Bilaminar Mechanics of the Human Optic Nerve Sheath

Abstract

Purpose/Aim:

The adult human optic nerve sheath has recently been recognized to be bilaminar, consisting of inner and outer layers. Since the optic nerve and sheath exert tension on the globe in large angle adduction as these structures transmit reaction force of the medial rectus muscle to the globe, this study investigated the laminar biomechanics of the human optic nerve sheath.

Materials and Methods:

Biomechanical characterization was performed in optic nerve sheath specimens from 12 pairs of fresh, post-mortem adult eyes. Some optic nerve sheath specimens were tested whole, while others were separated into inner and outer layers. Uniaxial tensile loading under physiological temperature and humidity was used to characterize a linear approximation as Young’s modulus, and hyperelastic non-linear behavior using the formulation of Ogden. Micro-indentation was performed by imposing small compressive deformations with small, hard spheres. Specimens of the same sheaths were paraffin embedded, sectioned at 10 micron thickness, and stained with van Gieson’s stain for anatomical correlation.

Results:

Mean (± standard error of the mean, SEM) tensile Young’s modulus of the inner sheath at 19.8±1.6 MPa significantly exceeded that for outer layer at 9.7±1.2 MPa; the whole sheath showed intermediate modulus of 15.4±1.1 MPa. Under compression, the inner sheath was stiffer (7.9±0.5 vs 5.2±0.5 kPa) and more viscous (150.8±10.6 vs 75.6±6 kPa·s) than outer sheath. The inner sheath had denser elastin fibers than outer sheath, correlating with greater stiffness.

Conclusions:

We conclude that maximum tensile stiffness occurs in the elastin-rich optic nerve sheath inner layer that inserts near the lamina cribrosa where tension in the sheath exerted during adduction tethering may be concentrated adjacent the optic nerve head.

Introduction

Although previously neglected as a mechanical factor, it has recently been recognized that the optic nerve (ON) and its sheath contribute to mechanical loading on both extraocular muscles and the ocular globe¹. We recently demonstrated by magnetic resonance imaging (MRI) that ON length is insufficient to avoid tethering the globe when, for normal subjects, adduction exceeds about 26°². Further adduction approaching the much greater limit of the oculomotor range requires strains in the orbital tissues, including translation of the whole globe. This translation in adduction is mainly in the nasal direction in healthy people, but the globe retracts posteriorly in primary open angle glaucoma³. While medial rectus (MR) muscle force acts across a broad tendon insertion⁴, reaction force to it is concentrated on the smaller ON canal and peripapillary sclera. Optical coherence tomography (OCT) shows that deformations of the ON head and Bruch’s membrane produced by adduction exceed many-fold those resulting from extreme intraocular pressure (IOP) elevation⁵, or IOP-related deformations recently proposed as pathological to retina⁶. Sibony et al. used OCT to show in patients with elevated intracranial pressure that adduction induces folds extending from the optic disc to the retinal macula⁷.

Eye movements occur as often as several times per second⁸ during wakefulness⁹, but persist even during sleep¹⁰. Rapid eye movements called saccades that change gaze direction by 25-45° frequently occur when the head and body are both free to move naturally¹¹. Typical gaze shifts that occur during everyday head movement include eye movements averaging about 30 deg ^{12, 13}. This means that the optic disc and peripapillary retina and sclera experience relentless, transient mechanical loadings by the ON. It has even been proposed that that many cases of primary open angle glaucoma may represent repetitive stress injury to the optic disc induced by tractional force exerted as the ON and its sheath become tethered during large adduction movements³. Finite element modeling (FEM) based upon bovine tissue properties suggests that reaction force to MR contraction is concentrated in the temporal peripapillary region and lamina cribrosa, sites of early damage in glaucoma¹⁴. It is evident from such modeling that the mechanical consequences of adduction tethering depend to a large extent on the local mechanical properties of the ON, its enclosing dural sheath, and the posterior sclera. The term “ON sheath” is widely used in the clinical and surgical literature for this mobile connective tissue structure, where it is preferred to the anatomically synonymous term “optic nerve dura mater,” probably because all other dura mater is tightly constrained by anchoring bone or intervertebral discs¹⁵.

While biomechanical characterization has been performed of the lamina cribrosa^{16, 17}, and sclera, especially peripapillary sclera^{18, 19}, the ON and its enclosing sheath have been comparatively neglected. Consequently, prominent FEMs have either omitted the sheath entirely²⁰, or parameterized the human ON sheath as identical to the dura mater of brain²¹. The biomechanical properties of the ON sheath have recently been characterized in mammals such as cow¹⁴ and pig^{22, 23}. Based on measured mechanical properties of ocular tissues, FEM has been performed to predict stress and strain distribution around ONH due to eye movement^22,14. Although the foregoing studies have consistently suggested that the stress and strain are induced near the ONH by eye movement, the accuracy of such FEM simulations is questionable since mechanical properties of the non-human ON sheath were employed. Biomechanical measurement of the human ON sheath is clearly warranted to make accurate inferences about the role of eye movements in human ON sheath biomechanics. Moreover, it has recently been found that the adult human ON sheath consists of distinct inner (IL) and outer layers (OL), with the IL containing more dense collagen and abundant elastin, and the OL containing less of each of these important structural proteins²⁴. The differing composition of the two ON sheath layers leads to the obvious supposition that they might have differing intrinsic mechanical properties. Therefore, current study aimed to investigate the laminar mechanical properties of the human ON sheath by employing some standard biomechanical characterization methods compared with histological evaluation of connective tissue composition in the same specimens.

Methods

Twelve pairs of whole human globes with long attached ONs were obtained in conformity with legal requirements from eye banks within three days of natural death, having been donated for research purposes. Specimens of cadaveric tissue, which were anonymous as received by our laboratory, are not further subject to ethical oversight since they are not defined under United States of America federal law as “living human subjects.” Attached optic nerves with enclosing sheaths were 10 to 20 mm long as supplied by the eye banks. The anterior 3 mm of the tissue closest to the globe-optic nerve junction was discarded because of thickness variation in transition to sclera, so that tested specimens were obtained posterior to that in the region of apparently uniform sheath dimensions. Average donor age was 66±15 (standard deviation, SD) years, and gender was equally balanced. Due to variations in technique by eye bank tissue harvesters, lengths of ON specimens varied so that it was impossible to perform every experimental test on each specimen. In the laboratory, the whole ON sheath was dissected from the ON, and then where possible the sheath was carefully separated into IL and OL using fine forceps, as illustrated in Fig. 1. When grossly distinct, only mild dissecting force was necessary to separate two layers because in suitable cases there was a weak boundary or spontaneous separation between the layers. Physical separation of the IL from OL was generally easiest in the most elderly specimens. In younger specimens where the two layers were strongly adherent, laminar separation was impossible. An industrial optical coherence tomography (OCT) scanner (OCS1300SS, Thorlabs Inc., Newton, NJ) was used to measure the cross sectional areas of specimens that averaged 0.67±0.08, 0.49±0.04, and 0.92±0.14 mm², respectively, for IL, OL, and whole sheath specimens.

Fig. 1.

Laminar separation of optic nerve (ON) sheath for tensile testing. Prior to this photo, the cylindrical sheath was incised longitudinally to form a flat sheet. A. ON sheath was separated into inner and outer layers via forceps distraction when the boundary between the two layers was weak. B. Three pairs of inner and outer layers from whole ON sheath prepared for tensile experiments. 1 cm calibration grid.

Specimen dimensions were constrained by lengths of ONs obtainable by the eye banks even when harvesters were instructed to maximize specimen length. Each ON sheath was cut into 2 × 6 mm (width × length) rectangular shape for tensile testing including 1 mm clamping margin from each end; thus, aspect ratio was maintained at 2:1 in order to avoid experimental artifact due to aspect ratio variation²⁵. Specimens were oriented in the longitudinal or circumferential directions for testing in separate orientations. Cross sectional dimensions of specimens were measured by OCT immediately after applying pre-loading for each tensile test. For micro-indentation, the whole sheath was trimmed to approximately 6-7 mm square shape for indentation at 4-5 widely-spaced locations.

A horizontal tensile loads cell was constructed, incorporating a linear motor (Ibex Engineering, Newbury Park, CA) having 100 mm/s maximum speed, a quadrature optical position encoder having 1 nm resolution, and a sensitive force sensor (LSB200, FUTEK, Irvine, CA) having 5 mN resolution. Via a frictionless air bearing, the linear motor shaft was coupled to an environmental chamber maintaining the tested specimens in approximately physiological temperature and humidity. Details of the load cell are published elsewhere²⁶. Specimens were glued using cyanoacrylate between 5 mm long layers of thin cardboard that was anchored in serrated clamps to prevent slip. Pre-loading of 0.05 N, which is just above the noise level of force measurements, was applied by fine incremental elongation to avoid slackness, after which a constant elongation rate (0.01 mm/s) was imposed until failure as tensile force was recorded to characterize stress-strain behavior. Preliminary experiments were conducted with specimens preconditioned by cyclic loading up to 5% strain, but since tensile data obtained after this preconditioning did not differ significantly from data without preconditioning¹⁴, preconditioning was thereafter omitted as unnecessary. Specimens were loaded to failure, although of course with failure often at nonphysiologic strains. Full data sets with loading to failure are nevertheless presented with this understanding. In order to examine the possible anisotropic mechanical behavior, tensile loading in orthogonal longitudinal and circumferential loading directions was performed in separate specimens. Three different type of specimens were tested as available from each donated eye: 1) IL only; 2) OL only; and 3) the whole ON sheath consisting of both layers. Numbers of each type of specimen are reported in Results section 3.

While it is recognized that the relationship between stress and strain in most biological materials is non-linear, much of the existing literature and many published simulations of ocular biomechanics have approximated tensile properties using Young’s modulus (YM), a linear relationship between stress and strain. For convenient comparison with the literature, we identified in every tensile stress-strain curve for each of the three ON sheath specimen types a linear region midway between the low strain “toe” and the failure region whose slope was considered to be YM. The reporting of YM values below should not be interpreted as a claim for linear behavior of the ON sheath, nor as contradiction to our additional reporting of better-fitting non-linear material laws. In addition to the computation of YM, we also performed non-linear fitting to the Ogden hyperelastic model that gives excellent agreement with experimental data for brain and fat tissues ²⁷, and that fit the current data better than alternative hyperelastic models. The hyperelastic model of Ogden ²⁸ (N = 3) was employed having the strain energy potential form:

$$\mathrm{W}=\sum _{i=1}^N\frac{2{\mu }_i}{{\alpha }_i{}^2}({\overline{{\lambda }_1}}^{{\alpha }_i}+{\overline{{\lambda }_2}}^{{\alpha }_i}+{\overline{{\lambda }_3}}^{{\alpha }_i}-3)+\sum _{i=1}^N\frac{1}{D_i}(J^{el}-1{)}^{2i}$$ Eq. 1

where ${\overline{\lambda }}_J$, (j = 1, 2, 3) are the deviatoric principal stretches, J^el is the elastic volume ratio, μ_i (MPa) and α_i are material constants, and N is the number of terms in the material parameter descriptions of each layer determined by curve fitting to the tensile data under the assumption of near-incompressibility. D_i determine specimen compressibility.

Micro-indentation is a method of locally indenting the surface of a material with a probe of known shape to a depth that is small in relationship to overall specimen thickness. Provided that the specimen in relatively thick in comparison to the indentation depth, the result of microindentation is independent of total specimen thickness, and of whatever material may lie on the material’s opposite side. Thus, for example, with a sufficiently small indenter, the IL or OL of the ON sheath could be tested by indentation of each respective surface without separating the two layers. A displacement-controlled indentation load cell (Fig. 2A) was constructed by synchronizing a linear stepper motor (LNR50 Series, Thorlabs, Newton, NJ) and analytical balance (ML54, Mettler-Toledo, Columbus, OH)²⁹. Specimens were immersed in Ringer’s lactate solution in a Petri dish placed on the pan of the analytical balance. The apparatus is sufficiently sensitive to detect the changing weight due to evaporation of water from the Petri dish. In order to minimize changes in load due to evaporation of water from the Petri dish, the load cell was surrounded by a glass chamber closed on all sides except for a small slit through which passed the indenter shaft. Downward indentation was created by a linear motor displacement having 100 nm precision, as force measurement was performed by analytical balance with 100 μg-force resolution, so loading could be recorded in a time series. Position of the balance pan was maintained by a servo within the balance, so that indenter position was equal to indenter depth. A 1 mm in diameter stainless steel sphere was used to indent samples to 50 μm depth. Because indentation depth was small relative to layer thickness, this method thus permitted local measurement of the properties of only the layer in direct with the indenter. In most cases the whole ON sheath specimen was simply placed in a Petri dish on the balance pan with the OL upward facing the indenter probe sphere to test the OL, or the specimen was inverted to test the IL. It was not possible to interpret microindentation of the whole ON sheath, because the technique only evaluates the one surface layer contacting the indenter.

Fig. 2.

Micro-indentation compressive load cell. A 100 nm precision stepper motor was synchronized with a digital analytical balance whose pan was maintained in constant position by electromechanical feedback. B. Typical relaxation data by micro-indentation. After the spherical tip contacts the specimen surface, force rapidly increases due to the indentation that is smaller than the indenter radius. Force relaxation occurs during maintained deformation, after which then the indenter was lifted from the specimen.

For each indentation, surface tension of the liquid layer on the specimen transiently created tractional (negative) force upon initial probe contact. Further incremental indentation was then applied to generate about 100~200 μg compressive (positive) force, indicating onset of contact with the specimen itself. Following contact, the probe was accelerated at 0.5 mm/s² to a speed of 1 mm/s until 50 μm indentation was achieved, after which the indenter was maintained in that fixed position as the specimen relaxed for approximately 80 seconds during applied force measurement. Figure 2B illustrates the stress relaxation testing procedure.

The Hertzian linear viscoelastic model was chosen for microindentation data analysis since a spherical-tip micro-indenter was used, a previous study successfully characterized bovine ocular tissues using this model²⁹, and insufficient data could be collected to characterize a nonlinear model in these tissues. The full derivation of the model is included in the Appendix.

We show in a companion paper that the ON sheath contains collagen and abundant embedded elastin fibers²⁴. A general rule of mixture provides a reasonable estimate of the YM of a composite material composed of parallel fibers of uniform direction embedded in a matrix³⁰. There are two extreme cases: the upper-bound modulus corresponding to loading parallel to the fibers, and lower-bound corresponding to loading transverse to the embedded fibers. The rule of mixture was applied to the two layers of the ON sheath as follows:

$$E_{w,upper}=f{E}_i+(1-f)E_0$$ Eq. 2

$$E_{w,lower}=(\frac{f}{E_i}+\frac{1-f}{E_0}{)}^{-1}$$ Eq. 3

where E_i, E_o, and E_w are YM’s for IL, OL, and whole ON sheath, respectively, and f is the volume fraction of the IL in proportion to the whole ON sheath. The rule of mixtures was used to make an inference about isotropy based upon comparison of whole ON sheath YM, and corresponding individual values for the IL and OL.

The companion paper provides detailed anatomical data on multiple ON sheath specimens that were subjected to histological examination, but for which overall ON dimensions were insufficient for both anatomical and biomechanical study in the same eye. Histological analysis is reported here for two globes with long enough ONs that they could also be utilized for biomechanical study in the same eye. The donors had no known histories of glaucoma. In both cases, ONs were sufficiently long that a histological specimen of retrobulbar ON and its enclosing ON sheath of approximately 2-3 mm length could be excised just anterior to a specimen sufficiently long for biomechanical testing. Histological specimens were processed as described elsewhere²⁴. Briefly, specimens were fixed in 10% neutral buffered formalin, dehydrated in graded alcohol solutions, embedded in paraffin, and sectioned at 10 μm thickness before staining with van Gieson’s elastin stain. Elastin density, defined as the number of pixels in an image occupied by black elastin fibers divided by the total number of pixels occupied by the ON sheath²⁴ in each IL and OL was determined for comparison with tensile and micro-indentation data in the same specimens.

To account for possible interocular correlation between the two eyes of each subject, statistical analysis was performed using generalized estimating equations (GEE) implemented in SPSS software (Version 24.0. Armonk, New York, USA: IBM Corporation). This was considered more rigorous than parametric statistical approaches such as analysis of variance that might be confounded by intrasubject correlations.

Results

Specimens were loaded to failure, which occurred above 30% strain. While this extremely high ultimate strain is nonphysiologic, complete stress-strain curves are illustrated in Fig. 3 for completeness. As a measure of approximate linear behavior and for convenient comparison to the literature, YM was determined in the range of 10- 15% strain for all specimen types (Table 1). The YM of the IL was about twice that of the OL (P<0.001, GEE), with the whole sheath YM intermediate between the two but not significantly different from the IL (P>0.2) but greater than the OL (P<0.001, GEE).

Fig. 3.

Nonlinear mechanical behavior of the human optic nerve sheath. A. Mean uniaxial tensile data. Behavior of the inner layer did not differ significantly from the whole sheath (P>0.38), but significantly differed from the outer layer. (P<0.02) The number of specimens and globes is indicated in Table 1. B. Three-parameter Ogden hyperelastic model closely fits the data in A.

Table 1.

Linear Elastic Properties of Optic Nerve Sheath

Tensile		Inner Layer	Outer Layer	Whole
Number of Subjects		6	5	6
Number of Specimens		13	6	37
Young’s Modulus (MPa)	Mean	19.8	9.7*	15.4
	SEM	1.6	1.2	1.1

Compressive		Inner Layer	Outer Layer
Number of Subjects		10	10
Number of Specimens		58	44
Stiffness (kPa)	Mean	7.9	5.2*
	SEM	0.5	0.5
Viscosity (kPa·s)	Mean	150.8	75.6*
	SEM	10.6	6.0
Time Constant (Sec)	Mean	19.4	14.6*
	SEM	0.7	0.9

Multiple specimens were extracted from the same eyes where possible. SEM: Standard error of the mean.

^*- Outer layer different at P < 0.001 by generalized estimating equations.

To investigate possible anisotropy, whole ON sheath specimens were elongated in longitudinal (N=16) vs. circumferential (N=15) directions, for which YM values were 15.2±1.2 MPa and 15.1±2.1 MPa, respectively (P>0.94, GEE). This similarity indicates that the sheath may be regarded as approximately isotropic with respect to the longitudinal and circumferential directions.

In order to obtain nonlinear characterizations, mean stress values at each corresponding strain were plotted for both layers, and for the whole ON sheath. Mean stress-strain plots averaging data for all specimens in Table 1 are shown in Fig. 3A. Whole ON sheath behavior was similar to the IL (P>0.38), but different from the OL (P<0.02), consistent with YM behavior in the linear region. Nonlinear curve fitting was performed for each layer, achieving an excellent match to the data (Fig. 3B).

Resulting parameters are illustrated in Table 2. Ogden functions with three strain energy terms (N=3) adequately described each layer’s mechanical behavior within the entire strain range, including both the physiological lower part and the extremely high part approaching failure, thus providing quantitative descriptors that can be directly implemented in FEM.

Table 2.

Mean Ogden Hyperelastic Parameters for Optic Nerve (ON) Sheath.

	μ1 (MPa)	μ2 (MPa)	μ3 (MPa)	α1	α2	α3
Inner Layer	445.65	−188.30	−256.60	3.09	4.12	1.92
Whole ON Sheath	40.65	−19.60	−20.21	6.01	7.81	1.07
Outer Layer	45.52	−19.10	−24.20	1.89	3.91	−2.10

The rule of mixtures was used to estimate theoretically the combined properties of the whole ON sheath based upon measured YM for each layer, by this comparison to make an inference about predominant orientation of connective tissue fibers embedded in the sheath. In Eqns. 2 and 3, the mean f was calculated as 0.57 from OCT cross sectional measurements assuming constant thickness throughout the specimen length. By employing measured YM (E_i=19.8 MPa for IL, and E_o=9.7 MPa for OL), the rule of mixture predicts for whole sheath YM an upper bound of 15.5 MPa for parallel fiber loading and a lower bound of 13.7 MPa for transverse fiber loading, nicely matching the measured whole sheath YM of 15.4 MPa that corresponds to parallel loading.

For every indentation, stress relaxation data was fitted to the Hertzian linear viscoelastic formulation (averaged in Fig. 4), from which instantaneous stiffness (analogous to YM), viscosity, and time constants were calculated. Table 1 shows averaged results for the IL and OL. Results showed the IL to be stiffer than the OL, and the IL about twice as viscous with about a 30% a longer time constant (P<0.001, GEE for all three comparisons).

Fig. 4.

Mean relaxation behavior of inner (IL, 58 samples from 10 subjects) and outer layer (OL, 44 samples from 10 subjects) of the optic nerve sheath. The IL exhibited longer time constant (τ_inner = 19.4 sec) than the OL (τ_outer = 14.6 sec) corresponding to greater IL viscosity.

Two globes with very long attached ONs were used to explore a possible relationship between mechanical properties and histology suggested by differences in laminar anatomical characteristics of the ON sheath. It was seldom possible to perform both anatomical and biomechanical analyses in the same sheaths, since both types of tests are destructive and only occasionally was overall ON length sufficient to provide samples for both analyses. For one such long specimen, micro-indentation was performed to obtain local stiffness and viscosity of the two ON sheath layers. For this specimen (66 year old female), the IL was about 30% stiffer (12.6 vs 8.9 kPa) and about three-fold more viscous (296 vs 109kPa·s) than the OL. This may be compared with about 40% higher elastin density in the IL than OL (9.8 vs 7.2%) of this same sheath. Uniaxial tensile loading was performed for the second specimen (26 year old female). The IL exhibited about five-fold greater YM (8.1 vs 2.9 MPa) and about 60% greater elastin density (8.8 vs 5.5%) than the OL in this specimen, as illustrated in Fig. 5.

Fig. 5.

Mechanical properties of the ON sheath compared with elastin density in the same specimens. A. Compressive testing in one specimen aged 66 years. The inner layer (IL) had higher elastin density (9.8%) than the outer layer (OL, 7.2%), corresponding greater stiffness (12.6 vs 8.9 kPa) and viscosity (296 vs 109 kPa·s). B. Tensile testing of a different specimen aged 26 years. Young’s modulus was 8.1 MPa for the IL and 2.9 MPa for the OL, which may be compared with 8.8% and 5.5% elastin density in the two layers, respectively.

Discussion

The current study demonstrated that inner (IL) and outer (OL) layers of the adult human ON sheath, comprising the dural investiture of the ON, have distinct tensile and compressive mechanical properties. In both tension and compression, the IL is stiffer, and its tensile modulus dominates behavior of the sheath as a whole. Local micro-indentation demonstrated that ON sheath IL is significantly stiffer and more viscous than the OL.

The ON sheath exhibited similar mechanical behavior under both longitudinal and circumferential tensile loading, and this similarity is supported by application of the rules of mixtures to compare the YM of the whole sheath with YM values of each of its layers that are readily-separable in older specimens. Therefore, it is reasonable to regard the ON sheath as a transverse isotropic material (Fig. 6), a special case of orthotropic material, that is symmetric about an axis normal to the plane of isotropy³¹, along the radial axis of the ON and sheath. Transverse isotropy is common in geophysics and biological membranes, wherein the properties in the plane of the membrane differ from those perpendicular to it.

Fig. 6.

Transverse isotropy of optic nerve (ON) sheath that consists of inner and outer layers weakly attached along the radial boundary, so that the whole sheath has isotropic properties in the longitudinal and circumferential directions.

Examination of mechanical and anatomical characteristics in two of the same specimens suggests that the IL might be stiffer, at least in part, because it has denser elastin. Elastin is a protein with reversible extensibility preventing dynamic tissue creep and permitting tissues to resume their shapes after loading³². Elastin is abundant in artery, lung, elastic ligament, cartilage, and skin^{33, 34}, as well as in the lamina cribrosa^35-37, the peripapillary scleral ring³⁶, and the connective pulley tissue system of the orbit at sites of concentrated mechanical stress³⁸. Elastin resists tissue deformation and has a memory permitting it to return to its undeformed shape after low stress loading^{32, 39}. Elastin and collagen fibers comprise parallel mechanical elements when tissue undergoes strain: at low strain, collagen fibers extend easily with most of the load borne by elastic fibers, but at high strain, collagen fibers become limiting⁴⁰. For example, in rat aorta, collagen digestion has little effect on stress levels at up to about 30% strain, implicating a major role for elastin in this loading regime⁴¹ that is most relevant to the strains used in the current paper. We describe in greater detail elsewhere the laminar differences in elastin abundance and fiber orientation in the human ON sheath²⁴. Cross linking and absence of elastin turnover with age leads to tissue stiffening⁴⁰. The higher elastin density in the ON sheath IL is consistent with its stiffer mechanical properties than the OL.

The biomechanical properties of the human ON sheath are important to understanding of the mechanical loads on the ON and its critical juncture with the globe at the elastic fiber ring that borders the lamina cribrosa (LC)⁴². Finite element models of this mechanical behavior were first developed to clarify the effects of IOP and intracranial pressure on the LC and surrounding ocular tissues^{43, 44}, but have recently been expanded to include effects of eye movements^{14, 22, 45}. Since MRI demonstrates that the ON and sheath exhaust their redundancy and tether the globe against the medial rectus muscle in adduction exceeding about 26 degrees¹ in a manner that correlates with OCT evidence of peripapillary deformation^{22, 46, 47}, ON sheath mechanical behavior has become of particular interest. It has recently been proposed that eye movement, particularly adduction, may produce repetitive strain injury to the ON and thus constitute an IOP-independent mechanism of optic neuropathy in glaucoma^{3, 22}. This has been proposed to be a major contributor to so-called “normal tension glaucoma” that now represents the most prevalent form of primary open angle glaucoma worldwide³.

A FEM of the mechanical effects of adduction tethering of the ON has been developed based on the local material properties of bovine tissues¹⁴. This FEM incorporates human anatomical dimensions, and predicts effects of 6° incremental adduction past the first point at which ON path redundancy is exhausted at a typical 26° adduction angle². Although predicted effects vary depending on use of linear elastic or hyperelastic material properties, the FEM suggests that the temporal ON sheath posterior to the scleral may experience 3 – 7% strain during adduction tethering to an angle 6° beyond exhaustion of ON path redundancy. A more recent FEM sensitivity analysis employing hyperelastic characterization of human ocular tissues suggests that maximum strains in the ON sheath could be in the range of 5 – 8% for this incremental adduction⁴⁸. It should be recognized that the physiologic limit of adduction exceeds 45°, but the ON becomes tethered for all adduction angles exceeding about 26°. Thus, strains in the ON sheath are likely to exceed 10% during larger physiological adductions. It is notable from Fig. 3 that below about 10% strain, stress in the OL exceeds that in the IL, but this relationship is reversed for greater strains. The stress-strain relationship for the OL is nearly linear up to about 30% strain for the OL, but has a nonlinearly increasing slope for the IL (Fig. 3). Thus, the tangent stiffnesses of the IL is lower than that of OL at less than 10% strain, but exceeds it for more than 10% strain. The differential behavior of the two layers would tend to shift stress from the OL to the IL during larger adductions exceeding about 32°. The current data probably represent reasonable characterizations of ON sheath behavior in vivo.

We have elsewhere reported the anatomical finding that the adult (although not child) human ON sheath is distinctly bilaminar. While elastin fibers do not bridge the junction between the two layers, collagen or some other material does so in younger specimens; this interlaminar bridging appears to degenerate with advancing age, allowing the two layers to become more readily separable and presumably more mechanically independent. The present findings that the ON sheath is transversely isotropic with a stiffer and more viscous IL, have important implications for understanding the role of the ON sheath in glaucoma. No prior finite element model has represented the bilaminar nature of the ON sheath, even though the IL inserts closest to the LC and scleral canal through which the ON axons traverse into the eye. There is published data on the biomechanical properties of the ON sheath only for pig, whose sheath is much thinner than that of human²³. In the absence of measured data on the human ON sheath, commonly published FEMs have assumed that it is identical to the ON sheath of the cow at 45 MPa ¹⁴ (three-fold greater than measured here for human), or fitted a Yeoh model to porcine data resulting in an approximate YM of about 4 MPa²³ (about one fourth that than measured here for human). It is obvious that neither of the foregoing values of YM accurately represents human ON sheath, determined here to be about 15 MPa overall, but 19 MPa in the relatively critical IL, about twice that of 10 MPa in the less critical OL. It is possible that the relatively-low 2:1 specimen aspect ratio employed in the current experiments out of anatomical necessity might have caused a modest under-estimation of YM. It is known that aspect ratio is positively correlated with tensile stiffness, so that as a specimen aspect ratio increases from 1 to 5, stiffness increases by 36%²⁵. While aspect ratio might account for as much as 30% differences in YM, it seems unlikely to account for the three- to four-fold differences between the current values for human, and those published for pig.

No prior model has considered ON sheath viscosity, which should eventually be considered to account for human eye saccadic eye movements typically reaching hundreds of deg/s⁹ and occur about 180,000 times daily⁸. Greater IL stiffness might concentrate the stress generated by ON sheath tethering in adduction into the critical ONH area¹. Further FEM incorporating all these aspects would provide more precise predictions of the mechanical behavior of the posterior eye, both as influenced by IOP and by eye movement.

Due to limited length of ON tissues attached to the donated globes, it was not possible to determine if the mechanical properties of the ON sheath might vary according to distance along its entire length in the orbit. However, specimens avoided the sheath region immediately adjacent the sclera and were from a region of the sheath in which thickness appeared uniform on inspection. In general, younger people may have more compliant peripapillary tissues than older people because of age-related stiffening in sclera^49-51 and Bruch’s membrane ⁵². Elastosis of the lamina cribrosa progresses with age^53-56, particularly people who have open angle glaucoma^{40, 57, 58}. Scanning laser ophthalmoscopy demonstrates that optic disc and peripapillary tissue deformation during adduction tethering is less in older than younger healthy subjects⁵⁹. Elastin density in the human ON sheath increases significantly with age²⁴. Although specimens in the current study lacked sufficient age variation to address this question, it would of course be important to determine the relative laminar effects of age-related elastosis on the ON sheath, as sheath elastosis with advancing has been proposed as a likely etiologic factor in primary open angle glaucoma without elevated IOP³. The current study had a paucity of specimens available for concurrent biomechanical and anatomical study, limiting quantitative interpretation of anatomic-mechanical correlations. Future studies would benefit from performing such correlative studies in additional specimens with sufficiently long optic nerves, and preferably in donors both affected and unaffected by glaucoma.

Conclusions

Under tensile loading, the stiffness of the inner layer of the human optic nerve sheath exceeds that of the outer layer, with the sheath as a whole exhibiting an overall stiffness intermediate between the two layers. Under compression, the inner layer of the human optic nerve sheath is stiffer and more viscous than outer layer. The inner layer contains denser elastin than the outer sheath. The greater stiffness and viscosity of the elastin-rich inner layer of the optic nerve sheath may be important in concentrating mechanical stress near the lamina cribrosa when the sheath is under tension during eye movement.

Appendix

Hertzian Viscoelastic Model

The Hertzian viscoelastic contact model relates the force, P, exerted by a rigid sphere of radius, R, to the elastic modulus, E, and the Poisson ratio, ν, of an incompressible material indented to a displacement, h.

$$P=\frac{4\sqrt{R}}{3}\frac{E}{(1-v^2)}{h}^{3∕2}=\frac{8\sqrt{R}}{3}(2G)h^{3∕2}$$ (1)

The relationship between elastic and shear modulus values is simply E = 3 G for an incompressible solid (ν = 0.5). The principle of viscoelasticity is used for modeling time dependence⁶⁰, and a viscoelastic operator is substituted for the elastic modulus, E or shear modulus, G, in equation 1^{61, 62}. The relaxation response of a step-load, rigid, spherical-tip indenter to the material as a function of time and shear modulus, G(t), is illustrated by Eq 2.

$$P(t)=\frac{8\sqrt{R}}{3}{h}_0^{3∕2}[G(t)]$$ (2)

where G(t) is the time-dependent shear relaxation modulus. The observed rise time (t_R) in real instances cannot be instantaneous; hence the ramp correction factor (RCF) is used to account for a physically achievable loading rate^{61, 63}. A viscoelastic integral operator for relaxation, where u is a strain function in terms of time dummy variable τ, is shown by Eq. 3.

$$P(t)={\int }_0^{t}G(t-\tau )\left[\frac{du(\tau )}{d\tau }\right]d\tau$$ (3)

By combing Eq. 2 and 3, the real exerted force as a function of time, P(t) is

$$P(t)=\frac{8\sqrt{R}}{3}{\int }_0^{t}G(t-\tau )\left[\frac{d}{du}{h}^{3∕2}(u)\right]du$$ (4)

As shown by Mattice et al.⁶¹, the Boltzmann integral method is used to solve Eq. 4 and for ramp-loading rate, k, displacement for ramp-hold relaxation can be written as

$$h(t)=kt0\le t\le t_R$$ (5)

$$h(t)=k{t}_R=h_{max}t\ge t_R$$ (6)

Since the load, P(t), exponentially decays, the solution is expressed as the step-loading relaxation solution adjusted by an RCF due to non-instantaneous ramp loading. Only the first two terms are considered for simplicity, as shown in Eqns. 7 and 8.

$$P(t)=B_0+B_1\text{exp}(-t∕{\tau }_1)$$ (7)

$$G(t)=C_0+C_1\text{exp}(-t∕{\tau }_1)$$ (8)

where τ₁ represents the time constant for the first exponential decay. B_n, and C_n represent the fitting constants and the relaxation coefficients, respectively. Once fitting parameters ( B₀, and B₁) have been determined, they are converted to relaxation parameters (C₀, and C₁) using Eqns. 9, and 10.

$$C_0=\frac{B_0}{h_{max}^{3∕2}\left({}^{8\sqrt{R}}∕_3\right)}$$ (9)

$$C_1=\frac{B_1}{\left((RC{F}_1)h_{max}^{3∕2}{}^{8\sqrt{R}}∕_3\right)}$$ (10)

The equation for RCF₁ is illustrated by Eq. 11.

$$RC{F}_1={}^{{\tau }_1}∕_{t_R}[\text{exp}(t_R∕{\tau }_1)-1]$$ (11)

Instantaneous modulus, E₀ , can be computed from the fitted relaxation coefficients using Eq. 12.

$$E_0=\frac{3G(0)}{2}=\frac{3(C_0+C_1)}{2}$$ (12)

The viscosity, η, for each type of material can be calculated using Eq. 13⁶⁴.

$$\eta =E\cdot {\tau }_1$$ (13)