Maintenance of ongoing mammalian physiology demands adequate oxygen supply. The human cornea is known to receive its oxygen supply, under normal conditions, anteriorly through the tears (from the atmosphere when the eye is open and from the tarsal conjunctiva when the eye is closed, e.g. during sleep) and posteriorly from the eye’s anterior chamber.1,2
Smelser and Ozanics1 were the first to demonstrate that contact lenses, then large scleral lenses made of oxygen impermeable glass, would interfere with the cornea’s anterior oxygen supply leading to a number of complications (well defined over the past half century) including acute stromal swelling (and secondary disruption of optical clarity), chronic stromal thinning, epithelial microcysts and fragility, loss of corneal sensitivity, and eventual corneal vascularization.3–7
Contact lens designs have evolved from large scleral contact lenses to both rigid contact lenses smaller in overall diameter than the cornea and flexible (soft) contact lenses, larger than the cornea but usually smaller than most scleral lenses. Contact lens materials evolved over the same time period from oxygen impermeable glass and polymethylmethacrylate to various oxygen permeable plastics, both rigid and flexible, to address the oxygen supply restriction discussed above.
The term “Dk” is used to define oxygen permeability in the contact lens literature, whether considering a cornea or a gas permeable plastic contact lens; D is the oxygen diffusion coefficient (cm2 s−1) and k represents the Henry’s law oxygen solubility of a given material (cm3 O2/cm3 mmHg).8 Dk, however, incompletely describes the oxygen performance of contact lenses, which have various thickness values (t, cm). Oxygen transmissibility (Dk/t) has therefore been used to describe the oxygen performance of a contact lens.8,9 , a Variability in central contact lens thickness from one lens design to another is compounded, however, by considering that most optically powered lenses are individually not uniform in thickness (it is only reasonable to consider low minus powered lenses—approximately −0.75 DS—as uniformly thick11).
Uniformity in t is certainly not the case in modern scleral gas permeable lenses of complex designs. (While several studies suggested various ways to derive clinical average thickness values for optically powered contact lenses, it should be remembered that Fatt and Neumann12 proposed a conservative approach; they suggested that the site of most oxygen resistance (greatest t, or lowest Dk/t) is the parameter of concern when considering hypoxic stress to the cornea).
Our quantitative understanding of how much oxygen is required to maintain normal corneal metabolism has similarly evolved over the years, with various proposed metrics: corneal oxygen consumption (termed Q, ml O2 cm−3 s−1), flux (termed j, μl O2 cm−2 h−1), or tear layer oxygen partial pressure (pO2, mmHg), which maintains adequate j and Q.
As direct measurement of oxygen under in vivo contact lenses is technically quite challenging,13 several attempts have been made over the years to model the physical situation of contact lenses on eyes and through these models predict the likely oxygen metrics available at the anterior corneal surface: j, Q, and/or tear pO2.9,14–19
These models all depend on boundary conditions (e.g. oxygen partial pressures or tensions in the atmosphere, the anterior chamber, and in the tarsal conjunctivae) and the above defined physical intrinsic parameters of the cornea, contact lenses, and tears: oxygen consumptions (Q), oxygen diffusion coefficients (D), solubilities (k), and thicknesses (t), both of whole cornea and/or the various layers encountered.20
As our knowledge of the cornea has evolved over the decades, our ability to provide improved values for boundary conditions and improved values for Q, D, k, and t for use in these models, and the models themselves, continue to improve. Hence, it is valuable to revisit new models over time. The last 15 years have seen the publication of several serious efforts to develop improved quantification of Q for both whole cornea and the various layers.15,16,18,19,21,22 Compañ et al.,19 in particular, advanced the Monad equation model of Chhabra et al.,18 which develops a variable Q that disallows negative oxygen tensions in the corneal stroma (an acknowledged flaw of previous constant Q models).
Scleral contact lenses, now made of gas permeable plastics, have made a resurgence, being particularly useful in the treatment of eye diseases such as advanced keratoconus and severe dry eye. Modern scleral gas permeable lenses are fitted to vault the cornea, creating a layer of tears underneath. Tear exchange is believed reduced or disallowed with scleral contact lenses23 and therefore any oxygen reaching the anterior corneal surface must diffuse through two layers (similar to some hybrid and piggyback lens systems). This creates a situation of “resistance-in-series.”24 Most research in this area, not surprisingly, suggests that oxygen supply may be somewhat more questionable with contact lenses that create resistance-in-series than that found with solo modern high Dk rigid or soft contact lenses.17,19,25,26
The main goal of this current work is to use an updated paradigm to provide theoretical guidelines to assist the clinician to improve his or her patients’ corneal oxygen supply. Clinicians cannot change the patient’s corneal thickness or physiology but can practically modify only one or more of three parameters: lens Dk, lens thickness, or tear layer thickness (called “vault” with scleral lenses). Lens thickness is the least clinically usable of these three parameters, as both required optical power and the necessity to minimize on-eye lens flexure limit clinical manipulation of this parameter.
This study updates the model of Jaynes et al.26 who modeled scleral gas permeable contact lens-tear systems with a constant cornea Q; we used a Monad equation model with a variable Q and two maximum corneal oxygen consumption rates from the previous studies. As two groups have used differing Dk values for the whole cornea, we will also consider both here: Dkc = 8619 and Dkc = 24.79 Fatt Dk units, respectively, to gain perspective on the difference each will make in modeling pO2 at the corneal surface. We use updated corneal and tear vault thickness values. We also use a different, and we believe more reasonable, value for the Dk of tears (93 Fatt Dk units) rather than several previous values (80,9,20 86,27 or 9919 Fatt Dk units).
In summary, the model presented here predicts both oxygen tension and flux profiles through the contact lens, tears, and cornea system and is updated from previous studies.
Several models estimating the in vivo corneal oxygen supply associated with contact lens wear are in the literature. Each model has advantages and disadvantages.
Both Holden and Mertz,5 and later Harvitt and Bonanno,15 theorized that adverse corneal physiological changes would occur below a “critical” lens Dk/t value. Fatt, however, stated that Dk/t is an in vitro measurement and therefore is a “disappointment” as a metric of in vivo performance.28
Historically, in vivo oxygen flux (j) was considered a potential alternative useful metric, but Brennan showed that j changes minimally when considering contemporary higher Dk lenses (≥50 Fatt Dk/t units).16 Brennan also proposed a percent of normal corneal oxygen consumption (%Qc) as a potential metric, but %Qc has the same problem as j and has gained minimal acceptance.
Polse and Mandell proposed a “critical” tear layer oxygen tension value (COT) at the anterior corneal surface below which corneal physiology would be compromised and corneal swelling induced.3 Presumably, this is the tear layer oxygen tension below which both j and Q suffer. From their somewhat limited data, Polse and Mandell suggested a COT of 11–19 mmHg,3 but later researchers estimated COT at 70 to 100 mmHg or greater.29,30 In our opinion, tear pO2 is as good a metric as any proposed and will be used in this analysis. Weissman and Ye17 suggested that a reasonable COT would be a tear pO2 of 100 mmHg from a review of the previous literature wherein COT ranges from the oxygen tension found in palpebral conjunctiva during closed eye conditions (~50–60 mmHg) to 120 mmHg or greater.30
Weissman and Ye17 adapted Huang et al.’s31 interpretation of the Fatt model (but utilized most of Brennan’s16 boundary condition modifications), used tear pO2 as their outcome metric, and considered the situation of piggyback contact lenses as a resistance-in-series24,32 to oxygen diffusion. This same model was later applied to scleral gas permeable lenses by Jaynes et al.26
Michaud et al. calculated scleral gas permeable lens oxygen resistance-in-series Dk/t at both the lens center and periphery.25 Michaud et al.25 used the Holden and Mertz5 criterion (Dk/t of 24 Fatt units) for the central cornea and the Harvitt and Bonanno15 criterion (Dk/t of 35 Fatt units) for the peripheral cornea. As noted above, however, we believe that the metric used in this study (contact lens oxygen transmissibility or Dk/t) may not be the best possible metric to evaluate scleral gas permeable lens in vivo oxygen effectivity and that tear pO2 at the corneal surface could be a better metric.
In summary, our current model, discussed in more detail below, uses a variable value of Q developed through Monod equations (that disallows negative oxygen tensions within the stroma associated with the constant Q models) similar to the work of Chhabra et al.33 We also use a tear layer Dk of 93 Fatt Dk units. And, as mentioned above, as whole corneal Dk has been previously proposed at both 86 and 24.7 Fatt units, we will below consider both.
Although each layer of the cornea has individual values for oxygen consumption (due to the characteristic metabolism and cells in each living layer: epithelium, stroma, and endothelium), we here consider the cornea as a single layer and we assume that Qc only depends on oxygen tension; therefore, the complete cornea has the value of Qc,max. This value was calculated from the in vivo human data of Bonanno et al.13 regarding measurements of oxygen tension in the post-lens tear film as a function of time. Chhabra et al.18 developed a maximum corneal oxygen consumption rate Qc,max and then calculated the spatial-averaged in vivo human maximum corneal oxygen consumption rate of 1.05 × 10−4 ml·cm−3·s−1. This value corresponds to the average of the obtained values (Qc ,max(ave)), and it is practically the same as the value found by del Castillo et al.34 (On the other hand, this new value is 2.34 times higher than the one given by Brennan16: Qc ,max = 4.48 × 10−5 ml·cm−3 s−1, and it is 1.8 times higher than that found by Larrea et al.22: Qc,max = 5.75 × 10−5 ml·cm−3·s−1.)
To obtain oxygen tension and flux profiles through the total system cornea–post-lens tear film–scleral gas permeable lens, we used the above Chhabra et al.18 value of Qc ,max (1.05 × 10−4 ml·cm−3·s−1 and K = 2.2 mmHg for the Monod kinetics value). For comparison, however, we will also consider a Qc ,max = 5.0 × 10−5 ml O2 cm−3 s−1 (equivalent to 9.7 ml O2 cm−2 h−1 used previously by Jaynes et al.26 after the work of both Freeman35 and Weissman36).
When trying to solve these analyses, there are some difficulties in selecting the “best” boundary conditions to use, such as corneal and tear permeabilities or the partial pressure of oxygen at the different interfaces. During the last decades, there has been agreement in the partial pressure of oxygen at sea level for the open eye (155 mmHg). Other boundary conditions, however, have been re-evaluated and updated, such as the partial pressure of oxygen at the posterior (or endothelial) corneal surface. This was first considered to be 55 mmHg,14 but a more recent value is 24 mmHg.13,16 We also used the palpebral conjunctiva pO2 value of 60 mmHg15,18 for closed eye conditions rather than the previously used value of 55 mmHg.14
Thicknesses of endothelium, epithelium, and stroma have been taken at different values by different authors. We here used values of 2, 58, and 480 μm for endothelium, epithelium, and stroma thickness, respectively, from the average central cornea thickness of 540 μm provided by the meta-analysis of Doughty and Zaman37 also bearing in mind other data.38
Tear film oxygen permeability should be identical to water permeability, taking into account the values of the oxygen diffusion and solubility coefficients in water solution at 25°C of D = 3 × 10−5 cm2 s−1 and k = 3.1 × 10−6 cm3 O2/cm3 mmHg, respectively; a tear permeability of 93 Fatt units27,39 is therefore determined. Also, noting that the cornea is 78% water, we feel it is reasonable to use a value for oxygen permeability through the cornea tissue of Dkc = 86 Fatt Dk units (the product between the oxygen diffusivity D = 2.8 × 10−5 cm2 s−1 and k = 3.05 × 10−6 cm3 O2/cm3 mmHg measured for oxygen in water at 35°C). Because several groups have directly measured Dkc, however, we will also use a value of 24.7 Fatt Dk units and compare results of both values.
Importantly, it should also be noted, consistent with current research,40,41 that neither tear exchange, lateral diffusion, nor tear mixing under scleral gas permeable lenses were considered in our following analysis.
All the parameters considered in the calculations of oxygen tension at the interface cornea–post-lens scleral gas permeable tear film, and both oxygen tension and flux profiles in the cornea, discussed below, are given in Table 1.
We used the same technical procedures from previous studies19,34 employing FiPy 3.1, a finite volume PDE solver written in Python. We used a spatial grid with 103 points in all computations and time steps of 10−1 s for the time-dependent equations. All the computations were performed by a personal computer with an Intel Core i7-3770K under Debian Linux.
Our model is a 1D construct, which assumes that there are three layers (cornea, post-lens tear film layer, and scleral gas permeable lens) between the atmospheric air and the aqueous humor; resistance to oxygen flux in the very thin pre-lens tear film on the front surface of contact lenses is considered negligible (in comparison with the other three resistances). Only mono-dimensional oxygen flux is considered; diffusion parallel to the cornea is neglected, as it has been previously, because the cornea is very thin compared with its width.9
The non–steady-state diffusion equation that describes oxygen tension as a function of time and position for the cornea supposing 1D model is given:
where, pc is the partial pressure of oxygen in the cornea. As discussed above, Dc is the diffusion coefficient of oxygen in the corneal tissue, kc is the oxygen solubility coefficient, and x is the distance perpendicular to the surface (cm); Qc is the oxygen consumption rate into the cornea (ml O2/cm3 s) and here t is time (s). For the steady-state conditions, Eq. (1) becomes:
At steady-state conditions, the following expression holds at the tear film and gas permeable scleral lens, respectively:
where xc, x tear, and x lens are the thicknesses of the cornea, tear film, and lens, respectively.
The solutions of Eq. (2) for the cornea are functions of Qc(pc), which consider that corneal oxygen consumption is a function of oxygen partial pressure into the cornea as a consequence of the aerobic metabolism.18,20,42
It is clear that aerobic metabolism does not occur at zero oxygen tension and therefore Q is zero at 0 pO2. At high oxygen pressures, the reaction is limited by the equilibrium concentration of activated complexes formed by reactions between oxygen and the enzymes that act as catalysts; the reaction is then saturated and Q is independent of the oxygen partial pressure. In these cases, aerobic metabolism is quantified by Monod kinetics model (also known as Michaelis Menten Model21,43) and Q is related with oxygen tension by the expression:
Here, K is the Monod dissociation equilibrium constant, which represents the oxygen tension when corneal aerobic metabolism reaches maximum Q, that is, the oxygen pressure required for the cornea to be in equilibrium where the reaction is saturated and the system starts to have consumption of oxygen independent of partial pressure; Qc, max is the maximum corneal consumption of oxygen and pc is the partial pressure of oxygen at the cornea-tear interface. As noted above, from Bonanno et al.’s13 in vivo estimation of oxygen consumption Qc, Chhabra et al.18 obtained, using a Monod kinetics constant with K = 2.2 mmHg, Qc,max = 1.05 × 10−4 ml O2 cm−3 s−1. The inclusion of this Nonlinear Monod oxygen consumption for the cornea, described by Eq. (5), avoids aphysical oxygen partial pressures in the cornea (such can happen when models use a constant Q). In our study, the solution of Eq. (2), taking into account Eq. (5), was obtained following our previously described procedure.19,34
A more detailed description of our procedure is given in the appendix of Compañ et al.19 and discussed in Appendix A,44 available at http://links.lww.com/OPX/A261. Table 1 shows the different values of parameters used in the numerical solution of Eqs. (2) to (4) taking into account Eq. (5).
Several following figures plot results using our model. Oxygen tension isolated to the cornea–post-lens tear film interface (that is, at the anterior surface of the cornea just in contact with the tears, or x = xc) versus tear vault thickness for several specific scleral gas permeable lens Dk and thickness (L here) values is shown in Fig. 1. The reader should note that two of three scleral gas permeable lens situations in our Fig. 1 can be directly compared to figure 1 of Jaynes et al.26
If a COT of 100 mmHg is indeed a reasonable criteria, similar to the results of Jaynes et al.26 and Michaud et al.,25 our results again suggest that scleral gas permeable lenses for most tear film thickness combinations considered (except for perhaps the thinnest of tear vaults) are unlikely to provide a pO2 ≥100 mmHg, e.g. enough oxygen to fully avoid hypoxic effects. Electing a lower COT (e.g. 60 mmHg, such as provided by the inside of the lids during sleep), or a higher value such as 125 mmHg, would, of course, lead to a different prediction of oxygen-related contact lens physiological tolerability.
Oxygen tension (pO2) and flux (j) profiles across the system (cornea–tears–scleral gas permeable lens) for various tear vaults are shown in Figs. 2 to 5.
The simulated scleral gas permeable lens of Figs. 2 and 3 has a Dk of 140 Fatt Dk units and is 250 μm thick. The left-hand panels of Fig. 1 present oxygen tension profiles through the cornea, tear vault, and lens, with different profiles for tear vault thickness varying from 50 to 300 μm. (Similar results could be obtained for other combinations of lenses with different oxygen transmissibilities and ranges of tear thicknesses.) In the top panels, corneal Dkc = 24.7 whereas in the bottom panels, Dkc = 86 Fatt units. Q is a variable function of oxygen tension, obtained following the Monod kinetics model with a maximum corneal oxygen consumption rate Qc,max of 1.05 × 10−4 ml O2 cm−3 s−1 in Fig. 2.
From the calculated oxygen tensions (pO2) through cornea– tears–lens, we also calculated oxygen flux (j) using a combination of Eqs. (2), (5), and Fick’s law of diffusion across each layer. The right hand panels in Figs. 2 and 3 show oxygen flux profiles for the same scleral gas permeable lens–tear vault systems shown in the left panels. Because there can be no oxygen partial pressure discontinuity at the interfaces between layers, through the boundary layers: cornea–tears, tears–lens, and lens–atmosphere, the continuity between fluxes must be satisfied, such as has been observed in all figures.
Fig. 3 displays oxygen tension and flux profiles with maximum oxygen consumption Qc ,max of 5 × 10−5 ml O2 cm−3 s−1.
As can be seen in Figs. 2 and 3, distribution of oxygen tension and flux through the cornea, as well as pO2 at the corneal surface, are all functions of the thickness of the tear layer trapped between the cornea and the scleral lens.
Similar results could be shown for other scenarios (other gas permeable scleral lens oxygen permeabilities, lens thickness, and tear layer thickness). (All results for all the systems cornea– tears–gas permeable scleral lens analyzed here can be seen in Tables B1 to 4 in Appendix B, available at http://links.lww.com/OPX/A262.)
Figs. 4 and 5 similarly show results obtained from our simulations for oxygen tension (left panels) and flux (right panels) profiles where the maximum corneal oxygen consumption rate, Qc ,max, is 1.05 × 10−4 ml O2 cm−3 s−1 (Fig. 4) and 5 × 10−5 ml O2 cm−3 s−1 (Fig. 5), respectively, for scleral gas permeable lenses with tear vaults held constant at 250 μm. In Figs. 4 and 5, however, lens Dk is allowed to vary (three current clinically used values of 100, 140, and 170 Fatt units). Again, corneal oxygen permeability (Dkc) is 24.7 Fatt units in the top panels and 86 Fatt units in the bottom panels.
Comparing Figs. 2 and 3, where tear vault thickness is allowed to vary between clinically viable values (50–300 μm), to Figs. 4 and 5, where lens Dk is allowed to vary between clinically viable values (100, 140, and 170 Fatt Dk units), it is clear that both oxygen tension and oxygen flux profiles are far more clinically sensitive to changes in tear vault thicknesses than changes in these lens Dks. In Fig. 2, with Qc ,max of 1.05 × 10−4 ml O2 cm−3 s−1, and changes in tear vault thickness of 50 to 300 μm, anterior corneal surface tear pO2 ranges 36 to 38 mmHg (if Dkc is 24.7 Fatt units, pO2 ranges from 67 to 103 mmHg, and if Dkc is 86 Fatt units, pO2 ranges from 38 to 76 mmHg). Similarly, in Fig. 3 with Qc ,max of 5 × 10−5 ml O2 cm−3 s−1, and changes in tear vault thickness of 50 to 300 μm, anterior corneal surface tear pO2 ranges 31 to 36 mmHg (if Dkc is 24.7 Fatt units, pO2 ranges from 67 to 103 mmHg; and if Dkc is 86 Fatt units, pO2 ranges from 62 to 98 mmHg). On the other hand, in Fig. 4 with lens Dk ranging from 100 to 170 Fatt units, anterior corneal surface pO2 only ranges 10 to 12 mmHg (if Dkc is 24.7 Fatt units, pO2 is 65, 73, and 77 mmHg, respectively; and if Dkc is 86 Fatt units, pO2 is 36, 43, and 46 mmHg, respectively). Similarly, in Fig. 5, with lens Dk ranging from 100 to 170 Fatt units, anterior corneal surface pO2 only ranges 11 mmHg (if Dkc is 24.7 Fatt units, pO2 is 84, 91, and 95 mmHg, respectively; and if Dkc is 86 Fatt units, pO2 is 60, 67, and 71 mmHg, respectively).
First of all, despite much improved mathematics (although maintaining a single chamber corneal model), anterior corneal oxygen tension values during gas permeable scleral lens open eye wear predicted by this enhanced analysis differ little qualitatively (i.e. clinically) from that found by the simpler analysis of Jaynes et al.26 If one accepts a COT of 100 mmHg, the majority of current scleral gas permeable lenses should produce some levels of corneal hypoxia under open eye conditions regardless of use of enhanced paradigms for corneal oxygen consumption modeling, or corneal oxygen permeability, or updated boundary conditions.
Our results, as displayed in Fig. 1, suggest the same conclusion as that of both Michaud et al.25 and Jaynes et al.,26 namely that clinicians would be prudent to prescribe scleral gas permeable lenses manufactured from the highest oxygen permeable lens plastics and to fit with minimal reasonable corneal clearance.
It is interesting to compare our results with the recent data of Giasson et al.45 Equivalent oxygen percentages (EOP) were generated by gas permeable scleral lens wear on eight human corneas; a Dk 140 material lens at a vault of 200 to 240 μm produced an EOP of 9% (about 70 mmHg) (Giasson et al. do not provide lens thickness data). Our Fig. 1 suggests our calculation predicts an anterior corneal surface pO2 between 65 and 85 mmHg for a Dk of 140 Fatt units, 350-μm-thick gas permeable scleral lens with a 225-μm tear vault.
Additionally, and importantly, we find that both anterior corneal surface oxygen tension and flux are each more sensitive to modification in tear vault (Figs. 2 and 3) than they are to changes in lens material Dk (seen in Figs. 4 and 5), within the ranges of current clinical manipulation.
This observation suggests that clinicians can exercise more control over anterior corneal oxygen supply by varying tear vault thickness than by changing scleral gas permeable contact lens Dk, using the values for each of these clinically available at this time.
The question of tear exchange of course continues to lead to concerns however, as there are groups who believe that scleral contact lenses (and hybrid contact lenses as well) preclude any effective tear exchange whereas other groups believe tear exchange occurs with the wear of such lenses. As noted above, this study did not consider any tear exchange. Hopefully, upcoming research will provide data supporting one or the other of these arguments—and if there is tear exchange, allow some quantization of same. Any significant tear exchange, lateral diffusion, or tear mixing during scleral gas permeable lens wear could substantially affect the values we report.
In conclusion, we sincerely hope our study above will assist clinicians engaged in the care of scleral gas permeable contact lens wearing patients.
Barry A. Weissman
Southern California College of Optometry
at Marshall B. Ketchum University
2575 Yorba Linda Blvd
Fullerton, CA 92831
The authors declare they have not received any extramural funding for this project. The authors have no conflicts of interest to declare.
Received March 15, 2016; accepted June 6, 2016.
Appendix A, a more detailed description of our procedure, is available at http://links.lww.com/OPX/A261.
Appendix B, four tables showing oxygen tension (mmHg) calculated by the method described, at the cornea–tears interface [column 3 p(x = xc)] (i.e. at the anterior corneal surface), at the tears–lens interface [column 4 p(x = xc + xtears)] (i.e. at back surface of the lens), and at the position of minimum oxygen tension in the stroma [column 6 (p(x = xmin)], found for all the systems cornea–tears–gas permeable scleral lens analyzed in this study, developed from the values of the parameters in Table 1. Column 1 provides the lens thicknesses, column 2 the tear vault thicknesses, and column 5 the position of minimal oxygen tension in the cornea, available at http://links.lww.com/OPX/A262.
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aTo help simplify the units of oxygen permeability or Dk ([cm2 s−1] [ml O2/ml mmHg]) and oxygen transmissibility or Dk/t ([cm s−1] [ml O2/ml mmHg]), “Fatt Dk units” and “Fatt Dk/t units” have been respectively proposed11 and will be used in our manuscript, where 1 Fatt Dk unit is equivalent to 1 barrer = 10−11 (cm2 s−1) (ml O2[STP]/ml mmHg).
scleral lenses; oxygen tension; oxygen consumption rate; tear film; corneal hypoxia
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