**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.

^{ 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 }

^{ 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.

*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*(Fig. 1a). The unit vector normal to the tamponade interface is where the subscript

_{z}*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*(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.

_{h}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 |

^{ 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.

^{ 5 }

^{ 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.

^{ 5 }for air and silicon oil.

^{ 5 }

^{ 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).

^{ 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.

*z*=

*z*, the boundary conditions are The boundary conditions to this solution are Above the contact line,

_{c}*z*≥

*z*,

_{c}*R*=

*R*and Δρ. The volume of the tamponade is In the limit of

_{z}*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*= 0.96.

_{c}/a*z*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

_{h}*r*−

*z*plane, since this determines

*z*

_{c}> or <

*z*. The numerical technique is a shooting method where

_{h}*z*is estimated, equation 5 is integrated from

_{h}*z*, and the solution that satisfies the boundary condition equation 7 is calculated.

_{c}*Semin Ophthalmol*. 2000;15(2):65–77. [CrossRef] [PubMed]

*Surv Ophthalmol*. 1985;30(1):47–51. [CrossRef] [PubMed]

*Vitreoretinal Surgery*. New York: Springer; 2005:147–159.

*Graefes Arch Clin Exp Ophthalmol*. 1994;232:438–444. [CrossRef] [PubMed]

*Graefes Arch Clin Exp Ophthalmol*. 2008;246(9):1217–1224. [CrossRef] [PubMed]

*Invest Ophthalmol Vis Sci*. 1978;17(1):77–79. [PubMed]

*Br J Ophthalmol*. 2004;88(5):692–696. [CrossRef] [PubMed]

*Graefes Arch Clin Exp Ophthalmol*. 2004;242(3):250–254. [CrossRef] [PubMed]