The Stokes−Einstein equation and the physiological effects of vitreous surgery
Removal of the vitreous humour influences the physiology of the eye. The diffusion characteristics of small molecules in the vitreous cavity are changed dramatically by the removal of vitreous gel and its replacement by aqueous humour. This effect is predicted by the Stokes−Einstein equation (Sinko 2006).
In vitrectomy the vitreous gel is replaced by water. As vitreous humour is 99% water, the chemical change is not terribly great, but there is an enormous change in viscosity. All liquids possess a definitive resistance to flow; viscosity is a measure of internal flow friction or the resistance of liquid molecules. The higher the magnitude of viscosity, the more resistant the liquid will be to flow. The viscosity of water is 1.00 centipoise (cp) at 20 °, whereas that of vitreous gel is 300–2000 cp (Lee et al. 1992; Soman & Banerjee 2003). The change in viscosity has a major effect on diffusion and thereby on the transport of all substances through the vitreous cavity. The amount (M) of compound flowing through a unit cross-section (S) of a flow barrier in unit time (t) is known as the flux (J):
The flux, in turn, is proportional to the concentration gradient (dC/dx) of the compound within a given medium (Fick's first law):
where D is the diffusion coefficient (or diffusivity) of the diffusing molecules and dx is the distance travelled. The diffusion coefficient is affected by various factors such as the size and chemical properties of the diffusing molecule, the properties of the diffusion medium (the liquid) and the temperature. The effects of some of these factors on the diffusion constants are described by the Stokes−Einstein equation:
where R is molar gas constant, T is temperature in Kelvin, η is viscosity of medium, r is radius of diffusing molecule and N is Avogadro's number (Sinko 2006). The key here is the viscosity. Vitrectomy decreases the viscosity within the vitreous cavity 300–2000-fold and consequently the diffusion coefficient is increased 300–2000-fold for all molecules. This means that after vitrectomy (or posterior vitreous detachment) all molecules diffuse much faster in the vitreous cavity.
Removal of the vitreous gel speeds up the transport of oxygen within the eye. Animal studies have shown that oxygen diffuses from the anterior segment to the retina following vitrectomy (and lens extraction) (Stefánsson et al. 1981). After vitrectomy, oxygen is transported more effectively from well perfused to ischaemic areas of the retina (Stefánsson et al. 1990). We initially predicted that this effect was due to fluid currents within the water-filled vitreous cavity. This may be true in part and would add to the effect of the viscosity change on diffusion. Human studies have shown that oxygen gradients in the vitreous cavity flatten out after vitrectomy and more oxygen is transported to the posterior pole of the crystalline lens (Stefánsson 2001; Holekamp et al. 2005), as the Stokes−Einstein equation predicts.
Various growth factors, such as vascular endothelial growth factor (VEGF), are produced in the retina in response to hypoxia. The growth factors are able to diffuse into the vitreous cavity and can be measured there (Aiello et al. 1994). The removal of the vitreous will increase the diffusion of growth factors and clear them from the retina faster. The rate of clearance by diffusion will be proportional to the change in diffusion coefficient and will be 300–2000-fold greater than before vitrectomy.
The increased diffusion is beneficial to the retina in many ways. The hypoxic areas of the retina receive extra amounts of oxygen from the fluid in the vitreous cavity, thus reducing growth factor production. At the same time, the clearance of growth and permeability factors from the retina is enhanced. This may help explain why retinal neovascularization stops after vitrectomy (Blankenship & Machemer 1985) and the beneficial effect of vitrectomy on diabetic macular oedema.
Meanwhile, in the vitrectomized eye oxygen diffuses away from the anterior segment towards the retina faster, especially if the lens is also absent. This creates hypoxia in the anterior segment and may contribute to iris neovascularization (Stefánsson et al. 1981, 1984). The growth factors produced in the hypoxic retina diffuse faster towards the anterior segment after vitrectomy and may stimulate iris neovascularization.
The increased transportation of oxygen and growth factors may help explain the clinical effects of vitrectomy: decreased risk of retinal neovascularization and increased risk of iris neovascularization. The beneficial effects of filling the vitreous cavity with silicone oil on iris neovascularization may be partly due to the retardation of diffusion by the viscous oil.
The increased diffusion after vitrectomy also explains the findings of Holekamp et al. (2005), who observed a flattened oxygen tension profile in the vitreous cavity and increased transport to the posterior pole of the lens, which they propose leads to cataract formation after vitrectomy.
The clearance of most molecules from the retina into the vitreous cavity will increase dramatically following vitrectomy or posterior vitreous detachment. The biological effects of this clearance need further study.
Silicone oil generally has a higher viscosity than vitreous gel. Commonly used silicone oils in vitreous surgery have viscosity of 1000 cp or 5000 cp. It may be concluded that 1000-cp oil will not change the diffusion characteristics dramatically from vitreous gel, whereas 5000-cp oil will slow down diffusion compared with vitreous gel, not to mention aqueous humour. Accordingly, silicone oil filling was found to prevent the effect of vitrectomy on oxygen transportation between the anterior and posterior segments of the eye and to normalize oxygen tension in the aqueous humour following vitrectomy and lens extraction in the cat (de Juan et al. 1986). When silicone oil fills the anterior chamber, it prevents the diffusion of nutrients to the corneal endothelium and stroma; this may be the reason for corneal complications in the silicone oil-filled eye where silicone oil touches the cornea (Abrams et al. 1995).
The clearance of intravitreal drugs can be increased or decreased after vitrectomy. In general, clearance increases when vitreous is replaced by water (Doft et al. 1985). This cannot be explained by the solubility of the drugs, which is presumably very similar in water and vitreous gel, which is 99% water. However, the increased rate of drug diffusion will hasten the removal of soluble drug away from the reservoir and therefore deplete the reservoir much faster. The viscosity does not change much when vitreous is replaced by silicone oil and under such conditions the clearance will be affected by drug solubility in the oil. Thus, the clearance of a lipophilic 5-fluorouracil prodrug is decreased when vitreous is replaced by silicone oil (Schmidt Laugesen et al. 2005), as the clearance of the hydrophilic drug ganciclovir is decreased (Perkins et al. 2001).
The Stokes−Einstein equation sheds light on the consequences of vitreous removal and possibly posterior vitreous detachment. The discoveries of classical physicists are useful and probably underutilized in ophthalmology.