Intraocular Fluid Dynamics and Retinal Shear Stress after Vitrectomy and Gas Tamponade

Abstract

Purpose.: To evaluate fluid dynamics and fluid shear stress on the retinal wall in a model eye after vitrectomy and gas tamponade in relation to saccadic eye movements and sudden head movements and to correlate the results with gas fill fraction (GF).

Methods.: Analyses was undertaken using high-resolution computational fluid dynamic software. The fluid volume within the eye was discretized using 6 × 10⁵ elements and solved with a volume-of-fluid method. The eye was abstracted to a sphere. Vertical and horizontal saccades and sudden rectilinear displacement of the head were examined. GF was varied from 20% to 80% of the eye height filled with gas.

Results.: Maximum shear stress during horizontal and vertical saccades was 1.0 Pa (Pascal) and 2.5 Pa, respectively, and was dependent on GF. Rapid rectilinear acceleration of the head caused a maximum shear stress of 16 Pa, largely independent of GF. Fluid sloshing within the eye decayed within 0.1 second. Stresses were maximum at the contact line and equator of the eye and were parallel to the direction of motion.

Conclusions.: This study predicts that saccadic eye movements and normal head movements after vitrectomy and gas tamponade generate only small fluid shear stresses on the retina that are below published norms for retinal adhesion strength. Sudden, jerking head movements generate fluid shear forces similar to retinal adhesion strength that localize to the area of gas–fluid interface. Fluid sloshing occurs after movement, but rapidly decays on cessation of movement. These results suggest that restrictive posturing after vitrectomy and gas tamponade may be unnecessary. Patients should avoid sudden head movements.

Vitrectomy and endotamponade is a frequent surgical choice for the treatment of rhegmatogenous retinal detachment (RRD) and advances in technique, technology, and instrumentation have led to broader indications and better outcomes for patients. ¹ Despite this, there is still considerable debate and insufficient clinical (no randomized control trials) and scientific evidence surrounding the role of postoperative posturing in terms of the final functional and anatomic outcomes after surgery. The issue of patient compliance with posturing after surgery is a significant factor when considering available evidence and in determining best practice. ²  

Retinal redetachment after surgery is likely to arise from several complementary mechanisms:

1.  
   Continued vitreoretinal traction
2.  
   Passage of fluid through a retinal break (either the primary break or a new break), leading to an area of localized detachment
3.  
   Propagation of fluid through the retinal break, leading to a more widespread retinal detachment, possibly due to intraocular fluid currents

In regard to these mechanisms, the conventional rationale for using a tamponade is based on the concept that the gas/liquid bubble prevents flow of fluid through tears in the retina. This blockade allows retinal reattachment to be temporarily maintained while a permanent chorioretinal adhesion forms in response to photocoagulation or cryotherapy (usually within 7 days). ³ In this context posturing may be beneficial in terms of maintaining contact of the tamponade with the retinal break and theoretically by reducing intraocular fluid movements (as the patient would theoretically remain still during this period). 

While this rationale appears logical, it does not sufficiently explain the success of surgery in cases of inferior RRDs, where contact between the tamponade and retina is difficult to establish. ⁴ It is common practice to ask patients with inferior retinal detachments to posture. Uncertainty about patient compliance with posturing leads some surgeons to use supplementary scleral buckling procedures in addition to vitrectomy. Despite these measures, there is no conclusive evidence that posturing and/or supplementary buckling improves outcomes in this group of patients, and recent publications have reported surgical success with vitrectomy and gas without posturing. ^{5 –7} These reports suggest that factors other than the barrier function of the tamponading agent is responsible for surgical success in these sorts of cases. 

For some time, intraocular fluid currents have been mooted as a significant pathogenic mechanism in the propagation of retinal detachment. ⁸ In 1934, Lindner ⁹ demonstrated the contribution of intraocular fluid currents in a simple experiment when he coated the inner surface of a glass flask with an alginate gel, and made a nick in the gel, lifting up a small flap to simulate a horseshoe retinal tear. The flask was then filled with water. While the fluid remained static, the gel remained attached to the flask. When the fluid was agitated by spinning the flask, the gel gradually detached from the wall of the flask in a manner similar to a retinal detachment. Spinning the fluid without a simulated tear did not lead to gel separation. In this context, saccadic and pursuit movements of the eyes and movements of the head may cause intraocular fluid currents that exert shear stresses on the retina. 

In this study, we used the finite volume method (FVM) and advanced computational fluid dynamic (CFD) modeling to compute the relationship between different fractions of gas tamponade fill (GF), fluid dynamics, and fluid shear stress on the retina in response to saccadic eye movements and rectilinear head movements. 

Methods

Experimental Design

We used CFD software to analyze and predict the fluid flow and shear stress on the retinal wall with different amounts of gas tamponade during horizontal and vertical saccades and with rectilinear head movement. The eye was abstracted to a sphere. The head position was modeled as upright. 

CFD is a discipline of fluid mechanics that utilizes numerical methods and algorithms to solve and analyze problems that involve fluid flows. CFD consists of three common steps;

1.  Preprocessing  
   •  
     The geometry of the problem is defined.
   •  
     The volume of the fluid is divided into discrete cells (mesh)
   •  
     Physical modeling is defined (equations of motion etc.)
   •  
     Boundary conditions are defined
2.  
   Processing: the simulation is started and equations are solved iteratively
3.  
   Post processing: analysis and visualization of the resulting solutions

Definition of Ocular Geometry

The eye was treated as a sphere of radius a = 12.5 mm. The sphere was partially filled with water to model partial filling of the eye after vitreoretinal surgery. CFD codes were applied to accurately calculate the unsteady flow within the partially filled sphere, which underwent rotational or rectilinear displacement to mimic respectively saccadic eye movements and head movement. 

Definition of Ocular Movement Parameters

Three types of eye movement were examined:

1.  
   Rotation around a vertical axis (horizontal saccadic movements)
2.  
   Rotation around a horizontal axis (vertical saccadic movements)
3.  
   Movement in the horizontal plane of the whole eye (rectilinear head movement)

The values used for vertical and horizontal saccades were based on previously published data. ^{10 –12} The angular displacement (for rotation around a vertical and horizontal axis) was fixed at Δθ = 30°, whereas the maximum angular velocity was θ̇_m = 500 deg/s.

We describe the eye's angular displacement, as a function of time to be  

 

To reduce the complexity of the model, we approximated the angular velocity of the eye to be weakly dependent on the eye displacement (which measurements support for Δθ < 10°). We have applied the same values for both horizontal and vertical saccades. 

When the head moves, the eye has both components of rectilinear and angular acceleration. The contribution of rectilinear acceleration is examined separately by considering the case of an eye moving a distance ΔX ∼0.1 m, with a maximum velocity of Ẋ _m = 1 m/s, over a time period of ΔX/Ẋ _m ∼0.05 seconds. This expression represents a sudden exaggerated jerking or whiplash motion of the head. ¹³  

To describe the rectilinear, unsteady movement of the eye caused by the movement of the head, we applied a similar methodology as for the angular rotation  

 

 

Computational Model

Three-dimensional numerical calculations were performed using the commercial CFD software (CFX; Ansys, Inc., Canonsburg, PA), to model the flow behavior inside a moving eye. This software solves Navier-Stokes equations (Newton's second law applied to fluid motion) based on the finite volume method (FVM) and for multiphase flow modeling employs the Eulerian-Eulerian (simplifying the liquid as inviscid) approach. This software is capable of describing the movement of mixtures of immiscible fluids. 

The eyeball was modeled as a sphere, which was filled with water and air. The flow domain was discretized (volume occupied by the fluid is divided into discrete cells) to more than 600,000 elements and the flow was modeled as viscous, incompressible, laminar, and unsteady. The Reynolds number associated with this is Re = u _E a/ν ∼1000 where u _E is the speed at the eye surface and ν is the kinematic viscosity of water. The effect of the GF on the eye was examined by varying the water depth 2af from 0.4a to 1.6a. In the simulation, water properties were viscosity μ = 8.9 × 10⁻⁴ kg m⁻¹s⁻¹ and density 997 kg m⁻³ and air properties were dynamic viscosity of 1.83 × 10⁻⁵ kg m⁻¹s⁻¹ and density of 1.185 kg m⁻³. The air–water interface was characterized by a surface tension of σ = 75 × 10⁻³ mN/m. The temperature was 37°C. 

Processing

Calculations were run on the University College of London's supercomputer, Legion (http://www.ucl.ac.uk/media/library/Dell), employing 32 cores and taking 30 days of computation (23,040 computer hours). Since the estimated boundary layer thickness is at most 5% of the sphere radius, this duration of computation is sufficiently resolved. 

Postprocessing

The measure of the influence of movement of fluid on the retina and the propensity for the retina to be disturbed is the tangential shear stress. The magnitude of the shear stress is τ_S = |n̂ × τ · n̂| where τ is the stress tensor and n̂ is the unit vector normal to the surface of the retina. For each GF, the maximum tangential shear stress as a factor of time was calculated in relation to:

1.  
   Maximum shear stress on the retinal wall
2.  
   Position of air–water interface
3.  
   Interpretation of flow fields

The maximum stress corresponded, in all cases, to the period of maximum velocity. The location of the maximum shear stress was obtained from shear stress contour plots.

Results

Horizontal Saccades

Figure 1a shows the variation of the maximum shear stress as a function of time for different GFs. Slightly trailing the time at which the angular velocity is maximum (t = 0.2 seconds), the maximum shear stress suddenly drops to 0. This sudden decline occurs because, during the deceleration phase, the relative velocity between the eye wall and interior fluid is momentarily small. After t = 0.3 seconds, the eye has stopped moving, but the residual motion decays over ∼0.1 seconds. The angular velocity of the fluid dragged by the sphere is largest at the extreme positions (for f < 0.5) and at the widest width of the sphere (for f > 0.5). The maximum shear stress occurs at t = 0.2 seconds. To interpret the location, the shear stress contours on the sphere are plotted at t = 0.2 seconds (Fig. 2a). Beyond the contact line, the shear stress increases from 0, up to a maximum, where the sphere is widest, and then decreases. The maximum shear stress is estimated (from the analysis in the Appendix) to be 0.6 Pa and is weakly dependent on f for f > 0.2 (range, 0.6–1.0 Pa; Fig. 3a). The calculations confirm values of similar magnitude, which vary between 0.6 and 1.0 Pa (Fig. 3a). 

Figure 1.

 

Variation of maximum shear stress as a function of time for different fluid fill fractions (f = 0.2–0.8). (a, b) Horizontal and vertical saccades, respectively. (c) Rectilinear acceleration of the eye. Stress measurements are at the fluid contact line.

Figure 2.

 

Shear stress simulation plots at t = 0.2 seconds, showing areas of maximum and shear stress as a function of time for different fractions of fluid fill (%). (a, b) Horizontal and vertical saccades, respectively. (c) Rectilinear acceleration of the eye. The color scale shows shear stress distribution in pascals.

Figure 3.

 

Maximum values of shear stress (Pa) on the retina for (a) horizontal saccades, (b) vertical saccades, and (c) rectilinear acceleration at different fluid fill fractions. Maximum values of shear stress occur at the fluid–gas contact line.

Vertical Saccades

Figure 1b shows the variation of the maximum shear stress as a function of time for vertical saccadic movement of the eye. The controlling aspect during rotation is the maximum velocity of the fluid, which is independent of GF, largely confirmed by the calculations. When the eye rotates and stops, the liquid sloshes within the eye. The free surface sloshes with a 0.03- to 0.1-second period, depending on whether the eye is almost filled or almost empty. This result is consistent with our estimates. The shear stress plot at t = 0.2 seconds shows maximum stress at the contact line and a force distribution that was focused as a vertical equatorial band (Fig. 2b). The maximum shear stress is of a magnitude similar to that for horizontal saccades but is slightly larger in value, ranging from 1 to 2.5 Pa (Fig. 3b). 

Rectilinear Acceleration

Figure 1c shows the maximum shear stress as a function of time for rectilinear displacement. The shear stress is an order of magnitude larger than for rotational displacement of the eye and shows no systematic trend with GF. The shear stress plot at t = 0.2 seconds shows that the maximum shear stress occurs either at the front of the sphere (where splashing can occur) or in a band at the equator, where the surface of the sphere is parallel to the direction of motion (which gives the largest slip; Fig. 2b). The thicker band is asymmetrically displaced to the back of the sphere as a consequence of inertia in the fluid. The shear stress is estimated (in the Appendix) to be ∼3 Pa over most of the sphere surface, it is ∼8 Pa at the moving contact line, and the numerical results indicate a higher value of 10 to 16 Pa. The variation of the maximum shear stress with GF is plotted in Figure 3c. 

Conclusions

The results of this study predict that both saccadic eye movements and normal head movements generate low fluid shear forces on the retinal wall after vitrectomy and gas tamponade (maximum, 1–2.5 Pa for horizontal and vertical saccades). This shear stress has some dependence on GF, with maximum stresses occurring at around 50% GF. Only those areas in contact with the fluid experience the fluid's shearing force. The gas-filled area experiences shear stresses that are two to three orders of magnitude less than those in the fluid-filled area and are hence negligible. 

Rapid rectilinear acceleration of the eye generates a shear stress that is an order of magnitude greater than that generated by normal saccadic movements and that is sensitive to the maximum acceleration of the eye and more weakly dependent on displacement and GF. When the eye moves with a maximum speed of 1 m/s, for a period of 0.05 second, the maximum shear stress is ∼16 Pa. These values represent a sudden rapid jerking or whiplash movement of the head, and in reality, such rapid head displacements are unlikely to occur during normal daily function. ¹³ Hence, actual shear stresses on the retina are likely to be considerably lower during normal activities. Halving the maximum speed used here reduces the shear stress by 75%. 

In this study, we calculated maximum shear stress distribution plots on the inner surface of the model eye. For horizontal saccades, the maximum shear stress occurs in a horizontal equatorial band, whereas for vertical saccades and rectilinear movements, the maximum stress distribution occurred as a vertical equatorial band. In addition, rectilinear displacement caused further stress localization at the anterior and posterior areas of the sphere. These patterns of stress distribution may be relevant to the location of retinal break formation in the eye. In all three types of analyzed eye movement, liquid in the eye is set into motion, and this motion persists for a time ∼a ²/v ∼100 seconds after the eye movement has stopped. When the gas–liquid interface is disturbed, it generates waves and sloshing, which oscillates with a period a/ $\sqrt{\sigma /\rho g}$ ∼0.15 seconds, which is consistent with the numerical results. ¹⁴  

Retinal redetachment after vitrectomy must logically originate from a retinal break at an area of retinal traction and/or weak chorioretinal adhesion that allows passage of fluid under the retina, leading to retinal–choroidal separation. In the context of a vitrectomized eye, if we assume that all tractional forces generated by the vitreous on the retina have been removed and that what remains is the interaction of forces between fluid, gas, and the retina, the fluid shear force on the retina must overcome the adhesive forces between the retina and retinal pigment epithelium if retinal detachment is to occur. As redetachment is likely to propagate from a previous retinal break treated with either cryotherapy or photocoagulation, or from a new or unidentified retinal break, it is useful to consider available values of normal adhesion strength in these situations. 

Different methods have been used to measure retinal adhesion, including peel testing and subretinal fluid injection. The available published normal values for adhesive strength in rabbits, cats, and monkeys range between 4 and 266 Pa. ^{15 –18} After cryotherapy and photocoagulation, there is an initial decrease in retinal adhesive strength for 8 to 48 hours after treatment (3.9 Pa). This decline is followed by an increase to supranormal levels in the following weeks (13–39 Pa). ^3,19,20 As retinal adhesive strength is known to decrease rapidly after death in all animals, accurate measurements ex vivo in human eyes are difficult to obtain. 

In our computational model, shear stresses generated by saccadic eye movements were maximally 1 Pa for horizontal saccades and 2.5 Pa for vertical saccades. These stresses are roughly 74% to 35% lower than the lowest reported adhesion strength seen soon after photocoagulation and may be insufficient to detach the postoperative retina. When we considered a very rapid rectilinear acceleration of the head, maximum shear increased briefly up to 16 Pa and was localized either to the front of the eye or in a vertical equatorial band. This shear stress could be sufficient to cause detachment of the retina, particularly in the immediate period after photocoagulation or cryotherapy, in particular if it occurs over a treated or untreated retinal break. 

It is perhaps simplistic to consider fluid shear stress as the only factor relevant to retinal redetachment after vitrectomy. Residual vitreous has been neglected in this discussion, and we have necessarily assumed complete vitrectomy at the time of surgery for the purpose of computational modeling. It is probably reasonable to consider that the effect of eye and head movement on any residual vitreous is negligible and that the arguments in terms of retinal adhesion and the shear stress caused by fluid movement against any residual vitreous are still valid. Vitreous contraction and the presence or absence of proliferative vitreoretinopathy, if present, is likely to occur independent of these factors and do not detract from the findings of this study. This experimental model demonstrated sloshing of the fluid level within the eye during eye/head movements and although this decayed rapidly, it is possible that the sloshing motion could elevate a retinal tear if the sloshing became localized over that area. This in turn could allow fluid to propagate beneath the retina, although as previously discussed, the predicted fluid shear force for normal saccadic eye movements is considerably lower (1–2.5 Pa) than values for retinal adhesive strength. 

Despite demonstrating here that many of the fluid shear stresses on the retina during normal saccades and head movements are below the adhesion strength of the retina, this fails to explain why we still see redetachments of the retina after surgery. Several factors are likely to explain this phenomenon. Our hypothesis assumes attachment of the retina after surgery such that retinal–RPE adhesion (or areas of fibrosis) is present throughout. However in reality, there may be areas of subretinal fluid where retinal adhesion is absent, and these may be localized at or near treated or new retinal breaks. Under these circumstances, the low levels of shear stress predicted herein, may be sufficient to redetach the retina and possibly explains the occurrence of redetachment after surgery. The important role of residual vitreous and the presence or absence of proliferative vitreoretinopathy in retinal redetachment have not been addressed here and would be difficult to model as part of a global computational simulation. These factors should be considered independent of our findings. 

In summary, the purpose of this study was to evaluate intraocular fluid dynamics and retinal wall fluid shear stress after vitrectomy with gas tamponade in relation to saccades and head movement after surgery. We have found that normal saccadic eye movements and normal head movements cause shear stresses on the retina that are below the levels of reported normal retinal adhesive strength. Sudden whiplash-like head movements cause shear stresses that may exceed the strength of retinal adhesion but are unlikely to occur under normal circumstances. These shear stresses occur within the fluid and at the gas–fluid contact line. These findings appear to have a weak relationship to the fraction of GF after surgery. Fluid sloshing may play a further undefined role. 

Our findings are likely to help explain reported successes with inferior retinal detachment surgery where there is no contact between the retina and tamponading agent. Removal of the vitreous (traction) and replacing it with fluid appears to result in weak fluid shear stress on the retina. As normal eye and head movements appear to cause low levels of fluid shear stress on the retina, restrictive postoperative posturing of patients may be unnecessary based on the findings of this study. Patients should be advised to avoid sudden brusque head movements in the immediate postoperative period. 

Footnotes

 Supported by a Senior Leverhulme Trust Fellowship (IE), a Dorothy Hodgkin Postgraduate Award at University College London (AA), and a Royal College of Ophthalmologists Pfizer Ophthalmic Fellowship award (RIA).

Footnotes

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

Appendix

The maximum angular velocity of the liquid occurs either at the edge of the air-water interface (for f < 0.5) or at r = a for f > 0.5. Thus, the maximum speed of the fluid at the edge is u_max = aθ̇_mmax(1, $\sqrt{4f\left(1-f\right)}$. The maximum shear stress corresponding to this is  

 

where δ is a typical boundary layer thickness and μ is the dynamic viscosity of the fluid. Since the eye rotates over a period ∼ Δθ/2θ̇_m, the boundary layer thickness is estimated to be δ ∼ $\sqrt{\nu \Delta \theta /{\dot{\theta }}_{\text{m}}}$ where ν = μ/s.²¹ When these estimates are combined, the maximum shear stress is  

 

The effect of eye fill indicates that the maximum shear stress increases with fill fraction until f = 0.5 and then does not increase further. 

For rectilinear acceleration, the movement of the sphere occurs over a short period. In the absence of splashing, the maximum shear stress occurs where the surface of the sphere is (approximately) locally parallel to the direction of movement as this gives rise to the largest slip over the sphere surface. The maximum shear stress scales as (A1), but with u _max = Ẋ _m and δ ∼ $\sqrt{\nu \Delta X/{\dot{X}}_{\text{m}}}$ . Combined together,