**Purpose.**:
Phakic intraocular lenses (pIOLs) are used for correcting vision; in this paper we investigate the fluid dynamical effects of an iris-fixated lens in the anterior chamber. In particular, we focus on changes in the wall shear stress (WSS) on the cornea and iris, which could be responsible for endothelial and pigment cell loss, respectively, and also on the possible increase of the intraocular pressure, which is known to correlate with the incidence of secondary glaucoma.

**Methods.**:
We use a mathematical model to study fluid flow in the anterior chamber in the presence of a pIOL. The governing equations are solved numerically using the open source software OpenFOAM. We use an idealized standard geometry for the anterior chamber and a realistic geometric description of the pIOL.

**Results.**:
We consider separately the main mechanisms that produce fluid flow in the anterior chamber. The numerical simulations allow us to obtain a detailed description of the velocity and pressure distribution in the anterior chamber, and indicated that implantation of the pIOL significantly modifies the fluid dynamics in the anterior chamber. However, lens implantation has negligible influence on the intraocular pressure and does not produce a significant increase of the shear stress on the cornea, while the shear stress on the iris, although increased, is not enough to cause detachment of cells.

**Conclusions.**:
We conclude that alterations in the fluid dynamics in the anterior chamber as a result of lens implantation are unlikely to be the cause of medical complications associated with its use.

^{1}and Fechner et al.,

^{2}which, as their name suggests, are held in place relative to the iris by haptics with claws that attach to the iris. These devices are marketed by Ophtec BV (Groningen, The Netherlands).

^{3}include chronic subclinical inflammation, corneal endothelial cell loss, cataract formation, secondary glaucoma, iris atrophy, and dislocation. In this paper we focus on three risks associated with pIOLs that could have mechanical underpinnings:

- Possible increase of pressure in the posterior chamber: The presence of the pIOL might increase the resistance to flow from the posterior to the anterior chambers, increasing the pressure in the posterior chamber that is required to maintain the flow. This risk is commonly mitigated by introducing a laser iridotomy in the iris, and it would be useful to know which patients would benefit from this treatment.
- Reduction of endothelial cell density on the posterior surface of the cornea: This could occur due to excessive wall shear stress arising from the altered flow patterns in the anterior chamber.
^{4} - Loss of pigment cells from the iris: This could also be due to excessive wall shear stress due to changes in the fluid dynamics of the anterior chamber.
- Secondary glaucoma: This is possibly caused by increasing the resistance to the flow of aqueous humor, either within the anterior chamber or at the outflow.

^{5}However, to our knowledge the only work in which this problem was considered was done by Niazi et al.,

^{6}who proposed a numerical model to study how aqueous flow induced by temperature differences between the front and the back parts of the anterior chamber is modified after implantation of an iris-fixated pIOL. In the current paper we also present the results of a numerical model that allows us to predict the change in the pressure distribution due to the lens, as well as the changes in the shear stress on the surfaces of the iris and cornea. We extend the work by Niazi et al.

^{6}by considering various mechanisms that can produce fluid flow in the anterior chamber and, specifically, focusing on the flow due to the production and drainage of aqueous humor, the flow due to miosis of the pupil, the buoyancy-driven flow that arises from density variations due to the temperature gradient across the anterior chamber, and also the changes that are produced due to the flow generated during saccades of the eye. In each case we calculate the distributions of pressure and wall shear stress (WSS) to estimate whether there is cause for concern regarding the use of pIOLs.

*u*denotes the velocity vector,

*p*is pressure,

*t*time,

*ρ*fluid density,

*ν*the fluid kinematic viscosity, and

*g*the gravitational acceleration, with magnitude

*g*= |

*g*|. When we study the flow induced by rotations of the eye, the above equations are solved on a moving domain, and therefore have to be suitably modified (see the OpenFOAM website for further details: http://www.openfoam.org/features/ [in the public domain]).

*T*is temperature,

*k*is the thermal conductivity of the fluid, and

*c*is the specific heat at constant pressure. In Equation 2 we have neglected the heat production due to mechanical energy dissipation, which is negligible in the present context. We adopt the Boussinesq approximation (see, for instance, Drazin and Reid

_{p}^{7}), which is known to be very accurate for liquids undergoing small variations in temperature. In this approximation we neglect variations in the fluid density except in the gravitational forcing term, and we neglect variations in the kinematic viscosity

*ν*. In this term we assume that the aqueous humor density

*ρ*has a small linear dependency on temperature and is independent of pressure, leading to where

*ρ*

_{0}is the fluid density at a reference temperature

*T*

_{0}, and

*α*is the linear thermal expansion coefficient of the fluid.

^{8}which is based on the finite volume method. All meshes are generated using the snappyHexMesh tool by OpenFOAM, which produces unstructured meshes consisting of tetragonal and hexahedral volumes. We adopt meshes consisting on average of 1 to 2 million volumes, except when analyzing the lens including the haptics for which the mesh has 2.7 million volumes, and run the code in parallel on a 32 processor computer (32 processor work station from the E4 Company, Scandiano, Italy). For all simulations, careful mesh-independence tests have been carried out. In the Table, we report the parameter values used for the simulations.

**Table**

**Figure 1**

**Figure 1**

**Figure 2**

**Figure 2**

*Q*that is uniformly distributed over this surface. The outlet is at the trabecular meshwork, where we impose zero pressure. This means that the computed values of the pressure actually represent pressure differences with respect to the outlet. The pressure is, therefore, defined up to a constant that does not affect the results. All the other surfaces are rigid walls at which we impose a no-slip boundary condition. We obtain the solution using the simpleFoam solver, which uses a semi-implicit method to solve the Navier-Stokes equations iteratively.

^{15}used PIV (particle image velocimetry) to measure the flow of aqueous humor through a laser iridotomy hole during miosis in rabbits. These results were used by Yamamoto et al.

^{16}to prescribe the inlet velocity of an iridotomy jet in their numerical model of flow in the anterior chamber, which they assumed to be the first quarter-period of a cosine curve. They used a hole of diameter

*d*= 0.56 mm for which the measured maximum velocity was

_{LI}*u*= 9.39 mm/s, and took the duration to be the average duration of miosis in humans after light stimulation,

_{max}^{17}giving

*T*= 0.66 seconds.

_{jet}^{15}could be similar to the volumetric flux through the iris–lens channel in the natural eye. We therefore adopt the formula used by Yamamoto et al.

^{16}and impose the following time-dependent total volumetric flux through the iris–lens channel in our simulations of miosis in the natural eye: for

*t*≤

_{init}*t*≤

*t*+

_{init}*T*, where

_{jet}*t*is time. To avoid numerical artifacts, we also added a ramp during 0 ≤

*t*≤

*t*= 0.1 seconds during which the velocity grows sinusoidally from 0 to

_{init}*u*; see Figure 3a.

_{max}**Figure 3**

**Figure 3**

^{18}In the present case, owing to the fact that we consider an incompressible fluid and a noncompliant domain, the outlet flux is instantaneously equal to the inlet flux, as a consequence of mass conservation. This implies that the use of the boundary condition proposed by Heys et al.

^{18}would produce an unsteady pressure variation at the outlet section, which can be directly related, through an outflow resistance, to the inlet flux. Such pressure change, being uniform across the domain, does not modify the flow. In other words, using the zero pressure outlet condition implies that the computed pressure represents the instantaneous difference between the local and the outlet pressure, which is in any case the pressure difference of interest to us in this study.

^{6}showed that the typical velocities are significantly reduced for greater values of the conductivity of the device, and thus the WSS also decreases with increasing thermal conductivity. Since our aim is to determine the risk of excess WSS on the iris and cornea, we expect the present study to provide an estimate of the worst-case scenario regarding the WSS.

^{21}which provides the angular velocity of the eye as a function of time; see the examples in Figure 3b. As for the buoyancy-driven flow described in the previous section, since the fluid velocities induced by the production and drainage of aqueous humor are expected to be significantly smaller than those induced by saccades, we neglect production and drainage and impose a no-slip velocity on all boundaries. We use the unsteady solver pimpleDyMFoam to simulate the Navier-Stokes and continuity equations on a moving grid.

**Figure 4**

**Figure 4**

^{22}A study of this process is outside of the scope of the present paper but might deserve future attention.

^{−4}mm Hg) is too small to be clinically relevant.

**Figure 5**

**Figure 5**

*h*(

*r*) in the range

*r*

_{1}<

*r*<

*r*

_{2}, with

*r*being the radial coordinate,

*μ*is the shear viscosity of the aqueous humor, and

*Q*is the total volumetric flux through the anterior chamber. As a rough estimate, consider a device of radius 3 mm with a gap of uniform height and of width 1 mm at its outer edge. The formula above shows that, to obtain a pressure drop of 1 mm Hg across the gap, the gap height needs to be approximately 6 μm, which is less than 5% of the normal distance between the pIOL and the iris. This estimate assumes that the lens remains axisymmetrically positioned, which is unlikely in the case of mispositioning; however, if the lens does not maintain close proximity to the iris all the way around its circumference, it would be likely to reduce the pressure drop further, because the fluid would have an alternative path to the trabecular meshwork. Furthermore, in reality the pIOLs in use do not have a flat region of width 1 mm near their edge, and the width of iris over which a gap of a few micrometers between iris and device could be maintained is much less than a millimeter. In summary, these estimates all indicate that it is highly unlikely that the increase in the pressure drop due to the flow needing to get through a narrow gap between the iris and device could ever be large enough so as to make a clinically significant difference.

**Figure 6**

**Figure 6**

^{−2}Pa (pIOL with haptics) and 2.69 × 10

^{−2}Pa (pIOL with no haptics), whereas it is almost zero with no pIOL.

^{6,19}the flow is approximately confined to two-dimensional vertical planes. The maximum velocity is higher by approximately two orders of magnitude than that of the flow due to the production of aqueous humor. These results agree very well with the theoretical predictions by Canning et al.

^{19}and also justify the fact that we have neglected the flow due to the production of aqueous humor. In Figure 7b we show the corresponding corneal WSS stress, which attains a maximum value of 1.64 × 10

^{−3}Pa in the center of the cornea.

**Figure 7**

**Figure 7**

^{6}Even though the flow structure is significantly modified by the presence of the pIOL, the maximum absolute velocity is very similar to that without the pIOL. We note, however, that Niazi et al.

^{6}showed that if a pIOL with a high thermal diffusion coefficient is used, aqueous humor velocity can be significantly decreased.

**Figure 8**

**Figure 8**

**Figure 9**

**Figure 9**

**Figure 10**

**Figure 10**

^{23}(see fig. 4 of that paper, red line). We also note that the WSS distribution on the cornea is not significantly modified by the presence of the pIOL, and it is in agreement with the distribution shown by Abouali et al.

^{23}(see fig. 3 of that paper).

**Figure 11**

**Figure 11**

^{5}There is a strong clinical interest in this, because iris-fixated pIOL implantation is associated with increased risk of secondary glaucoma due to a possible increase in the pressure in the posterior chamber and also with increased rate of loss of endothelial corneal cells and pigment cells from the iris

^{3}; and these complications could arise as a result of changes in the fluid flow following pIOL placement. The risk of secondary glaucoma is typically mitigated by introducing an iridotomy into the iris; however, the risk of secondary glaucoma with no iridotomy is currently unknown. The only existing work in which the effect of iris-fixated pIOLs on the fluid dynamics of the aqueous humor is considered is that of Niazi et al.,

^{6}who investigated changes in the thermally driven flow. We have extended their work by also accounting for various other mechanisms of fluid motion in the anterior chamber. Since in vivo measurements of aqueous flow would be extremely challenging (and are not available at present), a numerical approach seems to be the best option. Owing to the lack of pressure and aqueous velocity measurements in vivo, our results cannot be validated against experimental data. However, owing to the laminar nature of the flows considered (numerical simulations of laminar flows are notoriously much easier than of turbulent flows), we believe our numerical results to be quite reliable.

- If the lens is properly placed, there is a negligible influence on the pressure in the posterior chamber.
- There is no significant increase of the WSS on the cornea.
- The WSS on the iris is significantly greater than in the case with no pIOL, but the increase is not likely to be sufficiently great so as to give a risk of cell detachment.

^{24}performed experiments on porcine corneal endothelial cells that were plated onto glass slides, and found that significant detachment was observed for shear stresses in excess of 0.03 Pa if the cells had had 1 hour of adhesion, rising to 0.1 Pa for 3 hours of adhesion. Thus in vivo, the critical WSS at which cells start to detach from the cornea is likely to be in excess of 0.1 Pa. Our model predicts that the actual values are significantly smaller than this, and so it is unlikely that WSS is responsible for detaching the cells from the cornea. Although we do not have corresponding data for the iris, the result is likely to be similar for that too.

**R. Repetto**, Ophtec BV (C);

**J.O. Pralits**, None;

**J.H. Siggers**, Ophtec BV (C);

**P. Soleri**, Ophtec BV (E)

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