Appendices

Useful Formulae and Rules

Cryotherapy

¥JouleÐThomson Effect. The JouleÐThomson effect arises because real gases (nonideal) exhibit molecular interactions. Each gas has a threshold temperature below which it cools when expanded and can drain energy from the surrounding environment, causing it to cool down. At room temperature nitrogen and oxygen cool on expansion, whereas helium, for example, warms.

Fluids (i.e. Both Gases and Liquids)

¥Surface Tension. The forces present on the surface of a liquid and a gas, produced by intermolecular bonds, which must be overcome to break the surface of the liquid in air (Fig. A.1).

Air

Oil

Fig. A.1 The intermolecular attractions are shown around the molecules in a liquid (e.g. oil) in contact with another liquid or a gas (e.g. air). Within the liquid each molecule is pulled equally in all directions by neighboring liquid molecules top left, resulting in a net force of zero. At the surface of the liquid, the molecules are more attracted to other molecules inside the liquid than outside the gas, producing an overall force inwards. The liquid would like to be a sphere but is usually distorted by other forces, for example, gravitational

ÐThe surface tension in surgery is used to keep the gas as one bubble, for example, avoiding the separation off of a bubble which might pass through a retinal break.

ÐAn air or gas bubble in the eye has a ßattened inferior aspect because the gravitational force of the liquid under the bubble, combined with the high buoyancy of the gas, is high enough to overcome the surface tension of the gas bubble (which without gravitational forces would create a sphere), thus causing the bubble to ßatten rather than achieve a sphere (Fig. A.2). Similarly the forces acting on the bubble are enough to overcome the surface tension to cause the bubble to conform to the shape of the eye superiorly. This causes a large surface area in contact with the retina superiorly but a gap in contact inferiorly.

ÐAs the bubble becomes smaller, the balance of gravitational forces relative to surface tension is changed. If a bubble separates off, it remains so because the surface tension effects overcome the gravitational effects of the ßuid, and the ßuid remains between the bubbles separating them; therefore, multiple separate bubbles appear just before the bubble disperses.

Gas

Fig. A.2 Gas in the vitreous cavity has a ßattened inferior meniscus

¥Interfacial Tension. As above but between two liquids:

ÐThe interfacial tension in surgery is used to keep a liquid as one bubble, for example, avoiding the separation off of a bubble which might pass through a retinal break. The interfacial tension of silicone oil (being less than the surface tension of a gas) may be overcome if a retinal tear is under high tension, for example, proliferative vitreoretinopathy, causing the silicone oil to pass through a tear into the subretinal space.

ÐA silicone oil bubble in the eye has a nearly spherical shape because the gravitational forces of the liquid around the bubble, which is quite weakly buoyant, are not enough to overcome the interfacial tension of the oil bubble thus giving the natural spherical form. As there is a sphere within a sphere, there is much less surface area in contact with the retina than with gas (in practise with a maximal Þll of oil, probably in contact from the horizontal meridian upwards) (see Figs. A.3, A.4).

Gases

¥Gases Are Compressible.

¥FickÕs Diffusion Equation. FickÕs diffusion equation states that the rate of diffusion of a gas through a thin membrane is increased by the concentration differential, the area of the membrane and the diffusivity of the gas, and reduced by the thickness of the membrane (Fig. A.7). This

Silicone oil

Fig. A.3 Silicone oil in the vitreous cavity has a more spherical inferior meniscus

Air

Oil

Fig. A.5 Notice the gas in the vitreous cavity visible on MRI with the patient face up. There is a ßattened inferior meniscus

Fig. A.6 Oil has a spherical inferior proÞle (CT scan patient face up)

Fick’s diffusion equation

Fig. A.4 Even a large bubble of a ßuid which takes up a spheroidal shape (e.g. oil) within a sphere (the eye) has a small contact area on the inside of the sphere (grey line and arrows), whereas a smaller bubble of distortable ßuid (e.g. air) with a ßat meniscus will have a large contact area (black dashed lines)

Fig. A.7 The large molecule gas has low diffusivity and passes across the membrane slowly. The small molecule gas moves rapidly across the membrane. Therefore, initially the gas bubble on the left expands in relation to the gas bubble on the right

equation explains the longevity of some gases, for example, perßuoropropane, in the eye, and why these can expand.

F = − D(c2 − c1) x

F = rate of passage of the gas D = diffusivity of the gas

c2 − c1 = gas concentration difference across the membrane x = thickness of thin membrane

¥BoyleÕs Law. BoyleÕs law states that, at a constant temperature, the volume of a given mass of gas varies inversely with pressure.

Liquids

¥Liquids Are Not Compressible for Practical Purposes

¥Infusion Heights When calculating infusion bottle heights in vitrectomy surgery, use

14mmH2O = 1 mmHg

¥BernoulliÕs Principle. BernoulliÕs principle states that as the speed of a moving liquid increases, the pressure within the liquid decreases. This principle may be the reason that a retinal break ßattens onto an indent in nondrain retinal detachment surgery.

Pressure × velocity = k

¥Blood Flow Rate = Blood Velocity × Cross-sectional Area of the Blood Vessel

¥LaplaceÕs Law for Pressure in a Tube Radius (r)

Transmural Pressure

= Wall Tension

r

and a Sphere

TransmuralPressure = 2 × WallTension

r

¥This demonstrates that the larger the ßuid Þlled cavity for the same pressure, the higher the tension on the wall. Theoretically a highly myopic eye is more vulnerable to wall rupture for the same pressure.

¥Hagan Poiseuille Law

Volume Flow Rate(Q)

= π d 4 (Pa − Pb)

L8n

d = diameter of the tube

Pa − Pb = pressure difference between ends

Fig. A.8 EmulsiÞed droplets of oil are visible in the vitreous cavity of this patient on ultrasound

L = length of the tube n = viscosity

¥A higher pressure is required to make highly viscous materials such as silicone oil to pass through a tube. 5,000 mPas oil is therefore more difÞcult to remove through a small hole or tube than 1,000 mPas oil.

¥Emulsion

¥This is a complex interaction of otherwise immiscible substances such as oil and water to create small droplets of one in the other (Fig. A.8). In the eye silicone oil emulsion in aqueous (water) is probably facilitated by the presence of proteins in the aqueous and the mechanical action of eye movements on the surface of the oil bubble. The protein is acting as an emulsiÞer.

¥The Bancroft rule applies, that is, the emulsiÞers and emulsifying particles tend to promote dispersion of the phase in which they do not dissolve very well.

¥The protein dissolves better in water than in oil and so tends to facilitate an oil-in-water emulsion (i.e. it promotes the dispersion of oil droplets throughout a continuous phase of water).

¥ReynoldÕs Number. This is an empirical number to calculate the likelihood of turbulence in a ßuid. The thin layer of ßuid between the retina and a silicone oil bubble is unlikely to allow the development of eddies (turbulence) which may be a factor for the effect of oil in retaining the attachment of inferior retinectomy despite the fact that the oil is not in contact with the retinectomy edge.

Re

= P2rV

N

Re = ReynoldÕs number P = pressure

r = radius of a tube

n = viscosity of the ßuid V = velocity of the ßuid

¥Vapor Pressure. Vapor pressure is the pressure exerted above a liquid by its own vapour. If a liquid has a high characteristic vapor pressure, then the liquid is more likely to evaporate.

¥PascalÕs Principle. Pressure is transmitted undiminished in an enclosed static ßuid. Therefore, a blow to the front of the eye allows damage to the retina at the back of the eye.

Ultrasound

¥Velocity of Sound in 1,000 mPas Silicone Oil = 986 m/s. An ultrasound of the eye with oil in situ appears to show an enlarged eye because the sound waves are slowed down and take longer to return to the transducer. The increased delay is falsely interpreted as from increased distance.

¥MeldrumÕs Formula. MeldrumÕs formula adjusts the axial length calculated by an ultrasound scan for the presence of 1,000 mPas silicone oil:

Axial length (mm) = length of (anterior chamber + lens) +(0.63 x vitreous length) + the retro-silicone space

¥ Doppler Equation (Fig. A.9)

V = Vsound × DFrequency

flow 2Fout cosA

Vßow = ßuid velocity

Vsound = velocity of sound DFrequency = change in frequency Fout = Transmit frequency

A = angle of incidence of the Doppler beam to the direction of ßow (should be small)

Angle correction (degrees)

Fig. A.9 The effect of angle of incidence of the Doppler beam to the velocity measurements is shown. At high angles the effect on the measurements is high

Diffusion and Viscosity

Viscosity of vitreous 5Ð2,000 cP (aqueous 1 cP)

¥Inversely related to diffusion of a molecule

ÐFickÕs law

¥Diffusion ßux (J)=D/Concentration gradient (dc/dx)

ÐStokes-Einstein

¥Diffusion CoefÞcient (D) = RT/6pnrN D = diffusion coefÞcient

R = molar gas constant

T = temperature in Kelvin n = viscosity of the medium

r = radius of the diffusing molecule N = AvogadroÕs number

¥Stefansson E (2009) Graefes Archives.