fixation and embedding. However, we have shown in a recent study that the 3D reconstructions yield morphology that is consistent with the tissues in vivo [85]. Third, our models do not include the vasculature, which is likely to influence the mechanics of the ONH in vivo, especially adjacent to the central retinal vascular truck in the laminar region. This will be addressed in future studies that incorporate hydrostatic pressure and compressive strain mapping as surrogates for capillary perfusion. Fourth, the element edge length of the LC and NT in the microstructural model was set to ~14–20 μm while the original mesh had an element edge length of ~ 5 μm. While the remeshing led to an alteration in the area and volume of the LC in the FE model of the ONH, our comparative analyses showed that the small changes in the area and volume of the re-meshed LC had negligible effects on the resultant stresses and strains, while allowing the simulations to run in a reasonable amount of time on a workstation computer. Fourth, the LC microstructure was simulated in the continuum models as connected beams, although it is possible that there might be some free or unconnected LC beams that could potentially alter the patterns of the stresses and strains in the models with a homogenized LC. However, the LC beam microstructure in the microstructural models was constructed based on high resolution binary images of the actual LC from donor eyes, so this is not a concern in the microstructural modeling approach. Fifth, the regionalization in the relatively large volumes presented in this study cannot capture the LC stresses and strains at their insertion into the scleral canal wall, which may be important in glaucoma pathogenesis and progression. The microstructural modeling approach presented herein can capture strains at the LC/scleral junction, which will be the subject of a future study. Finally, several interesting areas of investigation are beyond the scope of the present study and will be the subject of future reports. Future studies will focus on the roles of age, race, disease, IOP dynamics, cerebrospinal fluid pressure, LC remodeling, vascular perfusion, in ONH biomechanics and glaucoma pathophysiology.

5. Conclusions

We have developed a microstructural FE model of the LC microstructure, constructed directly from high-resolution binary images of the LC, to estimate the stresses and strains across the entire LC and NT microstructure resulting from an IOP elevation. The microstructural models were compared to identical models that represent the LC as a continuum material with mapped laminar CTVF/anisotropy, a continuum with isotropic neo-Hookean properties The continuum model with mapped CTVF/anisotropy underestimated the stresses and strains in the LC and NT by a factor of ~2–3 when compared to the LC microstructural models, which underscores the importance of employing a true microstructural model if the objective is to accurately calculate the stresses and strains in the LC or NT. Also, while LC biomechanics have been the focus of many prior studies, the high strains in the NT suggests that axonal strain may be a mechanism of damage to the visual pathway in glaucoma. On average, the strains were highest in the peripheral LC and regional variation in LC strain is consistent with the pattern of axonal damage observed in glaucoma. The findings suggest a vital role for eye-specific models in predicting glaucoma risk and building a more thorough understanding of the complex mechanical response of the LC and NT, which drives glaucomatous remodeling and ultimately retinal ganglion cell death.

Statement of Significance.

Glaucoma is among the leading causes of blindness worldwide that is characterized by axon damage in the lamina cribrosa (LC) region of the eye. We present a new approach for finite element modeling the entire eye-specific 3D LC microstructure and the interspersed neural tissues, incorporated into an eye-specific posterior eye model that provides appropriate boundary and loading conditions. Results are presented for three human donor eyes, showing that prior modeling approaches underestimate the stresses and strains in the laminar microstructure. We constructed models from image stacks of the segmented microstructure (Matlab code included) using an approach that is ideal for modeling any structure with a complex microstructure composed of different materials, such as trabecular bone, lung, and tissue engineering scaffolds.

Data availability

The raw/processed data required to reproduce these findings cannot be shared at this time as the data is part of an ongoing study.