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We discuss some aspects of the fluid dynamics of vitreous substitutes in the vitreous chamber, focussing on the flow induced by rotations of the eye bulb. We use simple, yet not trivial, theoretical models to highlight mechanical concepts that are relevant to understand the dynamics of vitreous substitutes and also to identify ideal properties for vitreous replacement fluids. We first recall results by previous authors, showing that the maximum shear stress on the retina grows with increasing viscosity of the fluid up to a saturation value. We then investigate how the wall shear stress changes if a thin layer of aqueous humour is present in the vitreous chamber, separating the retina from the vitreous replacement fluid. The theoretical predictions show that the existence of a thin layer of aqueous is sufficient to substantially decrease the shear stress on the retina. We finally discuss a theoretical model that predicts the stability conditions of the interface between the aqueous and a vitreous substitute. We discuss the implications of this model to understand the mechanisms leading to the formation of emulsion in the vitreous chamber, showing that instability of the interface is possible in a range of parameters relevant for the human eye.

Retinal detachment is a serious, sight threatening condition that occurs when fluid enters the potential space between the neurosensory retina and the retinal pigment epithelium. Posterior vitreous detachment is primarily responsible for the generation of tractions on the retina that might produce retinal tears. These can possibly evolve into retinal detachment, since the detached vitreous often displays tight attachment points with the retina, where concentrated mechanical stimuli occur [

Surgery is the only viable way to treat retinal detachment [

Various vitreous substitutes are employed in the surgical practice, with largely different mechanical properties [

At present, the only long-term vitreous substitutes widely employed in the clinical practice are silicone oils. They have suitable properties of chemical stability and transparency and have a high surface tension with the aqueous humour, which is a desirable property. The rational of using silicone oil as intraocular tamponade is to interrupt the open communication between the subretinal space/retinal pigment epithelial cells and the preretinal space with the aim of securing, in the first few days after surgery, chorioretinal adhesion induced by cryo- or laser treatment. Depending on the location of the retinal break oils with different densities (either higher or lower than the aqueous) can be adopted [

The mechanical properties of tamponade fluids (density, viscosity, and surface tension with the aqueous) influence the efficiency of the treatment and, therefore, a full understanding of the mechanical implications associated with the surgery is desirable. With the present work we aim at clarifying, from a purely mechanical point of view, the implications of adopting tamponade fluids with different mechanical properties. The problem is extremely complex even if only mechanics is accounted for, and, therefore, we proceed in this paper by introducing simple theoretical models that shed some light on specific, yet crucial, aspects of the problem.

We start by considering the effect of viscosity of the tamponade fluid on the mechanical actions exerted on the retina during eye rotations.

Due to the limited tamponade effect of silicone oils we then investigate further factors leading to the successful surgery. In particular, we investigate the changes of the maximum wall shear stress when silicone oils are used, accounting for the possible presence of a thin layer of aqueous separating the retina from the tamponade fluid.

The success rate of surgery when silicone oils are used is about 70%. One of the common problems after vitrectomy, especially in the long run, is the formation of an oil emulsion. The reasons why this happens when silicone oils are used as tamponades are still unclear. A further aim of this paper is to present a simple theoretical model that predicts the role of oil properties (particularly, viscosity and surface tension) in the process of emulsion formation. To this end we study the stability of the interface between two superposed immiscible fluids set in motion by movements of the eye.

The results presented in this paper are based on solutions of the mathematical equations that govern the motion of fluids. Fluid dynamics is a very well developed branch of physics, the modern foundations of which date back to the 19th century. The so-called Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, are known to accurately model the motion of a viscous fluid described as a continuum body. These equations are mathematically very complex and admit closed-form solutions, that is, solutions that can be expressed analytically in terms of known functions, only in very special cases. If an analytical solution of a problem can be found, its dependency on the controlling parameters (e.g., in the present case the size of the vitreous chamber, the viscosity of the fluid, and so forth) can be easily determined, without the need of computational simulations, and physical insight on the problem is therefore effectively obtained. In this paper we discuss some analytical solutions of the Navier-Stokes equations, which are relevant to understanding the dynamics of vitreous substitutes.

We consider purely viscous fluids, that is, fluids whose mechanical properties are completely characterized by the density

Fluid motion in the vitreous chamber can be driven by different mechanisms, in particular, rotations of the eyeball or thermal differences between the anterior and posterior segments of the eye. However, it can be shown by simple order-of-magnitude arguments that the motion induced by eye rotations is much stronger than the thermally driven flow [

We consider three different, relatively simple, models that shed light on important aspects of the dynamics of vitreous substitutes in the vitreous chamber. Proper interpretation of results from experimental or more complex theoretical models requires a full understanding of the results presented here. The details of the mathematical models are briefly reported in the appendices.

Sketch of the three models adopted in the paper.

We first consider the motion of a fluid contained in a sphere of radius

Velocity profiles in radial direction.

In purely viscous fluids, whatever the value of the viscosity, the maximum of the velocity is invariably attained at the wall (

In Figure

Dependency of the maximum shear stress at the wall on the viscosity in the case of a purely viscous fluid. The two curves correspond to two different values of the frequency of eye rotations (dashed line 20 rad/s; solid line 10 rad/s; A = 20 deg = ^{3},

Finally, we report in Figure

In the previous section we have discussed how the stress on the retina depends on the viscosity of a vitreous substitute, under the assumption that the fluid completely fills the vitreous chamber. In particular, we have shown that the mechanical actions on the retina grow with increasing fluid viscosity. In reality the situation is more complicated than this because, owing to the hydrophobic nature of vitreous substitutes, a thin layer of aqueous may form between the retina and the vitreous substitute.

We therefore now consider how the scenario described in the previous section is modified when we account for the presence of a thin layer of aqueous close to the retina.

In Figures

Velocity profiles in radial direction in the case in which the vitreous chamber contains two immiscible fluids.

This has important implications for the wall shear stress at the wall, as it is shown in Figure

Maximum stress at the wall versus the thickness of the aqueous layer. The stress is normalized to 1, and the thickness of the layer

Length of the shortest unstable perturbation

The presence of an aqueous layer separating the vitreous substitute from the retina was shown in the previous section to have an important effect on the shear stress on the retina. It is also known that one of the main complications after injection of long-term vitreous replacement fluids (particularly silicon oils) is the possible occurrence of emulsification. This implies that the oil-aqueous interface might break, eventually leading to the formation of oil droplets dispersed in the aqueous. There are several possible causes of generation of an emulsion, with one of them being introduction of mechanical energy into the system that breaks down the oil aqueous interface [

In order to investigate the feasibility of this assumption and determine which parameters play a role in the breakdown of the interface, we present in this section results from an idealized, yet informative, theoretical model. As discussed in Section

The problem of the stability of the interface is governed by the four dimensionless parameters introduced and described in the appendices. Here we discuss the role of two of them:

Our stability analysis shows that very long waves on the interface are invariably unstable during certain phases of the oscillation cycle. In other words the amplitude of very long disturbances always grows in time. We note that in the absence of an interface this stability problem consists in the stability of the so-called “Stokes boundary layer,” that is, the flow of a single fluid over an oscillating wall. This problem has been largely studied in the literature [

Figure

In Figure

In the present paper we have discussed theoretical results from three different idealized mathematical models that, in our view, help in understanding some of the basic features of the fluid mechanics of vitreous substitutes in the eye. We have focused our attention on the flow generated in the vitreous chamber by rotations of the eye globe, which is by far the most important mechanism generating fluid motion.

We first have considered the case in which the whole vitreous chamber is filled with a single fluid and have modelled the chamber as a rigid sphere, performing sinusoidal small amplitude torsional oscillations, similar to what was done by previous authors [

We have also briefly recalled how flow characteristics change when a viscoelastic fluid fills the vitreous chamber. The real healthy vitreous has viscoelastic properties, and there is a large body of research devoted to the identification of vitreous replacement fluids with viscoelastic properties. We have recalled that the motion of a viscoelastic fluid can be resonantly excited by eye rotations and, if this happens, large values of the shear stress are expected to develop on the retina. This has important implications for the choice of the ideal properties of vitreous substitutes. Soman and Banerjee [

In the second part of the paper we considered the effect of a thin layer of aqueous separating the vitreous substitute from the retina. Since vitreous substitutes are normally hydrophobic fluids and complete filling of the vitreous chamber can be hardly obtained, a layer of aqueous in correspondence with the retina is likely to form. We have shown that, when this is the case, the maximum stress on the retina can be significantly reduced, even if the viscosity of the vitreous replacement fluid is very large. Therefore, the possible existence of an aqueous layer should be accounted for when estimating the mechanical stresses on the retina after injection of a vitreous substitute.

The presence of an aqueous layer and, consequently, of an interface between the aqueous and the vitreous substitute also has a crucial effect in the possible development of an emulsion, which is one of the main drawbacks associated with the use of silicon oils. Making use of a simple mathematical model we have studied the stability of the aqueous-vitreous substitute interface. The results show that the interface becomes more unstable if the surface tension decreases and it becomes more stable if the viscosity of the vitreous substitute is higher. Both results are in agreement with clinical observations. In fact there is evidence that the tendency to emulsification is significantly enhanced by the presence of surfactants that decrease the surface tension between the two fluids [

We consider a hollow rigid sphere with radius

The motion of a viscous fluid within the sphere is governed by the Navier-Stokes equations and the continuity equation, which read

subject to the following boundary conditions:

where

Taking advantage of the assumption of small amplitude eye rotations

where

We now take into account the presence of a thin layer of aqueous between the retina and the vitreous substitute fluid. We assume that the two fluids have the same density

The problem is still governed by the Navier-Stokes equations for the two fluids and, at the interface between the fluids, we impose the continuity of the velocity and the dynamic boundary condition. Assuming again that the sphere rotates according to (

where the subscripts

The wall shear stress on the equatorial plane is equal to

We now wish to study the stability of the interface between the aqueous layer and the vitreous substitute. For simplicity we assume that the thickness

We work in terms of the following dimensionless variables (denoted by superscript stars):

where

We decompose the flow in a basic state and infinitesimally small perturbation as follows:

where capital letters indicate the basic flow and small letters with a bar refer to perturbation quantities.

The basic flow is unidirectional (in the

For the stability analysis we consider two-dimensional perturbations

We adopt the quasi-steady approach; that is, we assume that perturbations evolve on a time scale that is much smaller than the characteristic time scale of the basic flow. This implies that we study the stability of a “frozen” basic flow at time

Taking advantage of the assumed infinite extension of the domain in the

The final system of the equations for the perturbation evolution is given by two Orr-Sommerfeld equations, one for each fluid, together with suitable boundary conditions [

The authors declare that there is no conflict of interests regarding the publication of this paper.