Patients With Intravitreal Gas Bubbles at Risk of High Intraocular Pressure Without Exceeding Elevation of Surgery: Theoretical Analysis

Abstract

Purpose: The purpose of this study was to show the mechanism responsible for high peak IOP in patients with intravitreal gas bubbles resulting from a descent to low elevation and a return ascent, without exceeding the surgical elevation.

Methods: A computational model reconstructed four clinical cases, using published elevations, ascent rates, and initial bubble sizes. In each case, patients first underwent surgery (790 m), then went home (790 m, 790 m, 325 m, 240 m). When returning for follow-up visits, patients descended to a low elevation (20 m, 0 m, 25 m, −310 m), then ascended to surgical elevation (790 m). The computational model output bubble size, aqueous humor volume, and IOP during the patients' travels. A parametric study was conducted to investigate the role of each modeling parameter.

Results: All four simulated cases showed increased peak IOP (34–50 mm Hg). Intraocular pressure returned to a normal value (15 mm Hg) after prolonged exposure to the surgical elevation. Over the course of the entire path, the gas bubble volume changed approximately 5%, decreasing in size during descent and then increasing during ascent.

Conclusions: In our simulations the change of bubble size outpaced the change of aqueous humor volume resulting in a 2-fold risk to patients. First, the bubble size reduction at the low elevation may increase the risk of ocular hypotony and postsurgical retinal detachment. Second, the combined increasing bubble size and accumulated aqueous humor puts patients at risk of high peak IOP after ascent even without exceeding the surgical elevation. The risks are primarily dependent on rates of elevation change and duration spent at the different elevations.

Insertion of an intravitreal gas bubble (i.e., pneumatic retinopexy) is a commonly used procedure for the repair of retinal detachment.^1,2 The procedure begins by removal of all or a portion of the vitreous humor from within the eye, which is then replaced with a gas. The gas bubble presses the detached portion of the retina back into position, expelling fluid from behind the retina. The reattached retina can then be left to heal. The gas bubble is eventually absorbed over a period of 1 to 8 weeks.³ 

The compressible intravitreal gas bubble is subject to Boyle's law, which means the gas bubble volume changes when the exterior air pressure is altered.⁴ Since the bubble is constrained in the ocular globe, its volume is also affected by the ocular globe deformation and changes in ocular fluid volume (i.e., aqueous humor, vitreous humor, and blood)⁴ (Fig. 1). The dynamic relationship results in changes to gauge IOP. It is important to distinguish between absolute and gauge pressure. Absolute pressure is the force exerted (by the gas in our case) on a surface per unit area. Gauge pressure is the difference between two absolute pressures. Clinically, the use of the term “intraocular pressure” is in reference to the gauge pressure (i.e., the pressure difference between the interior and exterior of the eye). In this work, we explicitly state whether we are speaking of absolute pressure or gauge pressure. The gauge IOP ultimately determines the stresses to which the ocular tissues are subjected and is measured clinically using methods such as tonometry. A normal gauge IOP (approximately 15 mm Hg) is essential for maintaining ocular shape and function, but an increase in gauge IOP may lead to discomfort and vision loss and is a risk factor for glaucoma.⁵ 

Figure 1

 

A schematic of the theoretical model. The change in bubble size ΔV_B was constrained by the change in globe volume ΔV_globe and the change in aqueous humor volume ΔV_Aqu. The dynamic relationship resulted in changes to gauge IOP.

It is commonplace to warn patients against large ascents, such as air travel and/or vehicular travel to higher altitudes, during their recovery time.^6–10 The same consideration, however, is not shared if the ascent does not bring the patient to a higher elevation than where the surgical procedure has taken place. It is well known that an ascent in elevation, which corresponds to a drop in exterior air pressure, will cause the gas bubble to expand.^6–10 The expanding bubble will result in higher gauge IOP values. The opposite is also true: A descent (increase in exterior air pressure) will decrease bubble size and gauge IOP values. Theoretically, if the patient were to first descend, then return to the initial elevation, the gas bubble should decrease in volume, then return to a similar volume, resulting in a relatively unchanged gauge IOP. However, an increase to peak gauge IOP was observed. 

Recent observations in four cases from a clinic in Jerusalem, Israel, initiated concerns that postsurgical travel without exceeding the surgical elevation may put patients at risk.¹¹ Each of the four patients underwent surgery in Jerusalem to insert the gas bubble. Then they went home to heal. During the recovery time, the patients returned to Jerusalem for follow-up visits. Each patient's travel included a descent and subsequent ascent that returned the patient to the elevation of surgery. Since patients returned to the same elevation at which the gas bubble was inserted, theoretically a normal IOP should have been the result. Instead, patients were rushed to the emergency room with pain and elevated gauge IOP. It was hypothesized that an accumulation of aqueous humor caused the increase in gauge IOP.¹¹ The purpose of this study was to show the mechanism responsible for high peak IOP in patients with intravitreal gas bubbles resulting from a descent to low elevation and a return ascent, without exceeding the surgical elevation. 

Methods

The model from our previous study⁶ showed increased bubble size and high peak gauge IOP values associated with ascents. Our previous model was limited to one direction, only concerned with patients ascending above surgical elevation. The simulation for this work takes into account both the descent and ascent. Our model⁶ was adapted to approximate the four published clinical cases reported from Shaare Zedek Medical Center, Jerusalem, Israel.¹¹ The model was designed to use the input of a time–elevation vector that corresponded to the travel of each patient. The path of each patient began at the patients' homes and was idealized into three linear sections: a descent from home elevation (790 m, 790 m, 325 m, 240 m) to a low elevation (20 m, 0 m, 25 m, −310 m) at a constant rate (40 m/min, 40 m/min, 40 m/min, 35 m/min), a period of time spent at the low elevation (60 min), and a return ascent to high elevation (Jerusalem 790 m) at a constant rate (40 m/min, 40 m/min, 40 m/min, 35 m/min). The patients' travel is shown in Table 1 and in the time–elevation plots of each simulated case (Fig. 2a–2d). The paths were representations of either a trip from the mountains to the coast with a return or a trip through a large valley. Both cases were plausible scenarios due to Jerusalem's close proximity to a variety of topographical regions. 

Table 1

 

Parameters Used in Simulated Patient Cases and the Predicted Peak IOPs for Each Patient

 

Figure 2

 

Changes in (a–d) the elevation, (e–h) bubble size, and (i–l) gauge IOP versus time for the four simulated patient cases¹¹ representing a descent from home to low elevation, duration spent at the low elevation, and return ascent to the elevation of surgery (patients from cases 3 and 4 lived at a lower elevation than that where surgery took place) are shown. Peak gauge IOP for each case (34 mm Hg, 36 mm Hg, 46 mm Hg, 50 mm Hg) was observed directly following ascent and is indicated with an arrow (i–l).

The values for home elevation, low elevation, high elevation, and maximum ascent rate were taken directly from the clinical data.¹¹ Initial bubble size (70%, 55%) was taken directly from the published data for patients 1 and 2 who lived in Jerusalem.¹¹ Patients 3 and 4 lived at a lower elevation than where the initial surgery took place, so it was reasonable to assume that the bubble size had changed during the trip from surgery in Jerusalem to their home elevation. Therefore, the trip from Jerusalem to home was first modeled to determine the initial bubble size for the follow-up visit. After a prolonged period of time at the home elevation, the gauge IOP stabilized and the resulting bubble size values (76%, 66%) were used as the initial conditions for the simulation of the follow-up visit for patients 3 and 4. Ascent rate and descent rate used in the model were taken as the maximum ascent rate reported clinically.¹¹ Parameter values for each case are shown in Table 1. All patients were assumed to be at normal gauge IOP of 15 mm Hg when beginning the modeled paths. 

Our model deals with the total globe volume as it applies to intraocular pressure of the entire eye. The ocular globe, as shown in the theoretical model (Fig. 1), does not differentiate between the fluid compartments in the anterior and posterior segment of the eye. Consequently, a separate mechanism for fluid connectivity between the two segments through the anterior hyaloid is not part of our model. The initial globe volume was 7211 μL, as discussed in our previous work.⁶ 

The model took into account aqueous flow changes due to the IOP control mechanisms⁶ as both inflow and outflow were pressure dependent.^12–14 The trabecular meshwork, which is the primary site of outflow, was treated as a pressure-dependent one-way valve.¹⁵ As follows, outflow decreased to zero when the IOP was equal to the episcleral venous pressure. 

Another noted modification from the previous model was the prevention of the extreme cases of ocular hypotony. Clinically, ocular hypotony is generally defined as IOP of less than 5 mm Hg.¹⁶ The most extreme cases of the ocular hypotony (i.e., negative gauge IOP) could only occur when absolute IOP is less than the absolute exterior air pressure. The ocular shell withstands a positive pressure difference but would collapse under a negative pressure difference.¹⁵ Consequently, it is not possible for extreme hypotony to be experienced in uninjured eyes. In our model, if the absolute IOP was reaching values less than absolute external pressure, the extreme hypotony was prevented by substituting the absolute external pressure for the absolute IOP, yielding a gauge IOP of zero. 

Ocular compliance has been shown to be pressure dependent and nonlinear, with ranges from approximately 1 to 4 μL/mm Hg.¹⁷ The full range of values was further examined in the parametric study as described below, but for modeling purposes, deformation of the ocular globe was approximated as linear elastic behavior as discussed previously,⁶ with a compliance of 3.115 μL/mm Hg. Outflow facility has also been shown to be pressure dependent at high IOP, but for modeling purposes a constant outflow facility of 0.25 μL/min mm Hg was used. A range of values for the outflow facility was evaluated in the parametric study. 

A complete list of input parameters and their corresponding values is presented in Tables 2 and 3. The governing equations and their derivations have been described in our previous work⁶ and are also summarized in the Appendix. The equations were solved simultaneously for absolute IOP and aqueous humor volume change using Euler's method and MatLab (The Math Works, Inc., Natick, MA, USA). 

Table 2

 

Description of Variables Used in Development of the Theoretical Model

 

Table 3

 

Constant Values Used in the Theoretical Model

 

A parametric study was also conducted to evaluate the effects of altering modeling parameters on the simulation predictions. Specifically, the descent rate, ascent rate, initial bubble size, duration at low elevation, aqueous humor production, aqueous humor outflow facility, and corneoscleral compliance were altered in the ranges of 10 to 80 m/min, 10 to 80 m/min, 0 to 100%, 0 to 240 min, 0 to 3 μL/min, 0 to 2.25 μL/min/mm Hg, and 0.5 to 4 μL/mm Hg, respectively. Case 1 in Table 1 was used as a base case for comparison. The simulation was conducted while all parameters of the base case were held constant except for the parameter being studied. The peak gauge IOP was recorded for each study. 

Results

All of the simulated patient cases resulted in high peak gauge IOP values ranging from 34 mm Hg to 50 mm Hg (Figs. 2i–l). The largest value (50 mm Hg) was calculated for case 4, which also experienced the largest overall change in elevation (traveling from 240 m to −310 m then to 790 m). Figure 2 shows the changes in the elevation, bubble size, and gauge IOP for all four simulated cases. Gauge IOP was reduced to zero during the descent but increased above the normal value of 15 mm Hg during the ascent. It peaked directly following the return to the surgical elevation (i.e., 790 m). Gauge IOP gradually returned to a value below 20 mm Hg after a period of approximately 40 min at the surgical elevation. Recovery to within 5% of the normal IOP (i.e., 15 mm Hg) took place after a period of approximately 100 min. When patients reached the low elevations, gauge IOP was reduced to values below 5 mm Hg, putting them at risk for ocular hypotony. 

The patient from case 1 traveled from 790 m to 20 m and back to 790 m (Fig. 2a). Over the course of the entire path, the gas bubble volume changed approximately 5% (initially filling 70% of the globe). The bubble decreased in size during descent and then increased during ascent, returning to its initial volume (Fig. 2e). The bubble decreased in size by a total of 344 μL and was the smallest during the period of time spent at low elevation. Comparatively, the volume of aqueous humor increased by 229 μL. In other words, the volume of aqueous humor nearly doubled before returning to the initial value of 300 μL, as shown in Figure 3. The peak in aqueous humor volume occurred during the return ascent and not during the period of time spent at low elevation. The two volumetric changes did not directly offset one another. During the ascent, there was both an increased volume of aqueous humor and an increasing bubble size. The combined volumetric change from both aqueous humor and bubble size ranged from −298 μL to 56 μL. 

Figure 3

 

Dynamic changes both in the intravitreal gas bubble volume and in the total aqueous humor volume in a typical simulated case (case 1 in Table 1) are shown. Increased fluid volume compensated for loss of bubble volume, but lagged behind. Aqueous humor volume peaked during the return ascent after the bubble had already started increasing in size.

Prevention of ocular hypotony (i.e., negative gauge IOP) resulted in the abrupt plateaus in the bubble size and gauge IOP plots (Figs. 2e–l). A comparison of the gauge IOP and bubble size plots, both with and without extreme hypotony prevention, can be seen in Figure 4. It was observed that prevention of extreme hypotony had little effect on the peak gauge IOP (Fig. 4a). The disallowing of extreme hypotony only lowered peak gauge IOP from approximately 38 mm Hg to approximately 34 mm Hg. The overall change in bubble size was not affected considerably either (from 5.5% to 5.3% change), but the transition from decreasing to increasing bubble size was smoother (Fig. 4b). 

Figure 4

 

(a) Gauge IOP versus time and (b) bubble size versus time are shown with and without extreme hypotony prevention in a typical simulated case (case 1 in Table 1). The corresponding elevations are marked on the top axes. Extreme hypotony prevention created a sharp plateau in both plots, but the overall trend remained the same. Peak gauge IOP and overall changes in bubble size were relatively unchanged.

All of the model inputs, perturbed in the parametric study, had significant impacts on the peak gauge IOP values, as shown in Figure 5. Modification of low time (i.e., the time spent at the low altitude before returning to the surgery altitude) resulted in the largest change in peak gauge IOP. Peak gauge IOP values ranged from 21 mm Hg to 46 mm Hg for low times of 0 and 240 min, respectively. There was less than a 3% change in peak gauge IOP for low times greater than 150 min, indicating that by 150 min the eye had fully adjusted to the low elevation. Alteration of ocular compliance (1–4 μL) resulted in a 15% change in peak gauge IOP. Higher values of ocular compliance resulted in decreased peak gauge IOP values. The relationship between the bubble size and peak IOP was not absolutely monotonic. In other words, for smaller initial bubble sizes, an increase in the bubble size led to an increase in the peak IOP, but such a trend was reversed for the larger bubbles. Over the ranges studied, outflow facility modification was seen to have a slightly larger impact on peak gauge IOP (from 39 mm Hg to 19 mm Hg, 50% decrease) than aqueous humor production modification (from 24 mm Hg to 44 mm Hg, 45% increase). 

Figure 5

 

The outcomes of the parametric study show changes in predicted peak gauge IOP for alterations of (a) ascent rate, (b) descent rate, (c) initial bubble size ratio α₀, (d) duration of stay at low elevation (i.e., low time), (e) aqueous production rate Q_Aqu, (f) outflow facility μ, and (g) corneoscleral compliance κ (base case shown as enlarged red star in all graphs).

Discussion

We developed a theoretical model to investigate the effects of exposure to low altitude and the subsequent rise to the initial altitude in the presence of intravitreal gas bubbles. Our model's predictions were consistent with the clinical observation, indicating that patients could be at high risk of increased IOP without ever exceeding the initial elevation of the intravitreal gas injection. Our computational model described the mechanism of the IOP rise as the following: the reduction of bubble size and decreased outflow during the descent allowed for the accumulation of the aqueous humor in the eye. The aqueous humor volume continued to increase during the time spent at low elevation. Then, during the ascent, the expansion of the bubble outpaced the ability for fluid to be evacuated. The increased fluid volume combined with the expanding bubble resulted in an increased gauge IOP. 

Gauge IOP results from a number of complex relationships: The bubble is changing size, the ocular globe is deforming, and the volume of aqueous humor is also changing simultaneously. The rate at which all of such alterations are taking place determines the gauge IOP. Aqueous humor flow and corneoscleral deformation dampen the effects of the changing bubble size and eventually fully compensate for it, but flow modification cannot keep up with the change in bubble size that is instantaneous. For this reason, there is a lag observed before the eye reaches equilibrium at any particular elevation. Time becomes a vital factor. Slower descents and longer durations at low elevation allow for the eye to fully equilibrate at the low elevation by increasing the fluid volume. For the simulated cases, such interaction resulted in an increased peak gauge IOP after the ascent. A faster ascent resulted in higher peak gauge IOP because it increased the gap between the expanding bubble and the eye's ability to compensate. 

A plausible worst-case scenario could be predicted, where either a slow descent, long duration at low elevation, or combination of the two allows time for the eye to fully adjust to the low elevation. Such a scenario is essentially equivalent to the direct ascent from the surgical elevation that is currently warned against in the clinical practices. One such modeled case predicted a peak gauge IOP value of 53 mm Hg with a slow descent rate of 10 m/min and a rapid ascent rate of 80 m/min. 

The relationships governing gauge IOP during elevation change can be modified with medications that alter aqueous humor flow. Increasing inflow on the descent would help to fill the void from a shrinking bubble and increasing outflow on an ascent would help make room for an expanding bubble. It is not practical to selectively change both inflow and outflow according to patient travel. The parametric study, however, showed that outflow should be targeted if increases in gauge IOP are the primary concern. Outflow is reduced to near zero during descents due to trabecular meshwork collapse.¹⁵ Such a phenomenon mitigates the negative effects of gauge IOP reduction to extreme hypotony that could be caused by increasing outflow during a descent. Preemptively increasing the outflow helped to lower high peak gauge IOP values that were simulated. 

Some parameters were simplified for modeling purposes, which could pose potential limitations in interpreting the outcomes of this study. Ocular blood flow was not considered in the model. Changes to blood flow are overshadowed by the much larger changes in the aqueous humor volume. Further, experimental measurements have suggested that the ocular globe responds to changes in the gauge pressure in a nonlinear manner, but in our study the ocular shell was modeled as a linear elastic system. While such assumption could pose limitations, the wide range of compliance values used in the parametric study predicted a range of peak gauge IOP values that had only 15% difference between the maximum and minimum values. Viscoelastic effects were not considered due to the long period of time over which the tissue stresses are applied. Our method of prevention of extreme ocular hypotony at low elevations may not represent exactly in vivo scenarios as the clinical measurement of IOP values for patients at the low altitude with ocular hypotony is not currently available. Although the outflow facility was a parameter that was altered in the parametric studies, a fixed outflow facility value was used to allow for numerically solving the governing equations. There is evidence that outflow facility decreases at very high gauge IOP (Karyotakis, et al. IOVS 2009;50:ARVO E-Abstract 808).^18,19 Such change in the outflow facility would increase spikes of high gauge IOP and produce a slower return to the normal value, further exacerbating the trends seen in our study. Additionally, the effects of gas diffusion have not been incorporated into the model. Gas diffusion is dependent on the molecular weight of gas inserted and has been shown to affect the size of the intraocular bubble over the course of its life span within the eye. The changes to bubble size due to gas diffusion occur on the timescale of days,^3,20 rather than the scale of hours that this model deals with. Gas diffusion will have to be taken into account in long-term modeling of eye bubble dynamics. Despite modeling simplifications, it is highly likely that the trends predicted by the simulations represent those in vivo, particularly because the peak gauge IOP values obtained from our model were consistent with the range of values in recent clinical observations.¹¹ The discrepancies in exact values can also be attributed to the simplification of the patients' paths. 

Based on our simulation, patients with intravitreal gas bubbles may be at high risk of elevated IOP after a descent and subsequent ascent, even without ever exceeding the initial elevation of surgery. Our simulation suggests that medication regulating aqueous humor flow may help manage the risks. Further, the reduction in gas bubble size at the low elevation may increase the risk of ocular hypotony and postsurgical retinal detachment. The reduction of gauge IOP levels near extreme hypotony was observed during the descents in all four simulated clinical cases. Based on our theoretical study, recent clinical warnings¹¹ to avoid any rapid changes to altitude during the recovery of patients with intravitreal gas bubbles need to be taken seriously. 

Acknowledgments

The authors thank Darryl Overby and Rishi Singh for the helpful discussions. Computational work was facilitated by a supercomputing resource grant from the Ohio Supercomputer Center (Columbus, OH, USA).²¹ The authors are also grateful for support for Lucas Gsellman provided by the Department of Biomedical Engineering at the University of Akron. 

Disclosure: L. Gsellman, None; R. Amini, None 

Appendix

The two governing equations used in the simulations were  

 

 

 

 

The above system of governing equations was solved simultaneously in MatLab (The Math Works, Inc.) using the quadratic formula and Euler's method. The description of the parameters used in these two equations and the numerical values used for the constants are listed in Table 2 and Table 3, respectively. The development of the equations is explained below. 

Change in bubble volume ΔV_B was constrained by the change to ocular globe volume ΔV_globe and change to aqueous humor volume ΔV_Aqu:  

 

or  

 

 

Change in bubble volume was also defined as  

 

with V_B being the bubble volume and V0_B being the initial bubble volume.  

At constant temperature, the volume of the gas bubble V_B was governed by Boyle's law:  

 

 

The initial bubble volume V0_B was reported as a percentage of initial globe volume V0_globe:  

 

 

Equations (5), (6), and (7) were then combined:  

 

 

Change in globe volume V0_globe was related to change in GaugeIOP by κ, the compliance constant, obtained through a linearized pressure–volume curve from a finite element model⁶:  

 

 

GaugeIOP was the pressure difference between the absolute intraocular pressure P_in and the absolute exterior air pressure P_out, so changes in GaugeIOP from the initial elevation (i.e., ΔGaugeIOP) was defined as  

 

 

Combining equations (9) and (10) gave  

 

 

Substituting ΔV_B and ΔV_globe from equations (9) and (11) into equation (4) resulted in  

 

 

The quadratic governing equation of the system (i.e. equation [1]) was obtained by multiplying equation (12) by P_in, using equation (7), simplifying, and rearranging:  

 

or  

 

 

To obtain the second governing equation, the rate of aqueous humor volume lost was defined as  

 

where the inflow rate was calculated as aqueous production rate Q_Aqu minus a pressure-dependent term for pseudofacilityλ:  

 

 

The outflow rate was calculated using the outflow facility μ, absolute episcleral venous pressure P_e, and a pressure-independent tern for uveoscleral outflow U.  

 

 

The combination of equations (14), (15), and (16) gave the second governing equation:  

 

 

The absolute exterior air pressure P_out was calculated from the barometric formula²² using the elevation h obtained from patients' paths:  

 

 

The initial exterior pressure was calculated using the initial elevation. The absolute episcleral venous pressure P_e was calculated using the assumed constant gauge episcleral venous pressure of 9 mm Hg at all elevations:  

 

 

The initial absolute intraocular pressure P0_in was calculated based on the assumed normal gauge IOP of 15 mm Hg, which was the pressure difference between absolute interior and exterior pressures: