A Theoretical Model for Predicting Interfacial Relationships of Retinal Tamponades

Abstract

Purpose.: To establish a theoretical model to determine the relationship between retinal coverage and tamponade shape in relation to tamponade volume, for a variety of tamponades, and to test these relationships with a physical analogue of the human eye.

Methods.: The theoretical model was based on a static balance between buoyancy forces and surface tension forces, for an axisymmetrically shaped bubble or droplet. In the laboratory experiments, two hemispheres were cut into an acrylic block. The acrylic was soaked with bovine serum for 10 minutes to ensure that the wetting properties were similar to the human retina. Photographic images of various fractions of lighter-than-water (gas, silicone) and heavier-than-water (Oxane HD) tamponades were analyzed by using algorithms written in commercial image-processing software and compared with the theoretical predictions and published data.

Results.: The theoretical predictions of tamponade shape and retinal coverage agree closely with the results obtained from the analogue experiments.

Conclusions.: The theoretical model was validated against measurements in a human eye analogue and published data. The three key parameters that characterize the retinal coverage of any given tamponade are the bond number, the contact angle of the tamponade, and the volume used. The model may be used to predict the static properties of new tamponades without in vivo tests.

The use of various gases (SF6, C3F8) and silicone oils to provide tamponade in the treatment of retinal detachment is now common practice. ^1,2 The conventional rationale for using tamponading agents in the repair of retinal detachment is based on the concept that the gas/liquid bubble blocks the flow of fluid through the retinal breaks. ^3,4 The blockade allows retinal reattachment to be temporarily maintained while a permanent chorioretinal adhesion develops in response to laser photocoagulation or cryotherapy. In keeping with this paradigm, the contact angle of the bubble is thought to be of importance as an indicator of the relative area of contact that the bubble has with the retina. Thus, air and gas bubbles, which have a higher contact angle (38.8°), have a greater surface area of contact with the retina (the bubble tends to have a flat-bottomed shape) compared with silicone oil, which forms a more spherical bubble, has a lesser contact angle (16°), and has relatively less surface area in contact with the retina. ⁵ Silicone oil provides less contact area with the retina than does gas, but it remains in contact with the retina until removed. In contrast, gases dissolve over a period of several weeks. More recently, heavier-than-water tamponades (specific gravity greater than water tamponades; Oxane HD) have been introduced for tamponade of the inferior retina. ⁶  

Published data of the behavior of retinal tamponades have been based on measurements in physical analogues of the eye. ^5,7 Thus far, there have been no theoretical models to predict the behavior of tamponades within the eye. In this article, we introduce a new mathematical model that describes the interfacial shape of gas and liquid tamponades in triphasic contact. The model was compared against a series of experiments in which a physical analogue of the human eye was used, with various fractions of fill with lighter-than-water (specific gravity less than water; silicon oil, air) and denser-than-water (Oxane HD) tamponades. We also compared our theoretical projections against published data for gas and silicon oil. ⁵ We focused specifically on low fractional fills where the rate of change of angular coverage with volume injected is larger. 

Methods

Mathematical Model

The mathematical description of the shape of the tamponade interface was based on the assumption that the interface is axisymmetric around the vertical axis. The contact angle between the three-phase contact line (retina, aqueous, and tamponade) is denoted by θ, the density contrast between the tamponade and water by Δρ, and the eye radius by a. The Bond (Bo) number, as indicated in Table 1, characterizes the relative strength of buoyancy force to surface tension and is defined by  

 

The interface between tamponade and aqueous is defined in cylindrical polar coordinates by f = r − R_z (Fig. 1a). The unit vector normal to the tamponade interface is  

 

where the subscript z refers to differentiation with respect to z. The static balance is between the pressure difference due to surface tension and the hydrostatic force. Defining the position of the minimum height of the interface as z = z_h (Fig. 1) we obtain:  

 

which physically describes the balance between surface tension forces and buoyancy forces. The detailed mathematical analysis and numerical approach to solving equation 3 is described in the 1. 

Table 1.

 

Physical Properties of Commercially Available Tamponades

	Density, ρ	Surface Tension, σ (dyne/cm)	Viscosity, μ (Poise)	Contact Angle, θ (Degrees)	Bond Number, Bo
Gas/water	0	70	0	30.74 ± 4.24	14.27
Oxane 1300/water	0.98	44	1300	16.17 ± 1.23	0.6
Oxane HD/water	1.02	40	3300	20 ± 1.88	0.5

 Density and surface tension properties of Oxane are taken from data sheets (http://www.spiritmed.cz/Data/files/Katalogy/VR-PFCLSILOIL05.pdf). The contact angles for gas, Oxane 1300 and Oxane HD were obtained by averaging over 6, 5, and 6 measurements, respectively.

Figure 1.

 

(a) A (light) tamponade and water-filled sphere with the notation used in the analysis. The interface is assumed to be axisymmetric about the z-axis. The notation is the same as a dense tamponade. (b) Side image of an air bubble within an acrylic sphere that has been coated with a protein. The position of the air interface is indicated. In addition, the position of the internal surface of the sphere (below the air interface) and above the air interface is indicated.

Laboratory Study

The purpose of the laboratory study was to measure the properties of the tamponades and to examine their shape within a spherical container filled with water. To analyze the shape of tamponades injected into the eye, we undertook a series of experiments with an acrylic hollow sphere. The sphere was machined from two acrylic blocks and had an internal diameter of 21 mm, typical of the human eye. The internal surfaces were then hand polished. To ensure that the contact angle was close to that of a retina, we followed the approach of Fawcett et al., ⁵ and soaked the hemispheres (for each run of experiments) with bovine albumin (30% solution) to simulate the wetting properties of retinal tissue. The working fluid was water in all cases. Two lighter-than-water tamponades (air and Oxane 1300) and one denser-than-water tamponade (Oxane HD) were used for experimentation. Oxane 1300 and Oxane HD were obtained from Bausch & Lomb UK, Ltd. (Kingston-upon-Thames, UK). The tamponades were manually injected with a syringe. The volume of tamponade was varied to simulate different fractions of tamponade fill. Between injecting volumes of tamponade, the syringe was removed. 

Digital photographs were taken of the model sphere and tamponade for each series of experiments. Contact angles were determined by processing the images of the various tamponades within the analogue ocular cavity. Figure 1b shows a typical image of a tamponade (in this case an air bubble) within the acrylic sphere. The position of the interface is indicated, along with the internal position of the sphere above and below the air interface. The difference between the refractive index of water and acrylic gives rise to a slightly smaller sphere, and this effect is magnified for the gas pocket. Additional investigation confirmed that immediately below the interface, the stretching of the coordinates within the water region was uniform. An image processing algorithm (written in the Matlab Image Processing Toolbox; The MathWorks Ltd., Cambridge, UK) was used to capture the interface position and rescale it to give the interface shape. The volume of the tamponade was calculated numerically from the digitized images by using equation 9 in the 1. This procedure was adopted because of the difficulty in controlling the volume during manual injection. 

The angular coverage and the shape of the aqueous–tamponade interface, predicted by the mathematical model, were compared to the direct measurements obtained using the physical analogue of the eye for various degrees of tamponade fill. The relationship between the angular coverage and tamponade volume was compared against that of a theoretically ideal tamponade (equation 10, 1) and published data. ⁵  

Results

In the first series of experiments, we measured the contact angle at the three-phase contact line between the tamponades protein-coated acrylic and water. We measured θ = 16° for Oxane 1300, which is typical for silicon oils (for instance, Fawcett et al. ⁵ measured θ = 15°). Our measurements showed that Oxane HD had a larger contact angle θ = 20°. For air, we measured θ = 31°, again, in the range measured by Fawcett et al., but with a smaller variance. 

The variation of the angular retinal coverage with volume of tamponade is shown in Figure 2. The agreement between the theoretical model and the experimental measurements was good for the air and Oxane tamponades. The theoretical model showed that the retinal coverage by Oxane HD was (slightly) better than for silicon oil, owing to the larger contact angle. Figure 2b shows good agreement between the theoretical predictions and the measurements of Fawcett et al. ⁵ for air and silicon oil. 

Figure 2.

 

Variation of the angular coverage with tamponade volume for a spherical model. (a) Comparison between the theoretical calculations and results for air (×), Oxane 1300 (), and Oxane HD (▵) measurements of the interface shape and theoretical predictions. The full line corresponds to the limit of Bo = ∞. (b) Comparison with Fawcett et al.⁵ for air (○) and silicone oil (□).

Figure 3 shows a comparison between the measured tamponade shape and theoretical predictions for air, Oxane 1300, and Oxane HD. The comparison is based on equivalent volumes in both the experimental measurements and theory, using the measured values for contact angle and reported values for density. The agreement is excellent, confirming the salient features of the theoretical model. 

Figure 3.

 

The measured shape of the tamponades are compared with theoretical predictions for (a) air, (b) Oxane 1300, and (c) Oxane HD. The measurements are shown on the right side of each image with symbols while the left side corresponds to the theoretical predictions. The different lines correspond to different volumes of individual tamponades.

Discussion

The mathematical model of the interfacial relationships of intraocular tamponades developed in a physical analogue of the eye accurately predicted contact angles and degree of angular retinal coverage for several different retinal tamponades with close correlation to actual measurements. The theoretically predicted values for contact angles and angular retinal coverage dependent on fill fraction were close to the data published previously for air and silicone oil tamponades. ⁵  

It has been demonstrated that acrylic, when pretreated with protein, is comparable with retinal tissue in terms of its hydrophilic nature and results in contact angles with gas and silicone oil that are comparable with retinal tissue. ⁵ The acrylic analogue used in this study was thus a reasonable model for experimental measurement of contact angles and retinal coverage (although it assumed that the eye is a perfect sphere). 

In terms of the theoretical model, the agreement with the gas tamponade was very close, both for angular coverage of the retina and contact angle. We do not think that different types of gas tamponade (C3F8 or SF6) or different concentrations of gas would have any tangible difference in terms of interface characteristics, although this was not formally tested. For the liquid tamponades (silicone and Oxane HD), agreement with theoretical predictions while good for the contact angle, was less accurate for predicting angular coverage, owing to the strong affinity for the tamponades to adhere to the protein layer. The denser-than-water tamponade, Oxane HD has a larger contact angle than silicone and results in better retinal angular coverage which was predicted by the theoretical model and demonstrated in the analogue eye (Fig. 2). 

Using the mathematical model, we projected the angular retinal coverage for increasing fractions of fill, for theoretical tamponades with various density contrasts Δρ/ρ = 0.02–0.06 (Fig. 4a), and also for a tamponade with fixed density but various contact angles (15°, 20°, and 30°; Fig. 4b). These theoretical projections suggest that the contact angle, which can be controlled using contaminants, has as large an effect on retina covered as does tamponade density, for this low Bond number configuration. 

Figure 4.

 

(a) Effect of various tamponade densities (Δρ/ρ = 0.02, 0.05, and 0.06, with θ = 16°) and (b) effect of various contact angles (θ = 15°, 20°, and 30°, with Δρ/ρ = 0.03). The full line corresponds to the ideal tamponade (within the limit of Bo → ∞).

One of the difficulties of undertaking experiments with a spherical analogue of the eye is the contrasting refractive index between water and the constituent material. Most recent analogue studies have involved cylindrical models in which the interfacial relationships were easier to observe. ^8,9 In the cylindrical models, the tamponade interface has curvature perpendicular to the plane of observation, while in the spherical model, the tamponade interface has additional curvature around the z-axis. The consequence is that the tamponade appears more rounded in cylindrical models, and the angular coverage is smaller than for equivalent tamponade volumes in a spherical model. 

Conclusions

We have developed a theoretical model to examine the relationship between the angular retinal coverage and the physical properties of intraocular tamponades (density contrast, surface tension, and contact angle). The theoretical model accurately predicted these properties for both gas and liquid tamponades when compared with actual measurements with lighter-than-water (air, silicone) and denser-than-water (Oxane HD) tamponades in a physical analogue of the eye. 

The key determinants of retinal coverage by the tamponade are its Bond number, contact angle, and volume of tamponade. The theoretical model presented here may be used to accurately project the behavior of novel tamponades in the future. 

Appendix

Substituting equation 2 into equation 3 and using  

 

we obtain  

 

where the second term on the left-hand side of equation 5 is a consequence of the axisymmetric nature of the interface (and the two components of curvature). 

At the contact line z = z_c, the boundary conditions are  

 

The boundary conditions to this solution are  

 

Above the contact line, z ≥ z_c, R =

$\sqrt{a^2-z^2}$

The angular coverage of the retina by the tamponade is  

 

The evaluation of the volume of the tamponade depends on the initial angle R_z and Δρ. The volume of the tamponade is  

 

In the limit of Bo 1, the interface is flat except close to the retina. In this limit, the volume of tamponade is determined geometrically by  

 

The interface of the tamponade is completely flat when the height of the contact line satisfies  

 

For a gas–liquid interface, θ = 38°, z_c/a = 0.788, whereas for a silicon oil–liquid interface, it occurs at z_c/a = 0.96. 

The interface position z_h is unknown and is determined as part of the solution. The numerical method of solving equations 5, 6, and 7 involves first discriminating whether the gradient of the interface between the tamponade and water adjacent to the sphere wall is positive or negative in the r − z plane, since this determines z _c> or <z_h . The numerical technique is a shooting method where z_h is estimated, equation 5 is integrated from z_c , and the solution that satisfies the boundary condition equation 7 is calculated. 

Footnotes

 Disclosure: I. Eames, None; R.I. Angunawela, None; G.W. Aylward, None; A. Azarbadegan, None