## Abstract

Exposure to microgravity causes a bulk fluid shift toward the head, with concomitant changes in blood volume/pressure, and intraocular pressure (IOP). These and other factors, such as intracranial pressure (ICP) changes, are suspected to be involved in the degradation of visual function and ocular anatomical changes exhibited by some astronauts. This is a significant health concern. Here, we describe a lumped-parameter numerical model to simulate volume/pressure alterations in the eye during gravitational changes. The model includes the effects of blood and aqueous humor dynamics, ICP, and IOP-dependent ocular compliance. It is formulated as a series of coupled differential equations and was validated against four existing data sets on parabolic flight, body inversion, and head-down tilt (HDT). The model accurately predicted acute IOP changes in parabolic flight and HDT, and was satisfactory for the more extreme case of inversion. The short-term response to the changing gravitational field was dominated by ocular blood pressures and compliance, while longer-term responses were more dependent on aqueous humor dynamics. ICP had a negligible effect on acute IOP changes. This relatively simple numerical model shows promising predictive capability. To extend the model to more chronic conditions, additional data on longer-term autoregulation of blood and aqueous humor dynamics are needed.

**NEW & NOTEWORTHY** A significant percentage of astronauts present anatomical changes in the posterior eye tissues after spaceflight. Hypothesized increases in ocular blood volume and intracranial pressure (ICP) in space have been considered to be likely factors. In this work, we provide a novel numerical model of the eye that incorporates ocular hemodynamics, gravitational forces, and ICP changes. We find that changes in ocular hemodynamics govern the response of intraocular pressure during acute gravitational change.

- intraocular pressure
- ocular blood flow
- ocular compliance
- intracranial pressure
- visual impairment and intracranial pressure
- space physiology

approximately 60% of astronauts who spend more than one month in space suffer from visual impairment and intracranial pressure (VIIP) syndrome, a spectrum of ophthalmic changes, including gross distension of the optic nerve sheath, optic disk edema, optic nerve kinking, cotton wool spots, choroidal folds, and globe flattening (26, 33). Some of these changes are permanent, and thus, VIIP is a significant health concern for NASA (1). The etiology of VIIP is currently unknown, but choroidal engorgement, altered intracranial pressure (ICP) and/or posture-driven ICP variations are hypothesized to be among the most significant contributing factors (1, 29, 33). The potential importance of modified ocular volumes and pressures is consistent with the hypothesis that biomechanical factors acting on ocular tissues are, in part, responsible for the structural changes observed in VIIP (1).

One approach to improve understanding of the pathophysiology of VIIP is to numerically simulate the biomechanical environment in the posterior eye. Numerical simulations are now widely used to study the biomechanics of the optic nerve head in glaucoma (15, 18, 19, 45, 46). The results of the numerical model described here cannot directly predict functional changes in vision; instead, this model is one part of an integrated system to predict fluid shifts in microgravity and the resulting effects on ocular biomechanics. At the largest scale, our lumped-parameter whole body model (under separate development) calculates altered cranial and systemic pressures due to fluid redistribution caused by gravitational change. The lumped-parameter eye model uses these values as input and calculates intraocular pressure (IOP) and blood volume changes. Finally, a finite-element description of lamina cribrosa (LC) biomechanics (12), which is known to respond to changes in ICP and IOP (35, 36), calculates biomechanical strains in the tissues of the posterior eye, which can be correlated to anatomical pathologies and nerve damage in the eye (1). The relationship between LC deformation and optic nerve function is complex (53) and poorly understood; thus, the results of our modeling must be translated to laboratory and clinical studies to truly understand the pathophysiology of VIIP.

Here, we describe a lumped-parameter model of the eye, designed to predict the impact of increased ocular blood pressures and elevated ICP on IOP. Unlike other physiological systems, such as the cardiovascular and central nervous systems (43, 51), there are very few lumped-parameter models of the eye (20, 25). Neither model incorporates ICP dynamics. Further, choroidal engorgement is a frequently noted effect of microgravity (1), so models that do not include the choroid (20) or are formulated for species with a much simpler choroidal structure (25) do not provide sufficient insight into the major factors influencing the human eye. This model is, thus, novel in its formulation and is vetted by comparison against experimental data on human response to gravitational changes.

## METHODS

Lumped-parameter models represent the actual biological fluid and tissue systems as an ensemble of fluid-filled, deformable compartments. Each compartment is characterized by a uniform pressure that can change over time. The compartments interact through fluid exchange and/or pressure differences between compartments between adjacent compartments to cause variation in compartmental volumes and pressures with respect to time. Whether or not adjacent compartments exchange fluid, they can influence their neighbors’ pressures and volumes through their compliance (*C*), which is a measure of the ease with which a compartment can deform to accommodate excess fluid volume or an increase of applied pressure at the compartment’s boundary.

#### Governing equations.

The eye is characterized in this lumped-parameter model by five compartments (Fig. 1). Each compartment has a spatially uniform pressure and an initial volume. The retrobulbar subarachnoid space (rSAS, pressure P* _{csf}*, and volume V

*) at the posterior eye does not exchange fluid with other compartments but can influence compartmental pressures/volumes through deformation driven by the intercompartmental pressure difference acting in concert with a compliance,*

_{r}*C*

*. The passive compartment (pressure IOP and volume V*

_{rg}*) consists of intraocular components that do not change in volume, such as the lens and vitreous humor. The aqueous humor compartment (pressure IOP and volume V*

_{p}*) has inflow , representing aqueous humor formation, and outflow, . Ocular blood is contained in two compartments: arterial (pressure P*

_{aq}*and volume V*

_{a}*) and venous (pressure P*

_{a}*and volume V*

_{v}*). Blood flows into the arterial compartment at a rate of , and out from the venous compartment at . The capillary bed is assumed to be fully rigid and of fixed resistance, so that changes in the total volume of ocular blood are given by ΔV*

_{v}*= ΔV*

_{b}*+ ΔV*

_{a}*. Anatomically, the two reservoirs of ocular blood are the choroid and retina, each of which have arterial, capillary, and venous compartments. This model lumps the contributions from the retina and choroid together to form a generalized ocular blood volume. To calculate ΔV*

_{v}*, only the lumped arterial and venous volumes need to be considered. For blood pressures of interest to this model, the arterial compartment has a constant compliance*

_{b}*C*

_{ag}with respect to the globe since IOP is expected to remain well below P

*. On the other hand, the venous-globe compliance,*

_{a}*C*

_{vg}, can vary as a function of P

*and IOP. The net compliance between the blood and the other components of the eye is denoted as*

_{v}*C*

_{bg}=

*C*

_{ag}+

*C*

_{vg}. The globe is defined as the union of the passive, aqueous, and blood compartments, so that its volume V

*= V*

_{g}*+ V*

_{p}*+ V*

_{aq}*. Thus, changes in V*

_{b}*are due only to changes in V*

_{g}*and/or V*

_{aq}*(24).*

_{b}Using unsteady mass conservation for a generic compartment of volume V containing an incompressible fluid [aqueous humor, blood or cerebrospinal fluid (CSF)] and with inflows *Q _{in}* and

*Q*, we can write

_{out}Alternatively, by using the definition of the compartmental compliance, *C* = dV/dP, we can write the compartmental volume change in terms of rates of change of pressures:

While *Eqs. 1* *and* *2* are equivalent, one may be more convenient than the other, depending on whether information about flow rates or pressures are available^{1}. For the aqueous compartment, it is more convenient to use the formulation based on flow rates, and so we state

In the case of the blood and rSAS compartments, we use the formulation based on pressures, so that

(4)(5)Next, we write a rate of volume change equation for the globe. There are two ways to do so. The first is to consider the internal globe components, recalling that the passive (dark gray) components do not change their volume, to determine that

(6)The second is to treat the globe as a “black box” without consideration of its internal composition. From this external viewpoint, globe volume V* _{g}* can increase in two ways: by exchange of volume with the rSAS or by expansion of the corneoscleral shell. Thus, taking suitable account of signs, we can state that(7)where

*C*

_{shell}is the compliance of the corneoscleral shell, and we assume that the extraocular (orbital) pressure is constant. Equating between

*Eqs. 6*and

*7*and substituting appropriate terms from

*Eqs. 3*

*–*

*5*, we obtain:

We next note that *Eq. 8* can be rearranged to find that

To make further progress, we must specify various terms in *Eq. 9*.

#### Aqueous compartment.

We treat *Q _{aq,in}* as constant, since this quantity is little affected by gravitationally induced changes in blood pressure or IOP on time scales up to an hour (34). We assume aqueous humor drainage is the sum of a pressure-dependent (conventional) outflow through the trabecular meshwork (outflow facility =

*C*) and uveoscleral (unconventional) outflow,

_{tm}*Q*(10)where EVP is the episcleral venous pressure.

_{uv}*Eq. 10*is simply Goldmann’s equation (50).

In the above formulation, we must account for a complexity of aqueous outflow: the trabecular meshwork is a one-way valve, preventing blood reflux into the eye if EVP > IOP. We can represent this behavior by(11)where *C _{tm}*

_{,normal}is the outflow facility during the normal drainage of aqueous humor.

#### Blood compartment.

All simulations start from a baseline condition (typically upright or supine posture), in which we assume that arterial and venous volumes are in equilibrium. The blood-globe compliance is separated into arterial and venous components using a factor ω, i.e., and . For simplicity, we chose to take ω as the volume fraction of ocular blood that is distal to the arterioles, since this blood is (approximately) at venous pressure. From data provided by Guyton and Hall (21), ω lies in the range 0.7 to 0.8. In this work, we assumed that ω = 0.7, as this resulted in better agreement with available experimental data (data not shown). As noted above, for the blood pressures of interest in this work, all changes to the net pressure-dependent *C*_{bg} are ascribed to the venous compartment, i.e., .

Recall that ocular compliance is measured in vivo by infusion of fluid into the aqueous humor compartment and recording the associated change in IOP. In such an experiment, fluid infusion can change the globe volume, the intraocular blood volume, and (a small amount of) CSF volume in the rSAS due to posterior motion of the LC. Thus, in vivo globe compliance must be the sum of the individual compliances, namely, *C*_{g}_{,in vivo} = *C*_{shell} + *C*_{rg} + *C*_{bg}, allowing us to rewrite *Eq. 9* as

#### Reduced form of governing equations.

The terms explicitly containing IOP or derivatives of IOP in *Eq. 12* can be isolated to yield(13)where we have defined a “volume forcing” function for the globe as

Note that *F*_{g} has dimensions of flowrate and represents a volumetric inflow rate into the globe compartment due to (explicitly) non-IOP-related effects. Of course, we allow for the terms on the right-hand side (RHS) of *Eq. 14* to depend on IOP, so that *F _{g}* incorporates implicit IOP dependence. In particular, the values for the compliances

*C*

_{g,in vivo}and

*C*

_{bg}are dependent on IOP and on V

*, as considered further below.*

_{g}*Equations 13*and

*14*represent a nonlinear system that allows IOP to be determined if compliances, blood pressures and aqueous outflow parameters are known.

We will work with *Eqs. 3**–**6*, *13*, *14*, rewritten below for convenience:

#### Numerical implementation.

The six immediately preceding equations form the basis of a time-marching scheme, which we implemented in MATLAB (Mathworks, Natick, MA). *Equations 13* and *14* are first used to solve for IOP at time level *n* + 1 given the value at level *n*. The value of IOP at this time level is then plugged into the right-hand sides of *Eqs. 3**–**5* to advance V* _{aq}*, V

*, and V*

_{r}*to time level*

_{b}*n*+ 1. Finally,

*Eq. 6*is used to advance V

*to time level*

_{g}*n*+ 1.

If the time marching of *Eqs. 14* and *15* is done implicitly, this is a fully implicit scheme. However, we note that the terms *C*_{g}_{, in vivo} and *C*_{bg} depend on IOP, so that the RHS of *Eq. 13* implicitly depends on IOP. We can create a pseudo-implicit time-marching scheme by evaluating these terms at time level *n* in the RHS of *Eq. 13*, which is acceptable if *C*_{g,in vivo} and *C*_{bg} are not changing rapidly with respect to time. This is the approach that we adopted to solve the governing equations, as described in appendix a.

#### Determination of parameter values.

The majority of model parameter values were obtained directly from the literature (Table 1). However, several important parameters were unavailable, and it was necessary to deduce them from published studies. We describe this process below.

#### Compliance of the globe, C_{g}, *and compliance between blood and globe compartments, C*_{bg}.

_{bg}

Initial testing showed that the compliances of the globe, *C*_{g}_{,in vivo}, and of the blood compartment, *C*_{bg}, had a major influence on model performance. Therefore, we describe in some detail how these parameters were determined for our model. Silver and Geyer (47) carried out a meta-analysis of pressure-volume data for enucleated and living human eyes. Their expression for the volume of fluid that must be infused into a living eye to obtain a certain IOP, ΔV_{infused, in vivo}, (Fig. 2*A*) is(15)
where V_{g0} is the globe volume before injection and *C*_{0}, *C*_{1}, and *C*_{2} are empirically determined constants (Table 1) selected so that ΔV_{infused, in vivo} = 0 at IOP = 5 mmHg (47). In the above and subsequent equations, the volume increment is assumed to be small (ΔV_{infused} << V_{g0}), which is typically the case in practice. *Eq. 15* can be compared with the well-established Friedenwald relationship for enucleated human eyes:(16)where is an offset such that ΔV_{infused,enucleated} is zero at IOP = 5 mmHg (47) to match *Eq. 15*, and *k* is the nondimensional stiffness of the globe. We note that the practice of “zeroing” ΔV_{infused} at an IOP of 5 mmHg (47) is arbitrary; in practice, constant offsets have no effect on the compliances computed below, since our compliances are based on differentiating ΔV_{infused}. The value of *k* in *Eq. 16* is related to Friedenwald’s ocular rigidity coefficient^{2} *K* = 0.048 μl^{−1} by the expression *k* = *K*V_{g0}. Here, we focus on males, since that reflects the majority of the astronaut population. Taking an average globe starting volume of 6,500 μl (47), we obtain a value of *k* = 312. We determined that the solution was insensitive to V_{g0}; an increase of ± 10% resulted in a difference in final IOP of <<1% in all gravitational scenarios.

From the definition of compliance, we can compute the compliance of the globe, *C*_{g}, in both the in vivo and enucleated situations as (Fig. 2*B*):

The difference between the two curves in Fig. 2*A*, i.e., between *Eqs. 17* and *18*, can be attributed to the presence of blood in the living eye (11, 23). Specifically, fluid infused into an enucleated eye acts to expand the globe and displace the LC into the rSAS (see below), whereas fluid infused into the living eye also displaces blood from vascular beds in the eye. Therefore, we can estimate the compliance between the blood and globe compartments, *C*_{bg} (Fig. 2*B*), as

Note that *C*_{bg} is of order 1 μl/mmHg at a typical IOP of 15 mmHg, while the whole-globe compliance is ~2.5 times larger.

For this formulation, *C*_{bg} is a function of IOP alone. It can be argued that it is more appropriate for *C*_{bg} to depend on a transmural pressure difference, which can be estimated as IOP − P* _{v}*. appendix b discusses the implementation of this second strategy, but it does not yield solutions that are better matched to experimental data for our three test scenarios.

#### Compliance between the globe and retrobulbar subarachnoid space, C_{rg}.

_{rg}

Because of the anatomy of the posterior globe, we reasoned that we could estimate *C*_{rg} from information about the motion of the LC, which separates the rSAS from the globe contents, as the translaminar pressure difference was changed. Park et al. (40) used optical coherence tomography imaging to document the change in position of the anterior surface of the LC resulting from surgical intervention for IOP reduction in acute primary angle closure (APAC) (Fig. 3) and primary open-angle glaucoma (POAG) patients (not shown here). Despite a similar drop in IOP for both groups, the anterior displacements of the LC for the APAC patients were more than double that of the POAG patients. This difference is likely due to tissue remodeling which had occurred in POAG patients, but was absent in APAC patients who had an IOP elevation for only a few days. Therefore, we assumed that the response of the LC in the APAC patient group was reasonably representative of a normal population. The difference between the mean presurgery and postsurgery LC positions (Fig. 3) was swept through 180° to estimate a net globe volume change due to IOP reduction. From these data, as well as the mean IOP change in the APAC group, we estimated that *C*_{rg} = 1.1 × 10^{−3} μl/mmHg. Note that this compliance is three orders of magnitude smaller than *C*_{bg}, and, thus, is relatively unimportant in determining overall globe dynamics. Hence, any modest errors in the procedure described above were considered to be unimportant in the model performance.

#### Ocular blood pressures.

There are three blood pressures that must be specified/known to complete the simulations, namely P* _{a}*, P

*, and EVP. They have a major influence on the response of the model, especially as gravity is changed. Ultimately, this eye model will be integrated with a whole body model, under separate development, that will provide a set of consistent systemic pressures after fluid redistribution due to gravity changes. In the meantime, we adopted the following approach to specify these pressures.*

_{v}We specified the inflow arterial pressure, P* _{a}*, as a function of the systemic mean arterial pressure, MAP, accounting for a hydrostatic offset between the heart and the eye, as well as for viscous blood flow losses. In more detail, we first define , which for blood pressure of 120/80, gives MAP = 93 mmHg. We then write(20)where ρ is the density of blood, G

*is the component of gravity along the body axis, Δ*

_{z}*h*is the distance between the aortic root and the eye along the body axis, and ΔP

_{a}_{,losses}accounts for any viscous flow losses between the heart and the entry point of arterial blood into the eye. Note that in the upright or sitting posture on earth, , under inversion conditions, , and under head-down tilt (HDT) at angle θ, . We take the density of blood to be , and Δ

*h*to be 36.3 cm, based on a measured distance of 28 cm from the midpoint of the eye to the sternal angle (in a 185-cm tall male) and an additional 8.3 cm offset, as suggested by the results of Seth et al. (42) Exact figures for ΔP

_{a}_{,losses}do not appear to be available, but typically viscous losses in the prearteriolar circulation are small (21), and so we choose ΔP

_{a}_{,losses}= 2 mmHg. We found that the calculated IOP is not particularly sensitive to this value and would have no material effects on simulation output.

Under 1 G (baseline earth gravity, 9.81 m/s^{2}) upright conditions, *Eq. 20* yields a value of P* _{a}* = 66.5 mmHg, for a MAP = 93 mmHg. As a check on the above formula, it is commonly assumed that arterial pressure entering the eye is 2/3 of MAP (22), which yields P

*= 66.5 mmHg. These two values are relatively close, and we conclude that*

_{a}*Eq. 20*is sufficiently accurate for present purposes.

There is little empirical data that can guide the specification of the venous pressures, particularly for the vortex veins, which form the major drainage route for the choroidal blood. For these simulations, we calculated a venous pressure, , that is analogous to that of P* _{a}*, so that(21)where CVP is central venous pressure and ΔP

_{v}_{,losses}accounts for viscous losses. Note the reversed sign on the loss term due to the opposite direction for blood flow as compared with the arterial system. An additional complication arises in the veins, especially in high-G situations: it is possible for the venous pressure as predicted by

*Eq. 21*to drop below zero. Physiologically, this would lead to venous collapse, and so the hydrostatic offset predicted by

*Eq. 21*would be invalid. On the basis of the insight that P

*is not expected to drop below CVP, we remedy this nonphysiological behavior through a slight modification to*

_{v}*Eq. 21*:

We expect smaller viscous losses on the venous side than on the arterial side, due to larger venous calibers, and, therefore, took ΔP_{v}_{,losses} = 1 mmHg. As above, our results were not particularly sensitive to this value. Typical CVP measurements are in the range 3–8 mmHg (16); we took a baseline value of 7 mmHg. For the simulations shown in this article, we assumed that . (For an alternate strategy in assigning venous pressures, see appendix c). At 1 G in the upright posture, the above equation gives values of P* _{v}* and EVP of ~7 mmHg, which agrees well with in vivo EVP measurements (48, 49).

#### Aqueous humor dynamics.

The flow of aqueous humor is influenced by a number of parameters. Aqueous inflow rate (*Q _{aq,in}*) and trabecular outflow facility (

*C*

_{tm}) both decrease with IOP; however, the rates of decrease are small [0.5–1%/mmHg and 2%/mmHg, respectively (7, 8)] and, to a certain extent, offset each other. Thus, as a first approximation, we did not allow

*Q*or

_{aq,in}*C*

_{tm}to change with IOP or blood pressure. Uveoscleral drainage rate,

*Q*, is relatively independent of IOP change (37) and, thus, was also treated as constant in the model.

_{uv}The trabecular outflow facility, *C*_{tm}, varies appreciably from one individual to the next (17). Its value was adjusted so that aqueous inflow balanced net outflow at equilibrium, with a baseline value of 0.30 μl·min^{−1}·mmHg^{−1} (Table 1). In some simulations, we wished to allow for the possibility of different baseline IOPs between individuals. To do so, we chose to keep aqueous inflow rate and uveoscleral drainage rate constant and to adjust outflow facility so that aqueous humor dynamics were at steady equilibrium under 1-G seated conditions for the specified baseline IOP. This essentially assumed that the variability in baseline IOP from one person to the next was ascribed to differences in facility.

#### Miscellaneous parameters.

We used a starting globe volume of 6500 μl, suitable for adult males (47), a baseline blood volume of 193 μl based on the approach used by Kiel (23), and a starting aqueous humor volume of 202 μl (23). Our results are insensitive to these choices. In view of the very small value of *C*_{rg} relative to *C*_{bg}, and the absence of data on P* _{csf}* variations in the various test cases considered below, P

_{csf}was not changed in the simulations. Since P

*only influences conditions in the eye through temporal gradients acting with the (extremely small)*

_{csf}*C*

_{rg}, this simplification was unlikely to materially affect the outcome.

#### Test cases.

With the above formulation, if we specify the gravitational field, G_{z}, P* _{v}*, and P

*as functions of time, we have sufficient information to compute IOP, V*

_{a}*, V*

_{aq}*, and V*

_{b}*as functions of time. This is the approach that was taken for the test cases below.*

_{g}#### Parabolic flight.

In previous work, intraocular pressure measurements were taken on standing volunteers during parabolic flight, in which ~20 s of microgravity can be generated (31). The microgravity conditions produced during parabolic flights are essentially 0 G for the problem considered here. The arterial and venous blood pressures at the eye are expected to be influenced by the changing gravitational environment and by posture. Unfortunately, there are conflicting data about what happens to blood pressure in parabolic flight, with some researchers observing distinct changes (3, 41), while other studies indicate no change in MAP (39). To address these conflicting observations, we carried out simulations for both varying and constant MAPs. In the case of time-varying MAPs, we fit the raw data from Schlegel et al. (41) to obtain profiles of gravitational environment and MAP, as shown by the gray lines in Fig. 4. In the case of constant MAP, we used the average value during microgravity for comparison purposes (97.5 mmHg).

To specify CVP, we need data on CVP during parabolic flight, such as the measurements on supine subjects described in Lawley et al. (28). We used a piecewise linear fit to the time trace for a single subject (Fig. 5) to generate the timing of the CVP waveform, but we used the reported mean values over all subjects to specify the magnitude (7 ± 5 mmHg at 1.8 g; 4 ± 3 mmHg at 0 g). Because the subjects in the Mader et al. (31) study were in the standing, rather than supine, position throughout the parabolic flight, we incorporated an offset of −1.5 mmHg to account for the difference in posture between the two studies based on the study by Foldager et al. (13). However, because the solution is governed by the rate of pressure change with respect to time, rather than absolute pressures, the offset has little effect on predicted IOP or volume changes.

#### Head-down tilt.

Xu et al. (52) carried out studies in which 65 subjects underwent 15° HDT. The authors reported IOP, systolic and diastolic brachial artery pressures at baseline (supine), and then at selected time points up to 21 min after completion of tilt. Although data were presented for cohorts with different ocular refractive status, we pooled all subjects together and computed pooled average IOPs and mean arterial pressures vs. time from Fig. 2 in the original paper of Xu et al. (52). Standard deviations were estimated from graphs from each cohort, and then a pooled standard deviation was determined. This was inexact, as no tabular data were reported for blood pressures, and the error bars were plotted in such a way that it was difficult to unequivocally determine the standard deviation at every time point for every cohort. Since the error bars were generally of the same order between cohorts and between time points, the standard deviation was estimated from other time points for those points where error bars could not be read.

The model inputs were essentially the same as for the inversion study: we specified G* _{z}* and the measured MAP as a function of time, assuming a linear ramp in both quantities over a 10-s period as the tilt occurred. The pooled mean change in MAP was very modest (order of 1-mmHg change). To determine P

_{v}and EVP, we either used the data of Linder et al. (Fig. 4 and

*Eq. 26*) (30), or specified a time-varying CVP and used

*Eq. 24*. There were no data on CVP in the report of Xu et al. (52), so we used the measurement of CVP change during 6° HDT of −1 mmHg (44). We took the starting CVP as 4.2 mmHg, suitable for individuals in the supine position at 1 G (38).

#### Postural inversion.

The third test case considered the impact of a transient inverted (“upside down”) posture. Friberg and Weinreb (14) exposed 16 normal human volunteers to such an inverted posture, during which time they made a variety of ocular measurements. In both the sitting and inverted positions, they measured arterial blood pressure at the level of the eye by noting retinal artery collapse following ophthalmodynamometry and carried out tonometry to measure IOP. Subjects remained inverted for 1–5 min. They observed a significant increase in IOP (14.1 ± 2.8 mmHg to 35.6 ± 4.0 mmHg; mean ± SD; *n* = 16) and arterial blood pressure at the level of the eye. It was stated that the pressure increased rapidly, reaching its maximum within 30–45 s after inversion and remaining elevated for up to 5 min during inversion.

A larger study of 75 people measured IOP resulting from a change of posture from upright to a headstand (4). The baseline IOP was comparable to that of the previous study (14.2 ± 0.9 mmHg). Blood pressure was not reported, so we assumed that it was comparable to that used by of Friberg and Weinreb (14).

Inversion was simulated as follows: after a brief stabilization period in the normal 1-G upright position, the gravitational acceleration, G* _{z}*, was allowed to reverse sign over a 10-s interval, simulating the inversion process. Specifically, during this 10-s window, G

*was linearly interpolated from +9.81 to −9.81 m/s*

_{z}^{2}. Similarly, MAP was changed from 84 mmHg (upright) to 74 mmHg (inverted). These MAP values were “back-calculated” using

*Eq. 20*to match the experimentally measured mean values of P

*given in the table from Friberg and Weinreb*

_{a}^{3}(14), and so this aspect of the simulation matches the experiments well, by definition.

We used *Eq. 22* to calculate P* _{v}* and EVP, using a time-varying CVP to reflect the postural change. We could not find reports of CVP change during inversion, other than the expectation that CVP would increase. Therefore, we estimated the CVP change during inversion as follows: the upright CVP was taken as the baseline value of 7 mmHg, as above. To estimate the inverted CVP, we used the change in CVP observed during parabolic flights as a guideline. Specifically, in parabolic flight, the net change in G

*between microgravity and hypergravity phases is −1.8 G, while in inversion, the net change between upright and inverted postures is −2.0 G. Assuming that the change in CVP scales linearly with the change in effective G*

_{z}*and using the parabolic flight data in which the change in CVP was −3 mmHg (i.e., a decrease), we predict a CVP increase of 3.3 mmHg during inversion. This gives an inverted CVP of 10.3 mmHg.*

_{z}## RESULTS AND DISCUSSION

#### Parabolic flight simulations.

Here, we compare the results of the simulations based on the scheme described above to experimental data. The model predicts IOP increases during the microgravity portion of the parabolic flights. The behavior of MAP during the microgravity phase of flight has a marked effect on IOP (Fig. 6*A*). When MAP is constant during microgravity, IOP is also constant during microgravity. However, when MAP decreases during microgravity (see Fig. 4), there is a corresponding decrease in IOP. In effect, this introduces an additional variable that must be considered in specifying IOP: elapsed time since entry into microgravity. The volume changes in Fig. 6*B* show that aqueous dynamics play a minor role in comparison to ocular hemodynamics over the short time scales of parabolic flight.

From results such as those shown in Fig. 6, we extracted predictions of the IOP rise during microgravity for different starting values of IOP, corresponding to the range of baseline IOPs reported by Mader et al. (31). Because it is unclear exactly when IOP was measured during the microgravity phase of the experiment, we report here a range and average value for the predicted IOP rise during microgravity. The experimental data of Mader et al. (31) show a significant spread (mean response and prediction confidence intervals in Fig. 7); nonetheless, it can be seen that the linear regression of the data and most of the data points themselves fall within the range of simulated IOP during microgravity.

#### Head-down tilt simulations.

The model predicts the initial IOP increase following tilt very well (Fig. 8). It can be observed that, over time, there was a gradual increase in predicted IOP levels, while the experimental data showed a flat or slightly decreasing IOP trend. However, all predicted values lay within the estimated standard deviations for the data, and we judged the agreement to be good. As with the parabolic flight simulations, the acute phase of the IOP response was dominated by the change in blood pressures and volumes within the eye, while the IOP response for later times was governed by changes in aqueous humor volume.

#### Postural inversion simulations.

The simulation predicted a rapid increase in IOP to 21.9 mmHg due to postural inversion (Fig. 9), followed by a more gradual increase to 40.6 mmHg after the 5-min inversion period. These values were compared against two experiments with comparable starting IOPs (47). The simulations did an adequate job of predicting the initial jump in IOP, although they somewhat overpredicted IOP at 5 min. We judge that the model compared well to experimental data, at least over the most acute time scales.

The acute IOP response was dominated by the change in blood pressures and volumes within the eye, while later times were most affected by an increase in aqueous humor volume as the aqueous was prevented from draining against the large episcleral venous back-pressure. An uncertainty in this simulation was the duration of the inversion process itself, which we chose to be 10 s. However, choosing a longer time to complete the inversion had very little effect on the peak predicted IOP; for example, increasing this time to 20 s resulted in a difference in IOP of less than 0.2 mmHg, or less than 1% of the final IOP. Therefore, we conclude that our results are insensitive to this value.

## DISCUSSION AND CONCLUSIONS

Considering the conceptual simplicity of the five-compartment model that we have presented, it does a surprisingly good job of predicting changes in IOP due to variations in the gravitational environment, especially over short time scales of the order of tens of seconds. Over longer time scales of order minutes, the model performed adequately, albeit less well than over short time scales. We hope that the simplicity of the model will be attractive to researchers interested in understanding factors influencing IOP. In particular, we note that the short time-scale predictions of the model were almost completely dependent on parameters related to blood flow, which is consistent with a time-scale analysis of the governing equations (not shown). On the other hand, the longer-term model predictions were substantially influenced by aqueous humor dynamics. It is possible that the somewhat poorer agreement between model and experiment that we observed over long time scales was due to the fact that the model does not include any regulatory changes in aqueous humor and blood dynamics.

These results give guidance about future experimental studies and theoretical developments that would be useful. In particular, over short time scales, the model predictions are sensitive to the specified EVP profiles, the parameter ω (representing the effective fraction of ocular blood at venous pressure), and, to a lesser extent, the specified MAP profiles. We would also expect the model to depend on the hydrostatic venous pressure offset between the right atrium and the eye, as venous collapse occurs. Experimental measurements of these parameters under microgravity conditions would be a challenging, but valuable, addition to the model and would be useful for better understanding the short-term IOP response of the eye to changes in gravitational environment.

More challenging is to model the longer-term response of the eye to changes in gravitational environment. As noted above, model predictions over longer time scales were more dependent on parameters related to aqueous humor drainage, e.g., trabecular outflow facility, and aqueous inflow rate. In our model, we assumed that aqueous formation rate and trabecular facility were constant within a simulation. However, ignoring their dependence on IOP introduces a potential source of error in our results, although it is expected to be small and within the uncertainties in the experimental data for the acute situation (9, 34). For purposes of better understanding chronic changes in IOP, further studies of systemic pressures, ocular blood volume, anterior chamber volume (or at least depth), aqueous formation rate, and trabecular facility are warranted, particularly for long-duration microgravity and microgravity analogs. A better understanding of the interplay between aqueous and blood volumes is essential for building regulatory models that are representative of time scales beyond a few minutes.

Interestingly, the estimated globe-to-rSAS compliance, *C*_{rg}, was far smaller than other compliances in the system, indicating that changes in retrolaminar CSF pressure are relatively unimportant in directly affecting IOP when blood dynamics are in play. Elevated P* _{csf}* can be expected to have other, possibly indirect, consequences on ocular physiology (5, 6, 10, 44). We expect it to have a large influence on the biomechanical strain in the tissues of the posterior eye, so that precise calculation of P

*due to, for example, compartment syndrome (see Ref. 32) or hydrostatic variations, will be required for our finite-element model of the eye. However, for the purposes of the lumped-parameter modeling of IOP response to gravitational changes, P*

_{csf}*variations can be largely ignored.*

_{csf}This numerical model accounts for changes to pressures, flow rates, and volumes in a set of lumped compartments. It can be used in conjunction with a spatially accurate ocular model, e.g., a finite-element model (12), to determine the biomechanical stresses in the tissues of the posterior eye. It is important to understand that this model does not yet explicitly include autoregulation; incorporation of such effects is a topic of active research lying outside the scope of this paper.

The IOP response to gravitational change is interlinked with systemic pressure changes. Because there is substantial interindividual variation in response to gravitational changes (e.g., Ref. 1), it is vital to measure systemic pressures, as well as IOP during experiments to fully characterize physiological response. This numerical model was limited by the availability of such comprehensive experimental data. At minimum, simultaneous measurement of MAP and IOP is recommended. The literature is particularly sparse with respect to venous pressures and tissue compliance. Future work will incorporate such data as it becomes available, and we will continue to validate the code against studies such as Ref. 2. In the meantime, the next step is to examine the impact of these parameters through further sensitivity studies.

## GRANTS

This work was funded by NASA’s Human Research Program through the Digital Astronaut Project and NASA grant number NNX13AP91G.

## DISCLOSURES

The authors declare that there are no conflicts of interest, financial or otherwise.

## AUTHOR CONTRIBUTIONS

E.S.N., B.C.S., and C.R.E. conceived and designed research; E.S.N., L.M., A.F., J.G.M., B.C.S., and C.R.E. analyzed data; E.S.N., L.M., A.F., J.R., J.G.M., B.C.S., and C.R.E. interpreted results of experiments; E.S.N. and C.R.E. prepared figures; E.S.N. and C.R.E. drafted manuscript; E.S.N., L.M., A.F., J.R., J.G.M., B.C.S., and C.R.E. edited and revised manuscript; E.S.N., L.M., A.F., J.R., J.G.M., B.C.S., and C.R.E. approved final version of manuscript; C.R.E. performed experiments.

## ACKNOWLEDGMENTS

We would like to thank Marcia Stegenga and Wafa Taiym for data mining and library research. We also thank Dr. Julia Raykin for examining tissue properties of the rSAS and Dr. DeVon Griffin, Dr. Beth Lewandowski, Ms. Kelly Gilkey, and Dr. Paula Dempsey for administrative support.

## APPENDIX A: TIME-MARCHING SCHEME

For convenience, we reproduce the governing equation set below:

(13)(14)(3)(4)(5)(6)To solve these equations, we define

(A1)(A2)Then a Crank-Nicolson scheme for *Eq. 13* is(A3)with and , where superscripts denote time levels and the asterisk denotes lagged quantities in which *C*_{g,in vivo}, *C*_{bg}, and *F*_{g} are evaluated using IOP* ^{n}*. Note that an Euler forward scheme is obtained when α = 0 and β = 1, and Euler backward is achieved for α = 1 and β = 0. Rearranging, we obtain

Similarly, we can time-march *Eq. 5* by defining(A5)to obtain(A6)and(A7)where Δ(P* _{i}* − IOP) refers to the change in P

*− IOP between time levels*

_{i}*n*+1 and

*n*.

Finally, we can compute globe volume (equivalent to time marching *Eq. 6*), from

##### Comment.

It is possible to define different α and β for the different equations; for example, one could use Crank-Nicolson for time-marching the IOP equation, but Euler forward for the other equations. However, we chose here to use the same scheme for all equations.

## APPENDIX B: A MODIFIED OCULAR COMPLIANCE BASED ON VASCULAR TRANSMURAL PRESSURE DIFFERENCE

In this model, the ocular compliances for the living and enucleated globe (*C*_{g}_{,in vivo} and *C*_{g}_{,enucleated}), and for the blood-to-globe compliance (*C*_{bg}) were modeled as functions of intraocular pressure (IOP) alone, as derived from the meta-analysis of Silver and Geyer (47) on saline injection in living and enucleated eyes. As noted by Eisenlohr et al. (11), the difference between the compliances of living and enucleated eyes can reasonably be attributed to ocular hemodynamics.

However, the blood-to-globe compliance, an important part of overall compliance in vivo, could reasonably be expected to depend on the transmural pressure difference rather than IOP alone. Because the majority of the contribution to the blood “compartment” in our model comes from the venous blood, we can approximate the transmural pressure as difference P* _{v}* − IOP. Here, we consider the implications of such an approach.

One issue in such a formulation is that venous pressure has typically not been reported during in vivo ocular compliance studies. We attempted to overcome this problem as follows: A reasonable reference venous pressure can be estimated as 8.2 mmHg for supine subjects, appropriate for the condition in which ocular compliance measurements have been taken. We then define a modified blood-to-globe compliance to be numerically equal to the original blood-to-globe compliance when , i.e., so that . Here, the prime superscript denotes the modified compliance formulation, which can be written as:(B1) (B2) (B3)for the in vivo globe compliance, enucleated globe compliance, and the blood-to-globe compliance, respectively. Note that is identical to that defined in the original formulation. As expected, exhibits the expected nonlinearity about (Fig. B1).

When the modified compliance model is applied to all three scenarios described in this article, the effect of the compliance formulation is almost negligible with regard to the prediction of IOP.

Readers will note that the compliance formulation has a singularity as . Therefore, we tested additional formulations for compliance, including a cut-off on its upper bound and a curve fit. The specification of the former in vivo globe compliance () is given by(B4)where is the maximum permissible value of the compliance. The second approach uses discrete data obtained from *Eq. B4* to fit a curve of the form:(B5)where *A*_{1} to *A*_{4} are fitting coefficients. No benefit (improved fit to experimental data) was seen with the use of the curve fit, with the cost of additional complexity. The solutions using the bounded compliance formulation (*Eq. B4*) showed little to no difference with respect to the original formulation over a range of limiter values = 10 to 100 μl/mmHg. We concluded that the original compliance formulation was adequate for the purposes of this numerical model. Future test cases, in which the singularity could be approached, would need to reconsider the use of the bounded compliance formulation.

## APPENDIX C: THEORETICAL VS. EMPIRICAL DEFINITION FOR EVP

The second approach to specifying P_{v} was motivated by observations in the literature of a “dissociation” between values of EVP and CVP under different postures, implying a strong autoregulatory process (see e.g., Ref. 27). Therefore, we deduced P_{v} (and EVP) directly from experimental data. Linder et al. (30) measured IOP in a cohort of human subjects in a variety of postures (upright to inverted). We fit this experimental data with a piecewise linear function (Fig. C1) and then used values of outflow facility, aqueous inflow rate, and uveoscleral outflow rate (Table 1) as input to Goldmann’s equation to deduce values of EVP at different postures from the IOP fit. This “deduced EVP data” that followed was described by(C1)where α = 22.1 mmHg for θ ≤ 0, e.g., for HDT and inversion, and α = 2.23 mmHg for θ > 0, e.g., for upright posture. We can generalize *Eq. 25* to the case where subjects are in conditions other than 1 G by replacing sin (θ) by G* _{z}*; note that, as expected, this gives identical results to

*Eq. 25*under 1 G conditions, and predicts no effect of acute changes of posture on P

*= EVP under microgravity conditions.*

_{v}Both of the above approaches required assumptions. The approach outlined in the main body of the article lumped a complex autoregulatory process into a relatively simple *Eq. 22* that assumed complete venous collapse at a threshold pressure. The alternate approach is based on direct IOP measurements in humans undergoing tilt. The outflow facility, aqueous inflow rate, and uveoscleral outflow rate are assumed to be unaffected by changes in posture; we note that such assumptions were made when considering aqueous humor dynamics. As can be seen from Fig. C1, the two approaches gave EVP predictions that were similar, lying within 2 mmHg of one another, except at large negative values of θ (near-complete inversion), where the difference increased to 5 mmHg.

We carried out simulations using both approaches; below, we refer to predictions made using *Eqs. 20* *and* *22* as “hydrostatic”, while predictions using *Eqs. C1* *and* *C2* are denoted by “empirical”. While both conditions yielded an IOP that was within experimental error (Fig. C2), the empirically derived data produced somewhat better agreement with experiment, possibly because it implicitly includes regulatory effects associated with gravitational change.

## Appendix

## Footnotes

↵1 Of course, the inflow and outflow rates can depend on compartment pressures, and vice versa. This will become evident when we specify specific forms for the right-hand sides of

*Eqs. 1*and*2*for different compartments.↵2 The value for

*K*that is usually quoted in the literature is 0.021 μl^{−1}; however, this value uses base 10 logs rather than natural logs in*Eq. 6*. To use natural logs, the value of*K*= 0.021 μl^{−1}must be multiplied by ln(10) = 2.30, yielding*K*= 0.048 μl^{−1}. This value of*K*differs slightly from the value of 0.050 μl^{−1}apparently used by Silver and Geyer (45) to create their Fig. 2; however, the difference is modest and within the spread of the data.↵3 Note that Friberg and Weinreb’s tabulated values of MAP are based on a slightly different formula than used herein, namely, P

_{diastolic}+ 0.42 (P_{systolic}− P_{diastolic}). Using their reported systolic and diastolic pressures to compute mean pressures in the upright and inverted positions with our formula, we obtain 54 mmHg (upright) and 100 mmHg (inverted).

- Copyright © 2017 the American Physiological Society