Download Patients With Intravitreal Gas Bubbles at Risk of High Intraocular

Glaucoma
Patients With Intravitreal Gas Bubbles at Risk of High
Intraocular Pressure Without Exceeding Elevation of
Surgery: Theoretical Analysis
Lucas Gsellman and Rouzbeh Amini
Department of Biomedical Engineering, The University of Akron, Akron, Ohio, United States
Correspondence: Rouzbeh Amini,
Department of Biomedical Engineering, The University of Akron, Olson
Research Center, Room 301F, 260 S.
Forge Street, Akron, OH 44325, USA;
[email protected].
Submitted: August 20, 2015
Accepted: May 11, 2016
Citation: Gsellman L, Amini R. Patients
with intravitreal gas bubbles at risk of
high intraocular pressure without
exceeding elevation of surgery: theoretical analysis. Invest Ophthalmol Vis
Sci. 2016;57:3340–3347.
DOI:10.1167/iovs.15-18010
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 followup 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.
Keywords: glaucoma, vitrectomy, altitude, retinal detachment, ocular hypotony
nsertion 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
I
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
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
iovs.arvojournals.org j ISSN: 1552-5783
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3340
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Vitrectomy, High IOP, and No High Altitude Travel
FIGURE 1. A schematic of the theoretical model. The change in bubble
size DVB was constrained by the change in globe volume DVglobe and
the change in aqueous humor volume DVAqu. The dynamic relationship
resulted in changes to gauge IOP.
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.
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 lL, 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 pressuredependent 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
TABLE 1. Parameters Used in Simulated Patient Cases and the Predicted Peak IOPs for Each Patient
Patient
Case* †
1
2
3
4
Elevation at
Home, m
Low
Elevation, m
Surgical
Elevation, m
Descent
Rate, m/min
Ascent
Rate, m/min
Initial Gas
Bubble Size, %
Duration at Low
Elevation, min
Peak Gauge
IOP, mm Hg
790
790
325
240
20
0
25
310
790
790
790
790
40
40
40
35
40
40
40
35
70
55
76
66
60
60
60
60
34
36
46
50
Peak Gauge IOP values are included in bold as they are the primary recorded outputs of the simulations.
* Patients 1 and 2 lived at elevation of surgery, Jerusalem (790 m). Initial bubble size was modeled directly from patient data.11
† Patients 3 and 4 lived at lower elevations than surgery. The trip from surgical elevation to home elevation was first modeled to determine initial
bubble size for the return trip.
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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).
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 lL/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 lL/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 lL/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).
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 lL/min, 0 to 2.25 lL/
min/mm Hg, and 0.5 to 4 lL/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
TABLE 2. Description of Variables Used in Development of the
Theoretical Model
Symbol
a0
h
Pout
Pin
P0out
P0in
Pe
VB
Vglobe
VAqu
Description
Ratio of initial bubble volume to initial globe volume
Elevation
Absolute exterior air pressure
Absolute intraocular (interior) pressure
Initial absolute exterior air pressure
Initial absolute intraocular (interior) pressure
Absolute episcleral venous pressure
Bubble volume
Ocular globe volume
Aqueous humor volume
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TABLE 3. Constant Values Used in the Theoretical Model
Symbol
V0globe
j
l
U
QAqu
k
Description
Value
Comments
Initial globe volume
Corneoscleral shell compliance
Aqueous humor outflow facility
Uveoscleral outflow
Aqueous humor production rate
Aqueous humor pseudofacility
7211 mm3
3.115 lL/mm Hg
0.25 lL/(min mm Hg)
1.0 lL/min
2.5 lL/min
0.081 lL/(min mm Hg)
Calculated using a finite element model23
Calculated using a finite element model*
Experimental values12–14
Experimental values24
Experimental values12,14
Experimental values25
* Consistent with published data.17,23,26,27
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
lL and was the smallest during the period of time spent at low
elevation. Comparatively, the volume of aqueous humor
increased by 229 lL. In other words, the volume of aqueous
humor nearly doubled before returning to the initial value of
300 lL, 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 lL to
56 lL.
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
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.
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
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.
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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 a0, (d) duration of stay at low elevation (i.e., low time), (e) aqueous production rate QAqu, (f) outflow facility l, and (g)
corneoscleral compliance j (base case shown as enlarged red star in all graphs).
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).
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 lL) 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).
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.
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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
Quadratic:
2
ðjP0in þ jPout a0 V 0globe DVAqu ÞPin a0 V 0globe P0in
jPin
¼ 0;
ð1Þ
Differential:
d
ðDVAqu Þ ¼ lðPin Pe Þ þ U
dt
QAqu kðPin Pout P0in þ P0out Þ :
ð2Þ
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 DVB was constrained by the
change to ocular globe volume DVglobe and change to aqueous
humor volume DVAqu:
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DVB ¼ DVglobe þ DVAqu
ð3Þ
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inflow rate ¼ QAqu kðPin Pout P0in þ P0out Þ:
or
DVB DVglobe DVAqu ¼ 0
ð4Þ
The outflow rate was calculated using the outflow facility l,
absolute episcleral venous pressure Pe, and a pressureindependent tern for uveoscleral outflow U.
ð5Þ
outflow rate ¼ lðPin Pe Þ þ U:
Change in bubble volume was also defined as
DVB ¼ VB V 0B ;
ð15Þ
with VB being the bubble volume and V0B being the initial
bubble volume.
At constant temperature, the volume of the gas bubble VB
was governed by Boyle’s law:
ð16Þ
ð6Þ
The combination of equations (14), (15), and (16) gave the
second governing equation:
d
ðDVAqu Þ ¼ lðPin Pe Þ þ U
dt
ð2Þ
QAqu kðPin Pout P0in þ P0out Þ
The initial bubble volume V0B was reported as a percentage
of initial globe volume V0globe:
The absolute exterior air pressure Pout was calculated from
the barometric formula22 using the elevation h obtained from
patients’ paths:
V B ¼ V 0B
P0in
:
Pin
V 0B ¼ a0 V 0globe :
ð7Þ
Equations (5), (6), and (7) were then combined:
DVB ¼ a0 V 0globe
P0in
a0 V 0globe :
Pin
ð8Þ
Change in globe volume V0globe was related to change in
GaugeIOP by j, the compliance constant, obtained through a
linearized pressure–volume curve from a finite element
model6:
DVglobe ¼ jðDGaugeIOPÞ:
ð9Þ
GaugeIOP was the pressure difference between the
absolute intraocular pressure Pin and the absolute exterior air
pressure Pout, so changes in GaugeIOP from the initial
elevation (i.e., DGaugeIOP) was defined as
DGaugeIOP ¼ ðPin Pout Þ ðP0in P0out Þ:
ð11Þ
P0in
V 0B jðPin P0in Pout þ P0out Þ DVAqu ¼ 0:
Pin
ð12Þ
The quadratic governing equation of the system (i.e.
equation [1]) was obtained by multiplying equation (12) by
Pin, using equation (7), simplifying, and rearranging:
V 0B P0in V 0B Pin jPin ðPin P0in Pout þ P0out Þ
DVAqu Pin ¼ 0;
ð13Þ
or
2
jPin
ðjP0in þ jPout a0 V 0globe DVAqu ÞPin a0 V 0globe P0in
¼ 0:
ð1Þ
To obtain the second governing equation, the rate of
aqueous humor volume lost was defined as
d
ðVAqu Þ ¼ outflow rate inflow rate;
dt
The initial exterior pressure was calculated using the initial
elevation. The absolute episcleral venous pressure Pe was
calculated using the assumed constant gauge episcleral venous
pressure of 9 mm Hg at all elevations:
Pe ¼ Pout þ 9 mm Hg:
ð18Þ
The initial absolute intraocular pressure P0in was calculated
based on the assumed normal gauge IOP of 15 mm Hg, which
was the pressure difference between absolute interior and
exterior pressures:
P0in ¼ P0out þ 15 mm Hg:
ð19Þ
References
Substituting DVB and DVglobe from equations (9) and (11)
into equation (4) resulted in
V 0B
ð17Þ
ð10Þ
Combining equations (9) and (10) gave
DVglobe ¼ jðPin Pout P0in þ P0out Þ:
Pout ¼ 0:1013ð1 2:26 3 105 hÞ5:26 MPa
¼ 759:8ð1 2:26 3 105 hÞ5:26 mm Hg:
ð14Þ
where the inflow rate was calculated as aqueous production
rate QAqu minus a pressure-dependent term for pseudofacilityk:
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