purpose. Oxygen diffusivity and consumption in the human cornea have not been directly measured yet; current models rely on properties measured in vitro in rabbit corneas. The aim of this study was to present a mathematical model of time-dependent oxygen diffusion that permits the estimation of corneal consumption and diffusivity.

methods. The current oxygen diffusion model was extended to include the temporal domain and was used to simulate in vivo noninvasive measurements of tear oxygen tension in human corneas.

results. The new model reproduced experimental data successfully, provided values for corneal diffusivity and consumption, and described the relationship between oxygen consumption and oxygen tension in the cornea. Estimated values were three times higher than those reported previously in in vitro rabbit experiments.

conclusions. This model allowed for the further investigation of oxygen transport in the cornea, including a better mathematical description and a determination of the transport properties of the cornea and the specific oxygen uptake rate of the tissue. The combination of this model and tear oxygen tension measurements can be useful in determining the individual oxygen uptake rate and exploring the relationship between oxygen transport and corneal abnormalities.

^{ 1 }

^{ 2 }The cornea is an avascular tissue whose health depends on oxygen transport through its atmospheric boundary, but contact lenses provide additional resistance to this process.

^{ 3 }Further, the oxygen transmissibility (

*Dk/t*) of the lens is insufficient to evaluate the changes in oxygen supply to the cornea under contact lens wear because the relationship between these two variables is not linear.

^{ 4 }Other terms derived from mathematical models, such as the equivalent oxygen pressure (EOP) and biological apparent oxygen transmissibility (BOAT), have been introduced to better assess the performance of contact lenses with regard to the physiological changes in oxygen supply they induce.

^{ 5 }

^{ 6 }However, these measures rely heavily on the accuracy of the mathematical models and tissue properties used.

^{ 5 }

^{ 7 }

^{ 8 }

^{ 9 }

^{ 10 }Since its introduction, this model has seen some improvements by different researchers

^{ 11 }

^{ 12 }and has been extended to a two-dimensional model to determine oxygen transport through and around intracorneal lenses.

^{ 13 }Its main limitation, though, is its use of tissue properties derived from in vitro experiments on rabbit corneas. This limitation arises because of the experimental difficulty in obtaining analogous values for in vivo human corneas.

^{ 3 }published some novel measurements of tear film oxygen tension during contact lens wear on human eyes in vivo. Their measurements at the cornea-contact lens interface provide abundant and useful data to characterize oxygen diffusion in the cornea. The concentration underneath the contact lens depends not only on lens transmissibility but on corneal tissue properties (oxygen diffusivity and metabolic consumption of oxygen). Their experiments give open-eye (OE) and closed-eye (CE) steady state oxygen tension at the lens-cornea interface and the transient response of oxygen tension at this location. To make best use of this experimental data, the current diffusion models must be extended, making the diffusion equation time dependent and not steady state.

^{ 3 }were used to validate the model and to determine the oxygen consumption and diffusivity of the cornea by fitting the model to three different cases of contact lens transmissibility. This model, in conjunction with tear film oxygen tension measurement, could then be used for multiple purposes, such as patient-specific estimation of corneal oxygen uptake, comparison of contact lens performance, or study of oxygen transport in the cornea and its relation to different corneal abnormalities.

_{2}× k, which leads to the established equation for oxygen transport in the cornea (equation 2) , where

*D*is the diffusivity (mm

^{2}/s),

*x*is the position (mm), Po

_{2}is the partial pressure of oxygen (mm Hg),

*Q*is the consumption of oxygen (nmol/mm

^{3}· s),

*k*is the Henry’s solubility constant (nmol/mm

^{3}· mm Hg), and

*t*is time (s).

_{2}is zero, and yield increasing consumption with increasing Po

_{2}until the saturation level is reached. A Michaelis-Menten–type relationship for oxygen uptake kinetics was used for this purpose (equation 3 ; see 1 2 3 4 Fig. 5 ), where

*Q** is the oxygen consumption rate at the saturated oxygen tension

*p** (in this case, 155 mm Hg) and

*a*is a constant that determines the shape of the

*Q*vs. Po

_{2}curve.

^{ 5 }

^{ 7 }

^{ 8 }:

*Dk*= 99 × 10

^{−11}cm

^{2}· mlO

_{2}/s · mL · mm Hg;

*t*= 0.1 mm), a thin hydrogel lens (

*Dk*= 8.4 × 10

^{−11}cm

^{2}· mL O

_{2}/s · mL · mm Hg;

*t*= 0.06 mm), and a thick hydrogel lens (

*Dk*= 8.4 × 10

^{−11}cm

^{2}· mL O

_{2}/s · mL · mm Hg;

*t*= 0.2 mm).

*pdepe*in MATLAB (The MathWorks Inc., Natick, MA) was used to solve the diffusion equation. Bonnano’s experiments were simulated with the mathematical model by changing the boundary

^{3}condition at the anterior surface from CE (45 mm Hg) to OE (155 mm Hg) after 5 minutes of lens wear with the eye closed; during this time the CE steady state condition was reached. An example of the transient response of the whole simulation is shown in Figure 1 .

*Q*/k*,

*D*, and

*a*that fit best the data of Bonanno et al. This process minimized the difference between the tear film Po

_{2}of the transient model and the exponential fit proposed by Bonanno.

^{3}

*Q*/k*= 1.87 mm Hg/s,

*D*= 2.81 × 10

^{−3}mm

^{2}/s, and

*a*= 20. The calculated steady state Po

_{2}at the tear film was within the SD of the experiments by Bonanno et al.

^{ 3 }(Fig. 2) , and the model followed the experimental data closely for each contact lens.

*Q** and

*Dk*in converted units are reported in Table 1for direct comparison with the experimental results from Fatt et al.

^{ 10 }Oxygen solubility in water at 35°C of

*k*= 1.0268 · 10

^{−2}μmol/mL · kPa was used for this purpose.

^{ 14 }

_{2}(approximately 7 seconds) with the thick hydrogel after eye opening (Fig. 3C)and the time delay needed to reach steady state (Fig. 3) .

*IOVS*2008;49:ARVO E-Abstract 4846). The transient model presented here agrees with the new OE and CE steady state oxygen tension values over several lens transmissibilities. Unfortunately, the overall transient behavior could not be compared because these data were not explicitly presented

^{ 10 }

^{ 11 }

^{ 12 }

^{ 13 }The noninvasive in vivo experimental measures by Bonanno

^{ 3 }provide sufficient data to determine oxygen consumption and diffusivity of the human cornea. Although the relative merits of diffusion and tear oxygen experiments are debatable, the preference to acquire human in vivo data rather than rabbit in vitro data would seem under dispute. If the tissue properties from previous models are used, the oxygen tension is overestimated at the tears for CE and OE conditions. The initial time response is also faster, yet more time is needed to reach steady state (Fig. 4A) . Nevertheless, only small differences can be observed in the steady state oxygen distribution through the cornea (Fig. 4B) . The threefold increase in oxygen consumption has, therefore, little influence on the resultant oxygen distribution except at the epithelium and the anterior stroma. The calculated diffusivity is also approximately three times higher; hence, more oxygen is consumed, but it is also transported more quickly, leading to similar oxygen tension distribution. The oxygen supply from the aqueous humor explains why the difference is smaller at the endothelium and the posterior stroma.

*IOVS*2008;49:ARVO E-Abstract 4846),

^{ 3 }transient oxygen tension measures at the tears have the potential to provide accurate data for these models.

^{ 8 }and Freeman and Fatt

^{ 9 }previously measured an important difference in oxygen diffusivity and consumption between the layers of the rabbit cornea. In the present study, the assumption was made that the consumption/diffusivity ratio between the layers was the same as what had been measured in rabbits (equations 4and 5 ). This assumption was made for two reasons. First, calculating the “average” diffusivity and consumption for the whole cornea would discard the layer information, and the model would then be inadequate to predict the oxygen tension distribution in the cornea. Second, it is not possible to calculate the diffusivity and consumption of each layer based on tear oxygen tension. Intracorneal measurements would be necessary for this purpose.

^{ 10 }

^{ 11 }

^{ 12 }

^{ 15 }An appropriate relationship should give zero consumption at zero Po

_{2}and should increase

*Q*as Po

_{2}increases. In an initial optimization of the model, the consumption law was not taken into account. Only

*Q/k*and

*D*were optimized, and the Michaelis-Menten constant,

*a*, was set constant. This model reproduced individual experiments well (for a given lens), but it could not reach a good solution for the three lenses simultaneously. When the consumption law was included in the optimization loop, the model converged to a successful solution. By this iteration, we found that the oxygen consumption law was of key importance in reaching the solution and was a fundamental part of the oxygen transport model in the cornea. The resultant curve (Fig. 5)was close to the logarithmic relationship between oxygen flux to the cornea and postlens Po

_{2}proposed by Brennan.

^{ 16 }A similar relationship was found by Bibby et al.

^{ 17 }when they measured intervertebral disc (IVD) cell metabolism. The IVD is the largest avascular tissue in the human body; its nutrient transport and oxygen demands are similar to those of the cornea.

^{ 18 }found that corneal oxygen consumption increases under acute pH reductions in rabbit corneas in vitro. Although other researchers have reported decreased corneal oxygen uptake after prolonged use of contact lenses,

^{ 19 }

^{ 20 }this may correspond to a long-term adaptive change in cellular metabolism induced by an altered oxygen distribution. To develop the existing oxygen transport models, it is important to improve our understanding on how corneal cell metabolism behaves under the extreme environment found in the cornea during contact lens use.

**Figure 5.**

**Figure 5.**

**Figure 1.**

**Figure 1.**

**Figure 2.**

**Figure 2.**

Transient Model (human) | Fatt et al.^{ 10 }(rabbit) | |
---|---|---|

Q*_{stroma} | 5.75 · 10^{−5} \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(\frac{ml\ O_{2}}{ml\ {\cdot}\ s}\) \end{document} | 2.24 · 10^{−5} \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(\frac{ml\ O_{2}}{ml\ {\cdot}\ s}\) \end{document} |

Dk _{stroma} | 8.62 · 10^{−10} \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(\frac{cm^{2}\ {\cdot}\ ml\ O_{2}}{s\ {\cdot}\ ml\ {\cdot}\ mmHg}\) \end{document} | 3.0 · 10^{−10} \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(\frac{cm^{2}\ {\cdot}\ ml\ O_{2}}{s\ {\cdot}\ ml\ {\cdot}\ mmHg}\) \end{document} |

^{ 10 }Values for epithelium and endothelium can be easily calculated from equations 4and 5 .

**Figure 3.**

**Figure 3.**

**Figure 4.**

**Figure 4.**