An axial force balance on the eye was first undertaken to define the system and estimate the relative contributions of each potential source to the axial motion of the eye. Specifically, these included contributions from intracranial pressure (F _ICP ), fat swelling (F _fat ), extraocular muscle tension (F _EOM ), and optic nerve stretching (F _ON ), giving the net force balance ↑ a n t e r i o r + ∑ F z = F I C P + F f a t − F E O M − F O N = 0 . (1)

The homeostatic, terrestrial contributions from each source were estimated from literature sources and clinical experience as follows:

• F _EOM ∼ 1–4 mN in the posterior direction [passive contribution only; (Guo et al., 2016)]

Based on this analysis, we are left to conclude that Eq. 1 can be reduced to F E O M ≈ F f a t + F I C P , (2)

Indicating that EOM tension must increase by some amount unknown a priori and depend on the balance of volumetric swelling forces and must be estimated computationally. The model also has a boundary condition representing F _ON for completeness since these effects may come into play at larger deformations or during eye movements.

Based on the force balance, an axisymmetric geometric model (Figure 3) was constructed in COMSOL Multiphysics v5.6 (COMSOL, Inc., Burlington, MA) using nominal dimensions for the eye and orbit, based on the histological sections in Figure 2 (Ackerman, 1998). A baseline model intended to represent a normal human eye and orbit, in terms of anatomy and material properties was created and used as a basis for modeling predictions (Figure 3A). This model, referred to as the “Baseline Model,” was used as a starting point for all sensitivity analyses performed. This geometry was parametrized as shown in Figure 3B, while Table 1 gives the parameter values for the Baseline Model. An ICP of 10 mmHg (13.6 cmH₂O) was chosen for the Baseline Model as it corresponds to the middle of normal ICP [6.5–19.5 cmH₂O (Swyden et al., 2021)].

The living eye is loaded with intraocular pressure (IOP), intracranial pressure (ICP), tension from the extraocular muscles (F _EOM ), and the contact force generated by the swelling fat (F _fat ). All tissues were modeled using an incompressible, isotropic, neo-Hookean hyperelastic material model. Assumed values for elastic moduli are given the table below.

Each model was solved using a sequence of three steps (Figure 4). Model A modeled the zero-stress state of the eye and optic nerve in the presence of vitreous swelling and ICP while the orbit, fat, and all other external forces were absent. Model A was found using a coordinate search algorithm by varying the geometry of the zero-stress eye (i.e., the polar and equatorial elliptical radii) and the vitreous hygroscopic strain such that the resulting eye corresponded to the target geometry and IOP. Loading of Model A with vitreous swelling gives the residual stress state of the eye, Model B. Model B was then taken as the initial condition of the eye and optic nerve when simulating fat swelling as follows. Model B was set within the orbit and orbital fat, then the extraocular muscle tension was gradually added to find Model C—the terrestrial (homeostatic) state of the eye and orbit. Finally, orbital fat swelling and ICP were incremented to study the response of the eye and optic nerve, giving Model D as a final output. Results described below are for Model D relative to Model C rather than the intermediate models; it therefore represents the changes which might occur due to the hypothesized swelling of the orbital contents arising from the microgravity-induced fluid shift.

Fat swelling was modeled using a hygroscopic swelling model in which the hygroscopic volume ratio J H was varied as a modeling parameter. This was achieved in COMSOL by using the hygroscopic swelling model, where the hygroscopic strain ε H = β C W − C W , 0 , with β as the coefficient of hygroscopic swelling of the fat-water mixture (equivalent to the inverse of density, or about 0.001 m³/kg for water), C _W,0 as the baseline (terrestrial) water content, and C _W as the current water content. Thus, J H = 1 + ε H 3 . Values of J _H were studied to indicate how much fat swelling would be required to induce the ocular findings associated with SANS.

The IOP was also generated using a hygroscopic swelling model to mimic the physiological process which maintains IOP (i.e., the eye has more fluid in it than it does in its zero-stress state). In this case, the dimensions of the zero-stress geometry were estimated, swelling in the vitreous simulated, and the difference between the model predictions and target ellipse radii and target IOP were calculated using convex objective functions. COMSOL’s Optimization Module was used to determine the zero-stress geometry, the amount of “swelling” required to achieve IOP. Once this optimization was complete, the orbit and fat were constructed and assumed to be in contact with the eye and ON prior to simulating fat swelling.

This residually loaded state was then inserted into the model containing the fat, F _EOM , and ICP to find the terrestrial baseline geometry and stresses. The lateral and posterior boundaries of the orbital fat were held fixed to simulate attachment to the orbital bone. Intracranial pressure was applied to the exterior surface of the sclera over a segment representing the sub-arachnoid space with a radial distance of 270 µm [estimated from Figure 1 of (Lagrèze et al., 2009)].

Contact boundaries were established between the orbital fat and the sclera. This allowed sliding to occur at the sclera-fat interface, as would be expected in vivo as the eye must easily rotate during routine visual tasks.

Boundary conditions were applied as indicated in the free body diagrams shown in Figure 1B. The nerve is constrained using a spring boundary condition: the tension in the nerve increases with forward translation of the eye. Thus, tension could arise in the nerve if (A) significant proptosis occurred and/or (B) normal eye movements resulted in removal of slack. Demer (Demer, 2016) measured ON length in relation to orbit length, finding an increasing trend [Figure 11 of (Demer, 2016)]. To infer slack from these measurements, one would also need to know the axial length of the eye, allowing the slack to be approximated as O N   s l a c k = O N   l e n g t h − o r b i t   l e n g t h − 1 2 a x i a l   l e n g t h .

Taking the values for orbit length and axial length from Table 1, along with the regression equation from the referenced figure, gives a value for ON slack of 8.4 mm. We therefore chose a threshold slack of 6 mm as a conservative estimate of ON slack. This threshold was never exceeded during any simulations, so the optic nerve remained slack at all times.

The extraocular muscle tension was applied as a point source to the peripheral anterior globe. This tension was modeled as an analytical function representing the force required to achieve passive stretching of a single rectus muscle multiplied by the number of rectus muscles (i.e., four) [Eq. 10 of (Guo et al., 2016)]: F E O M = 4 M − 0.29 w + 4.79 e w 4.57 − 3.01 , (3) where w is the anterior displacement of the EOM insertion point in mm to give F _EOM in mN and M is a multiplier used to increment the loading as described below. Generally, M was incremented from 0–1 to simulate the passive muscle tension. Since the active forces in the muscle are unknown, an additional study was undertaken in which M was incremented from 0–2.25 to simulate additional EOM tension arising from active contributions. This simplifying approach was adopted since neither the active nor passive contributions are known for SANS, and an M value of 2.25 resulted in a net doubling of F _EOM .

The balance of forces was computed numerically using finite element analysis in COMSOL Multiphysics (Figure 3) to investigate the effect of an increase in retrobulbar fat water content on the eye. All simulations included geometric non-linearity. Following the aforementioned procedure for determining the terrestrial state of the eye, the ICP and J _H were systematically varied to simulate the effects of a microgravity-induced fluid shift on the eye and orbital tissues.

Changes to axial length and posterior radius of curvature (ROC) were calculated as a proxy for globe flattening. ROC was estimated by fitting a circle to the deformed scleral surface within a 4 mm radius of the axis of symmetry using the Pratt method (Pratt, 1987). Change in choroidal arc length was used to imply the formation of choroidal folds: shortening indicates a decrease in tension and the possibility of wrinkling (Ligarò and Barsotti, 2008; Barsotti and Ligarò, 2014). The peripapillary arc length is the arc length along the inner boundary of the sclera spanning the posterior-most quarter of the eye’s circumference in the zero-stress state. While this definition is somewhat arbitrary, we found that the results were robust when other definitions were considered (e.g., 1/8 or 1/2 rather than 1/4).

An additional series of simulations was conducted to determine whether and to what extent measurable anatomical variations may influence the risk of SANS based on a change in the water content of the orbital fat with exposure to microgravity. The varied parameters were polar radius R _p (where axial length = 2R _p ), equatorial radius R _eq , orbit depth L ₀ , radius of the orbit margin R _OM , and depth of orbit margin L _OM as shown in Figure 3B.

Iterative mesh refinement indicated that the solution was largely independent of mesh density using quartic Lagrangian shape functions, even with a coarse mesh. The mesh density used for the reported simulations (Figure 3C) was therefore selected on the basis that it offered improved stability for the non-linear solver for the variety of studies conducted.

The predicted deformations corresponding to 0%–7% increase in retrobulbar content volume are given in Table 2 and shown in Figure 5. Decreasing axial length and increasing posterior ROC correspond to physical signs of globe flattening. Decreased peripapillary arc length indicates compression of, or reduced tension in, the choroid which could result in choroidal folds. In addition, the model predicted low levels of proptosis. Although subtle proptosis has not been reported with SANS, it could be easily missed on clinical exam.

Sensitivity studies were performed to evaluate the model’s sensitivity to modeling assumptions. First, the boundary conditions were investigated. Since a single representative ICP value selected for the Baseline Model was selected from numerous literature values, a thorough sensitivity analysis was performed to compare the relative contributions of ICP and fat swelling to ocular deformation (Figure 6). Axial length, posterior ROC, peripapillary choroidal arc length, and proptosis depend very weakly on ICP but strongly on orbital fat swelling. Applying IOP using a hygroscopic swelling model in the vitreous gave results which were nearly identical (i.e., within 1%) to those applying a pressure boundary condition to simulate IOP. Initial simulations modeled the boundary between the eye and ON with the orbital fat as a tied boundary, as opposed to the more realistic frictionless contact boundary. This caused much larger ocular deformations but gave qualitatively similar results in all cases. Approximately doubling F _EOM to simulate active tension in the EOM decreased proptosis by 7% while resulting in a 63% larger drop in axial, 33% more decline in peripapillary arc length, and increased radius of curvature by 8%. This was achieved by setting M = 2.25 in Eq. 1, thereby increasing the maximum EOM force from 25.3 to 53.8 mN for the same extent of swelling.

Next, the elastic moduli E of the sclera, vitreous, and fat were varied over a broad range. Varying E _fat from 0.7–1.5 kPa and E _vitreous from 6.5–25 Pa had no effect on any modeling predictions over the entire range of E _sclera considered (0.25–5 MPa) as expected, since their incompressibility dominates their mechanical response. The fat and vitreous have a very low shear modulus all materials were modeled as incompressible and linearly elastic with the properties given in Table 3. Relative to bulk modulus, implying that they will effectively behave as incompressible fluids under the conditions studied: they change shape very easily but do not appreciably change volume due to loading. However, the response of the eye depended strongly on E _sclera over this range, with all SANS signs increased in magnitude in less stiff eyes (Figure 7).

Finally, sensitivity to orbital anatomy and ocular anatomical dimensions were examined (Figure 8). The model predicted orbit depth (L _O ) to be the most important feature of orbit anatomy, with a 10% increase resulting in ∼25% changes in the predicted responses. Increasing the eye’s equatorial diameter by 10% had a similar effect on globe flattening (i.e., change in axial length and posterior radius of curvature). Orbital depth varies considerably between individuals. Fat swelling-induced changes could therefore depend significantly on an individual’s orbital anatomy.

The terrestrial force balance (Eq. 2) for the Baseline Model indicates that F E O M ≈ 7.36   m N , F I C P ≈ 1.17   m N , and F f a t ≈ 5.6   m N . Increasing ICP to 20 cmH₂O–the lower limit of borderline intracranial hypertension (Swyden et al., 2021)—only increased F I C P ≈ 1.75   m N , while 7% swelling could increase F f a t ≈ 36.3   m N . Thus, even a pathological increase in ICP would be unlikely to contribute to a biomechanical mechanism of SANS.