Abstract
Purpose:
Corneal endothelial cell loss is one of the possible complications associated with phakic iris-fixated intraocular lens (PIOL) implantation. We postulate that this might be connected to the alteration of corneal metabolism secondary to the lens implantation.
Methods:
A mathematical model of transport and consumption/production of metabolic species in the cornea is proposed, coupled with a model of aqueous flow and transport of metabolic species in the anterior chamber.
Results:
Results are presented both for open and closed eyelids. We showed that, in the presence of a PIOL, glucose availability at the corneal endothelium decreases significantly during sleeping.
Conclusions:
Implantation of a PIOL significantly affects nutrient transport processes to the corneal endothelium especially during sleep. It must still be verified whether this finding has a clinical relevance.
In recent years, phakic iris-fixated intraocular lenses (PIOLs) have attracted widespread acceptance for refractive error correction and, in particular, for treating myopia. PIOLs are surgically implanted in the anterior chamber of the eye and fixated to the iris by haptics. Recent developments in lens manufacturing technologies, design, and surgical procedures have significantly decreased the risk of complications associated with PIOL implantation, and PIOLs are generally considered very safe. In some cases, a decrease in corneal endothelial cell density has been observed after PIOL implantation,
1,2 although this finding is still under debate and some authors have not observed evidence of it.
3
Various mechanisms have been proposed and investigated to explain the possible occurrence of corneal endothelium cell loss in the presence of PIOLs. The surgical procedure itself can cause a decrease in endothelial cell density. Mechanical friction between the endothelium and the PIOL has also been postulated to possibly cause cell detachment in patients with the habit of rubbing their eyes or sleeping face down. Various authors
4–6 have investigated through mathematical models the possible role of changes in the wall shear stress on the cornea due to aqueous humor motion on endothelial cell detachment. The authors have found that implantation of a PIOL does not induce a significant increase of the stress on the cornea and thus this is unlikely to cause cell loss.
Repetto et al.
4 and Davvalo Khongar et al.
7 have postulated that cell loss could be associated with a decrease in nutrient availability at the corneal endothelium in the presence of a PIOL. Since the cornea is an avascular tissue it relies on oxygen and nutrient supply from the surrounding tissues or from the external environment. In the healthy cornea, oxygen is primarily supplied by the surrounding air when the eyelids are open and from blood circulation in the palpebral conjunctiva when the eyelids are closed.
8 On the other hand, glucose primarily diffuses into the cornea from the aqueous humor.
9 The hypothesis put forward in the studies by Repetto et al.
4 and Davvalo Khongar et al.
7 is that the presence of a PIOL may interfere with the delivery of nutrients to the cornea. In particular, when the eyelids are open, owing to the existence of a temperature gradient across the anterior chamber, a buoyancy-driven aqueous flow is generated that quite efficiently allows mixing of the aqueous in the anterior chamber.
10 This flow also exists with a similar intensity in the presence of a PIOL.
4,11 Moreover, daytime eye rotations also generate significant velocities and mixing in the aqueous.
11,12 Thus, in this case the presence of the PIOL is not likely to interfere much with nutrient transport to the cornea. On the other hand, during sleep the eyelids are closed and, when rapid eye movements (REMs) do not occur, the eye globe is approximately fixed. In this case both of the above flows are suppressed and the only mechanism generating fluid flow in the anterior chamber is aqueous production at the ciliary processes. The presence of a PIOL diverts the flow passing through the pupil toward the periphery of the anterior chamber and effectively shields the central cornea. In fact, the flow anterior to the PIOL is very slow in this case
4 and nutrients have to rely mostly on molecular diffusion to reach the central region of the corneal epithelium, which is a very inefficient transport mechanism, on the spatial scale of millimeters. Thus during eyelid closure the cornea might experience nutrient depletion in the presence of a PIOL.
The aim of the present work was to explore this hypothesis. To this end we developed a mathematical model that accounts for aqueous flow in the anterior chamber and reaction-diffusion metabolic processes in the cornea.
Several mathematical models
8,13–16 and experiments
17–21 have been proposed, aimed at describing corneal metabolism. Taking advantage of the fact that the thickness of the cornea is much smaller than its lateral dimension, in most cases one-dimensional models have been adopted in which the transport of all species across the cornea is assumed to be much larger than in the other directions. Chhabra et al.
14 have proposed a coupled reaction-diffusion model for nutrient and oxygen delivery to the cornea, which together with the model proposed by Pinsky,
15 is the basis of the model used in this work. None of the above works have considered the direct effect of aqueous flow in delivering nutrients to the cornea.
As described in the previous section, corneal metabolism relies on the availability of oxygen and nutrients. Proper functioning of the metabolic processes described above is essential for corneal health. For example, accumulation of lactate ions due to lack of oxygen tension leads to edema.
16,25 In this article, following Pinsky,
15 we only considered the following three metabolic species: oxygen, glucose, and lactate.
As discussed in the introduction, when the eyelids are open the thermally driven buoyancy flow and eye rotations quite effectively mix the aqueous in the anterior chamber, smoothing out concentration gradients. This allowed us to avoid solving the problem of aqueous motion and transport in the anterior chamber. Rather, we assumed that concentration of all species is uniformly distributed in the anterior chamber and just imposed the corresponding concentration values as boundary conditions on the endothelium when solving the problem in the cornea.
On the contrary, when the eyelids are closed, aqueous velocities are much smaller and concentration gradients can develop in the anterior chamber. This implies that in order to study metabolic species transport to the cornea in this case, it is necessary to account for aqueous motion in the anterior chamber. This ingredient is essential for understanding the possible role of the presence of a PIOL.
We modeled the aqueous humor as an incompressible, Newtonian fluid. Fluid motion is governed by the Navier-Stokes and continuity equations that read as follows:
\begin{equation}\tag{3a}{{\partial {\bf{u}}} \over {\partial t}} + ({\bf{u}}\,\cdot\,\nabla ){\bf{u}} + {1 \over \rho }\nabla p - \nu {\nabla ^2}{\bf{u}} = {\bf{0}},\end{equation}
\begin{equation}\tag{3b}\nabla \,\cdot\,{\bf{u}} = 0,\end{equation}
where
t is time,
u denotes the velocity vector,
p is the departure of the pressure from the hydrostatic profile (thus we omit the gravitational acceleration),
ρ is the density, and
ν is the kinematic viscosity. The values of the density and viscosity adopted in the simulations are reported below.
Transport of metabolic species in the anterior chamber is governed by the advection/diffusion equation
\begin{equation}\tag{4}{{\partial {c_i}} \over {\partial t}} + ({\bf{u}}\,\cdot\,\nabla ){c_i} - D_i^{aq}{\nabla ^2}{c_i} = 0\qquad (i = O,G,L).\end{equation}
In the above equation subscripts
i =
O,
G, and
L refer to oxygen, glucose, and lactate ion, respectively.
ci and
Display Formula\(D_i^{aq}\) are the concentration and diffusion coefficients of the corresponding metabolic species in aqueous, respectively. In our model the oxygen concentration
cO will be represented as
kO pO where
kO is Henry's constant for oxygen (
kO = 1.283 × 10
−5 mM/Pa) and
pO is the oxygen tension.
Chhabra et al.
14 have proposed a one-dimensional metabolic model for the corneal layers, which is generalized to three dimensions by Pinsky.
15 Transport of oxygen, glucose, and lactate ion in the corneal layers is governed by the following diffusion/reaction equations:
\begin{equation}\tag{5}{{\partial {c_i}} \over {\partial t}} - D_i^{j}{\nabla ^2}{c_i} + mQ_i^{j} = 0\qquad (i = O,G,L).\end{equation}
In the above equations, the index
j refers to the three corneal layers of endothelium, stroma, and epithelium.
Display Formula\(Q_i^{j}(i = O,G,L)\) refers to oxygen, glucose consumption, and lactate production rate in each corneal layer
j, respectively. Moreover,
m is 1 for oxygen and glucose (consumption) and −1 for lactate ion (production).
Display Formula\(D_i^{j}\) refers to the diffusion coefficients of metabolic species
i, for each corneal layer, see
Table 1.
In the previous section we discussed the steady state results of our model. Obviously, for steady state concentrations to be reached a certain time is required, and it is of interest to estimate it. This allows us to verify whether the solutions described in the previous section can be reached in reality. We recall that when eyelids are open it is assumed that the aqueous humor is efficiently mixed by eye rotations and thermal flow. During sleep, these sources of fluid motion are suppressed and fluid motion is generated mainly by production/drainage flow. However, we know that at relatively regular intervals, REMs occur during sleep, which will induce fluid mixing. On average, the first REM phase occurs approximately 90 minutes after falling asleep, during the first sleep cycle. The following REM phases, which are increasingly longer, occur within each sleep cycle, which lasts approximately 1½ hours. Changes of posture during sleep also produce rotations of the eye bulb, which are, however, characterized by low angular velocities and short duration. Thus, they are unlikely to contribute much to fluid mixing.
We assumed that a fully developed production/drainage flow exists in the anterior chamber and we computed the temporal evolution of species' concentrations, imposing values at an initial time corresponding to the open eyelids case. Moreover, we assumed that at the initial time the concentrations of all species in the anterior chamber are uniformly distributed. With these initial conditions we ran unsteady simulations, using the same boundary conditions adopted in the previous section for the case of closed eyelids.
Figure 7 shows the time evolution of the concentration of all species at the anterior chamber–endothelium interface, on the axis of symmetry of the cornea. The red curve corresponds to the case without the PIOL; the green curve, the case in which a PIOL is present; and the blue curve, the case in which the PIOL has a central hole. In this figure we also marked with a vertical dashed black line the time at which the first REM cycle is expected to occur. The figure shows that the time scale of evolution is of the order of some hours: the final steady state is reached in approximately 4 hours. Since the first REM occurs after approximately 90 minutes and successive REMs occur within each sleep cycle, this final steady state is unlikely to be reached in normal conditions.
However, we found that when the first REM phase occurs, the glucose concentration is reduced from 6.9 mM to 4.1 mM (without PIOL), to 2.5 mM (PIOL with hole), and to 1.8 mM (with PIOL).
We proposed a mathematical model of transport of metabolic species to the cornea, considering the role of aqueous flow in the anterior chamber, which has been neglected in previous works. We considered both the case of open eyelids, when aqueous motion is enhanced by eye rotations and the onset of a thermally induced flow, and closed eyelids, in which case aqueous flow is much less intense. We considered both the physiological case and the case in which a PIOL is implanted in the eye, with the aim of understanding if and to what extent the PIOL affects corneal metabolism.
We accounted for the presence of three species (oxygen, glucose, and lactate) and modeled corneal metabolism, adopting the model proposed by Chhabra et al.,
14 Pinsky,
15 and Alvord et al.,
13 which we suitably modified to better reproduce available experimental data. In particular, we focused on glucose availability at the cornea. This is because oxygen is mostly delivered to the cornea from outside of the eye, from the air in the open eyelids condition and from the palpebral circulation in the closed eyelids case. On the other hand, glucose is transported to the cornea by the aqueous humor and its availability at the corneal endothelium is likely to be affected by the presence of an intraocular lens.
Steady state simulations show that the presence of a PIOL has a significant influence on glucose availability on the cornea, in sleep conditions. When a PIOL is implanted, aqueous flow is diverted toward the periphery of the anterior chamber and glucose concentration decreases in the region anterior to the PIOL. However, we showed that several hours are needed for this steady solution to be reached. Since during sleep REMs are expected to occur within every sleep cycle, which occurs every 90 minutes, the steady state solution predicted by our model is unlikely to ever be reached. Rather, the system is constantly in an unsteady state. Glucose concentration anterior to the PIOL progressively decreases until REMs occur; then eye rotations cause mixing of the aqueous humor, and the glucose concentration in front of the PIOL increases again. We found that the minimum value of glucose that is expected to be reached in the presence of a PIOL is approximately half of that corresponding to the nonsurgically altered case, which might be relevant for cell damage (4.1 mM without PIOL and 1.8 mM with PIOL). In the case of a perforated PIOL, glucose and oxygen availability on the cornea increases, compared to the case of a PIOL without a hole, and the lactate concentration correspondingly decreases. It is useful to keep in mind that the duration of the REM depends on age and sleep patterns and may therefore have an influence on glucose availability at the cornea after PIOL implantation. On this basis one could speculate that some people may be at higher risk of corneal endothelial cell damage than others.
Some possibly relevant effects were neglected in the present work, which are worth mentioning. Firstly, we neglected the possible presence of an iridotomy, which is often created when a PIOL is implanted. Iridotomy creates an additional passage from the posterior to the anterior chamber,
6,35 thus modifying fluid flow characteristics in the anterior chamber. We note, however, that iridotomies are typically produced in the periphery of the iris. Thus, the jet of aqueous through the iridotomy enters the anterior chamber close to the trabecular meshwork, where it drains, and is unlikely to significantly contribute to advective transport of nutrients to the central cornea. For this reason we do not think that our conclusions would be qualitatively different, even if the presence of an iridotomy was considered.
We also did not consider the effect of endogenous glycogen in our model. Thus, variations of glucose concentration in the cornea in vivo may not be as strong as predicted by our model.
In summary, we studied the effect of aqueous motion on nutrient transport to the cornea. We compared the physiological case to the case in which an iris-fixated PIOL is implanted in the anterior chamber. Our model suggests that implantation of a PIOL may produce a decrease in nutrient availability in the central region of the cornea during sleep that might be related to the possible loss of endothelial cells. We are unaware of data regarding the physiological threshold of glucose concentration for cell damage, and it must be verified whether our theoretical predictions have clinical relevance.
Supported in part by Ophtec BV (Groningen, The Netherlands).
Disclosure: P. Davvalo Khongar, None; J.O. Pralits, None; X. Cheng, None; P. Pinsky, None; P. Soleri, None; R. Repetto, None