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Vol. 8. Issue 1.
Pages 12-18 (January - March 2015)
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Vol. 8. Issue 1.
Pages 12-18 (January - March 2015)
Original Article
Open Access
Diffusion and Monod kinetics model to determine in vivo human corneal oxygen-consumption rate during soft contact lens wear
Modelo de difusión y cinética Monod para la determinación “in vivo” de la tasa de consumo de oxígeno de la córnea en pacientes con lentes de contacto blandas
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Luis F. Del Castilloa, Ana R. Ferreira da Silvac, Saul I. Hernándezb,1, M. Aguilellad, Andreu Andriod, Sergio Molláe, Vicente Compañe,
Corresponding author
vicommo@ter.upv.es

Corresponding author at: Dpto. Termodinámica Aplicada, Universidad Politécnica de Valencia, 46022 Valencia, Spain.
a Departamento de Polímeros, Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma de México (UNAM), Ciudad Universitaria, Apartado Postal 70-360, Coyoacán, México, DF 04510, Mexico
b Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, Apdo. Postal 55-534, 09340 México, DF, Mexico
c Clinical & Experimental Optometry Research Lab, Center of Physics (Optometry), School of Sciences, University of Minho, Braga, Portugal
d Departamento de Física Aplicada, Universitat Jaume I, 12080 Castellón, Spain
e Departamento de Termodinámica Aplicada, Escuela Técnica Superior de Ingenieros Industriales (ETSII), Universidad Politécnica de Valencia, Campus de Vera s/n, 46020 Valencia, Spain
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Tables (2)
Table 1. Parameters optimized by Chhabra et al.2
Table 2. Values of Qc,max obtained fitting the curves showed in Figs. 1–3 using Eqs. (1) and (3).
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Abstract
Purpose

We present an analysis of the corneal oxygen consumption Qc from non-linear models, using data of oxygen partial pressure or tension (pO2) obtained from in vivo estimation previously reported by other authors.1

Methods

Assuming that the cornea is a single homogeneous layer, the oxygen permeability through the cornea will be the same regardless of the type of lens that is available on it. The obtention of the real value of the maximum oxygen consumption rate Qc,max is very important because this parameter is directly related with the gradient pressure profile into the cornea and moreover, the real corneal oxygen consumption is influenced by both anterior and posterior oxygen fluxes.

Results

Our calculations give different values for the maximum oxygen consumption rate Qc,max, when different oxygen pressure values (high and low pO2) are considered at the interface cornea-tears film.

Conclusion

Present results are relevant for the calculation on the partial pressure of oxygen, available at different depths into the corneal tissue behind contact lenses of different oxygen transmissibility.

Keywords:
Corneal oxygen consumption
Corneal oxygen permeability
Monod kinetics model
Corneal oxygen pressure
Resumen
Objetivo

Presentamos un análisis del consumo de oxígeno de la córnea Qc a partir de modelos no lineales, utilizando los datos de presión parcial del oxígeno (pO2) obtenidos de la estimación in vivo obtenidas previamente por otros autores.1

Métodos

Asumiendo que la córnea es una capa homogénea única, la permeabilidad del oxígeno a través de la misma será la misma, independientemente del tipo de lentes que se utilicen sobre ella. La obtención del valor real de la tasa máxima de consumo de oxígeno Qc,max, es muy importante porque este parámetro está directamente relacionado con el el perfil del gradiente de presión dentro de la córnea y, además, el consumo real de oxígeno de la córnea está influenciado tanto por los flujos de oxígeno anterior como posterior.

Resultados

Nuestros cálculos aportan diferentes valores para la tasa de consumo máximo de oxígeno Qc,max, al considerar diferentes valores de la presión del oxígeno (pO2 elevado y bajo) en la interfase entre la córnea y la película lagrimal.

Conclusión

Estos resultados son relevantes para calcular la presión parcial del oxígeno, que se encuentra en diferentes profundidades dentro del tejido de la córnea por detrás de las lentes de contacto con diferente transmisibilidad al oxígeno.

Palabras clave:
Consumo de oxígeno de la córnea
permeabilidad al oxígeno de la córnea
modelo cinético de Monod
presión del oxígeno en la córnea
Full Text
Introduction

The rate of oxygen consumption in the cornea is an important parameter to guarantee its physiology, and it may be influenced by the use of contact lenses over the cornea. Estimation of tear oxygen pressure or tension (pc) behind hydrogel lenses in humans, using a time-domain phosphorescence measurement system, allowed to obtain the oxygen consumption from established oxygen diffusion models.1 Hovewer, previous papers have calculated oxygen consumption kinetics from transient post-lens tear-film oxygen tension, a method that relies on the simplistic assumption of a constant corneal-consumption rate that leads to negative oxygen tensions in the cornea which lacks physical meaning.2,3 Since oxygen diffusivity and consumption in the human cornea have not been directly measured, some authors, such as Larrea et al.,4 Alvord et al.,5 and Chhabra et al.,2 have proposed mathematical models of time-dependent oxygen diffusion that allows the estimation of corneal consumption and diffusivity. Such authors make use of the nonlinear Monod kinetics model to describe the local oxygen-consumption rate. Nevertheless, although the consumption of oxygen is a result of corneal cell metabolism that depends on a great number of factors, Chhabra et al. assume that the oxygen consumption only depends on the partial pressure of oxygen, and use as parameters Qc,max, (Dk)c and Km. Here Qc,max is the maximum corneal oxygen-consumption rate, kc is the corneal oxygen solubility, and Dc is the corneal oxygen diffusion coefficient. Km is the methabolic or Monod dissociation equilibrium constant, and is a parameter in the Monod kinetic model, which determines the shape of the Qcvs. pc curve, and represents the oxygen pressure when the aerobic metabolism in the cornea reaches the 90% of the maximum oxygen consumption.6,7

The appropriate relationship between oxygen consumption and pc into the cornea should be continuous, yielding a value of zero consumption when pc is zero. Moreover, oxygen consumption should increase with increasing pc until the saturation level is reached. Considering this, we proceeded with the analysis of the oxygen consumption using non-linear models, and also using data from in vivo estimations of partial oxygen pressure at the interface cornea-lens, provided by Bonanno (personal communication).

This work aims to present a single mathematical one-dimensional model of time-dependent oxygen diffusion through the cornea. The experimental data provided by Bonanno et al.1 were used to validate the methabolic model used previously by Chhabra et al.,2 and then to determine the oxygen consumption and metabolic constant Km. For this purpose, similar to Chhabra et al.,2 we fitted the model to three different cases of contact lenses: Balafilcon A, Polymacon1 (60¿m) and Polymacon2 (200¿m). In our calculations the oxygen permeability through the corneal tissue is considered constant, independent of the lens material situated onto the cornea, and the maximum oxygen consumption rate is also independent of the soft contact lens weared.

With the present work we intend to evaluate the impact of consider other values of Qc,max, taking into account that Km=2.2mmHg, which is the value reported by Chhabra et al.2 in the Monod kinetics model. From the values obtained for the parameter Qc,max, the oxygen pressure profile into the cornea has been calculated for the open eye and closed eye conditions. Finally, we have established a comparison with the oxygen tension profiles given by Chhabra et al.2

Methods

The non-steady state diffusion equation that gives oxygen tension as a function of time and position, for homogeneous slab of oxygen-consuming tissue (assuming a one-dimensional model for the cornea), is given by:

where pc(x) is the partial pressure or tension of oxygen into the cornea, Dc is the diffusion coefficient of oxygen in the tissue (cm2/s), kc is the oxygen solubility coefficient in the cornea tissue, i.e. Henry's law constant (cm3 of O2/cm3 layer/mm of Hg), and x is the distance perpendicular to the surface (cm). Qc is the oxygen consumption rate in the cornea (mL of O2/cm3 of tissue layer/s), and t is time (s). In steady-state conditions, Eq. (1) becomes

As we have mentioned, the aerobic metabolism is quantified by the Monod kinetics model, also known as Michaelis–Menton model,5,8 which relates the oxygen consumption with the oxygen tension by mean of the expression,

where Km is the Monod dissociation equilibrium constant to which we have referred above. For low oxygen partial pressure (pcKm), the oxygen consumption rate depends linearly on the oxygen tension, and tends to zero when the oxygen pressure approaches to zero. For large oxygen pressure (pcKm), the consumption also will be dependent on the oxygen tension, and it tends to a maximum value when the pressure is equal to 155mm Hg at sea level, in the case of open eyes. In such situation Qc(pc=155mmHg)=Qc,max.

By mean of the non-linear Monod kinetics model, Chhabra et al.,2 obtain different values for the maximum corneal oxygen-consumption rate Qc,max, depending on the contact lenses weared (see Table 1). The average of the values obtained is Qc,max(ave)=1.05×10−4mLcm−3s−1. This value is 2.34 times higher than the one given by Brennan,7 which is Qc,max=4.48×10−5mLcm−3s−1. Furthermore, Chhabra et al.2 propose a value of 2.2mmHg for the Km constant in Eq. (3) indicating that, in this case, Qc/Qc,max=0.9 when pc=20mmHg, as we have already mentioned. This value of Km has been given taken into account that the oxygen partial pressure for reaching 90% of the saturation oxygen consumption rate for various organism, is in the range of 12–25mmHg.

Table 1.

Parameters optimized by Chhabra et al.2

Lens  Qc,max×10−4 (mL(STp)cm−3s−1Km (mmHg)  Dc×10−5 (cm2/s)  kc×10−5 (mL/cm3/mmHg)  (Dk)c (Barrer)a 
Balafilconb  1.2  2.2  6.2  2.3  140 
Polymacon1c  0.9  2.2  5.9  1.5  90 
a

1 Barrer=10−11(cm2/s)(mL O2 (STp)/cm3/mmHg).

b

100¿m of thickness.

c

60¿m of thickness. All this parameters has been optimized (see Table III in Ref. 2).

From our opinion such value of Km is only valid for the assummed pressure value of 20mmHg in Chhabra's work. If we take the extreme values of 12 or 25mmHg for the oxygen partial pressure (Shoup,9 Amberson,10 Fatt,11 Tang,12 Takahashi et al.13), other values for Km could have been possible. Nevertheless, the Chhabra's value for Km tends to be an estimated average value, and in this way it may be perfectly acceptable. For this reason we have used the value of Km=2.2mmHg obtained by Chhabra et al.,2 in order to be able to stablish a comparison.

Anyway, our greater rejection to the results given by Chhabra et al.,2 is the use of two values for the corneal oxygen permeability, 140 and 90 barrers, when onto the cornea is weared a Balafilcon or a Polymacon lens, respectively (see Table 1). This is clear evidence that the values obtained by Chhabra et al.2 when fitting the experimental data provided by Bonanno et al.,1 should be reviewed. These values are in disagreement with the value of the cornea oxygen permeability used by the researchers during the last 30 years, which are of 24.5 barres or 28.5 barrers.5

Taking into account that the 78% of the cornea is composed basically for water, we have considered the apparent oxygen permeability through the cornea tissue as the value of the oxygen permeability in water at temperature of 35°C. This can be estimated as the product of the oxygen diffusion (DO2(water)=3.0×10−5cm2/s) and the oxygen solubility in water (k=3.1×10−5cm3of O2/cm3mmHg).14 Thus, as a result, we have used the value of 93 Barrers for the cornea oxygen permeability (Dk)c. For this reason, the values of the parameters in Monod kinetic model should be revisited and new fits should be obtained.

Bonnano et al.1 have determined the partial pressure of oxygen pc at the interface cornea-lens, using phosphorescence dye technique in the observation of the variation of oxygen partial pressure as a function of time, from steady state between close eye condition to open eyes condition. The analysis of the experimental transitory, in combination whith Eq. (1), have permitted to Bonanno et al.,1 and Chhabra et al.2 obtain the value of Qc(pc) in different situations. In this paper, following a similar procedure to that of Chhabra et al.,2 we have obtained in vivo human corneal oxygen-consumption rate from the data reported by Bonanno et al.,2 in their measurements of oxygen tension at the postlens-tear film as a function of time. In Appendix A we show the technical procedure followed for solving the partial differential equation (PDE), using FiPy (http://www.ctcms.nist.gov/fipy) which is a finite volume PDE solver written in Python.15

Results

The noninvasive in vivo experimental data provided by Bonanno et al.1 allowed to determine oxygen consumption rate and diffusivity of the human cornea.2,4 Considering the Monod kinetics model, and based on the experimental data given by Bonnano et al.,1 our calculations of the maximum corneal oxygen-consumption rate (Qc,max), are reported in Table 2, for the systems: cornea+Balafilcon lens, cornea+Polymacon1 lens and cornea+Polymacon2 lens.

Table 2.

Values of Qc,max obtained fitting the curves showed in Figs. 1–3 using Eqs. (1) and (3).

Lens  Qc,max×10−4 (mL(STp)cm−3s−1Km (mmHg)  Dc×10−5 (cm2/s)  Kc×10−5 (mL/cm3/mmHg)  (Dk)c (Barrer) 
Balafilcona  1.4  2.2  3.0  3.1  93 
Polymacon1b  0.9  2.2  3.0  3.1  93 
Polymacon2c  0.7  2.2  3.0  3.1  93 
a

100¿m of thickness.

b

60¿m of thickness.

c

200¿m thickness. In this work we have only optimized of Qc,max parameter. The rest or parameters have been taken from literature, as we point out through the text.

In Fig. 1 left we plot the postlens tear-film oxygen tension as a function of time for Balafilcon lens at 35°C, using the reactive diffusion model described and used in the work by Chhabra et al.2 In our calculations the cornea is assummed as a single homogeneous layer where the oxygen consumption rate represents an average of the oxygen consumption of three main layers (endothelium, stroma and epithelium). The lens is considered as a separated phase without oxygen consumption surrounded by two thin films of tears (prelens and postlens-tear films), where the resistence to the oxygen flux can be consider negligible because of their thickness (5–15¿m), in comparisson with the lens thickness (60 and 200¿m).16

Figure 1.

(Left) Representative results for the tear-film oxygen tension after 5min of close eyes (CE) lens wear and (right): the steady state oxygen tension profile through cornea thickness. Red color is the fit to Bonanno data proposed by Chhabra et al.2 Black color corresponds to our best fit to same data. Symbols are from Bonanno et al.,1 when the system is composed by cornea+Balafilcon lens.

(0.16MB).

A closer inspection of Fig. 1 left shows that the values of the parameters found by Chhabra et al. (Table 1) to fit the experimental data of postlens tear-film oxygen tension as a function of time at the interface corneal lens (red line), on wearing Balafilcon lens from Bonanno et al.1 data, appears as a best fit compared to the one achieved by us (black line), with parameters in Table 2 for both, open and closed eyes conditions. This is clearly related to the value of oxygen corneal oxygen permeability considered by Chhabra et al.2 (140 Barrer considered by them, instead of the 93 Barrer considered by us). However, in the case of cornea-Polymacon1 lens and cornea-Polymacon2 lens systems, the behavior of our calculated curve is arguably similar than Chhabra's curve, as can be seen in Figs. 2 and 3.

Figure 2.

(Left) Representative results for the tear-film oxygen tension after 5min of CE lens wear and (right): the steady state oxygen tension profile through cornea thickness. Red color is the fit to Bonanno data proposed by Chhabra et al.2 Black color corresponds to our best fit to same data. Symbols are from Bonanno et al.,1 when the system is composed by cornea+Polymacon1 lens.

(0.16MB).
Figure 3.

(Left) Representative results for the tear-film oxygen tension after 5min of CE lens wear and (right): the steady state oxygen tension profile through cornea thickness. Red color is the fit to Bonanno data proposed by Chhabra et al.2 Black color corresponds to our best fit to same data. Symbols are from Bonanno et al.,1 when the system is composed of cornea+Polymacon2 lens.

(0.14MB).
Discussion

We have fitted experimental in vivo postlens tear-film oxygen tension data as a function of time at tear-film temperature (35°C), on wearing Polymacon1 and Polymacon2 lenses from Bonanno et al.1 The only difference between Polymacon1 and Polymacon2 is the thickness, which are 60¿m and 200¿m, respectively. From Figs. 2 and 3, we can see that our best fits are at least similar to the fits of Chhabra et al.2 And, as can be seen from the values given in Table 1, the only parameter which differs from adjust performed by them is exclusively the value of the corneal oxygen permeability, which in our case we kept constant and approximately equal to that of the water oxygen permeability (93 Barrer), for all systems cornea-lens analyzed.

On the other hand, Fig. 3 shows the fit of Bonanno's data with the same parameters obtained by Chhabra et al.2 for Polymacon2-lens (Qc,max=0.9×10−4mL(STp)cm−3s−1; (Dk)c=90 Barrer and Km=2.2mmHg), but here with a thickness of 200¿m. However, it can be seen that our parameters (using the same value of the corneal oxygen permeability that in the other systems cornea-lens), allowed us to obtain a good fitting to the experimental data, taking the values of the parameters, Km and Qc,max in the case of polymacon of 200¿m of thickness, Qc,max=0.7×10−4(mL(STp)cm−3s−1), (Dk)c=93 Barrer and Km=2.2mmHg.

As can be seen, the metabolic model (Michaelis–Menton model), with our parameters successfully reproduces experimental results for transient oxygen tension during closed-eyes contact lens wear and steady state oxygen tension over several lens transmissibilities. The values of our parameters, while fitting the data by Bonanno, provides good results, and the best fits are obtained for Monod dissociation equilibrium constant Km, and corneal oxygen permeability constant for all systems analyzed. Thus, for a given lens on the cornea, our results reproduce individual experiments in an acceptable manner, maintaining constant the values of the parameters Km and (Dk)c. However, the maximum oxygen consumption rate diminishes when the oxygen tension at the interface cornea-lens diminish, contrary to what was expected (see Table 2). As occurs in other models, these results may be subject to certain limitations, like the uncertainties in experimental data, especially at high oxygen tensions and this could constitute as an intrinsic limitation of the model itself.

Considering the limitations of the model to explain the rate of change of the experimental data, which does not correspond to the tendency of the value reported by Bonanno et al.1 at moderate and high pressures (p100mmHg), where the maximum oxygen consumption rate should be constant independent of the lens wear onto the cornea, it is suggested the occurrence of a kinetic transition that should be assumed as continuous. This kinetics transition can be understood as a consequence of the existence of other effects into the cornea than those referred in the metabolic reactions that occur in the Krebs cycle. Bear in mind that, in the range from low to moderate other phenomena such as corneal swelling can occur. It should be noted that, when it has many parameters in an analysis of experimental data the physical meaning of the values obtained must be taken into account with caution. Particularly, in Chhabra et al., the value of the permeability to oxygen in cornea gives rise to an inappropriate value, which is remedied in our case when this value is given and consequently reducing the fitting parameters.

Conclusion

In this paper we present a procedure for solving the non-linear partial differential equation for the position and time depending pressure pc(x,t), for the oxygen diffusion model of the human cornea, which is an alternative solution respect to Chhabra's work. In this sense, the novelty of the results obtained here, consists in provide, previous to the solution of the model, the values of diffusion coeffcient Dc and solubility kc. Therefore, the only fitted value is the corneal oxygen-consumption rate Qc,max. Despite this limitation the present work shows a revision of the procedure described before by Chhabra et al.,2 using data previously obtained by Bonanno et al.,1 to determine the parameters Km and Qc,max by mean of the Monod kinetics model of oxygen difussion.

As can be seen, the metabolic model (Michaelis–Menton model), with our parameters, successfully reproduces experimental results for transient oxygen tension during closed-eyes contact lens wear and steady state oxygen tension over several lens transmissibilities. Our results reproduce individual experiments in an acceptable manner, maintaining constant the values of the parameters Km and (Dk)c. Moreover our main finding is that the maximum oxygen consumption rate is not a constant, but diminishes when the oxygen tension at the interface cornea-lens diminish.

Conflicts of interest

The authors have no conflicts of interest to declare.

Acknowledgements

We thank to Ministerio de Educación y Ciencia (MEC) of Spain for financial support through project ENE2011-24761. LFdC is grateful to SEP-CONACYT for financial support through Grant 154626 and project UNAM-DGAPA IN-102512. SIH is grateful to Red Temática de la Materia Condensada Blanda-CONACYT for a postdoctoral fellowship.

Appendix A

The general equation describing oxygen transport through the lens-corneal system, in one dimension, is Fick's second law with a reaction term,

where pc is the oxygen partial pressure in the lens-cornea system, t is time and x is the coordinate normal to the cornea, with x=0 at the interface between the anterior chamber and the cornea.

The second term on the right-hand side in Eq. (A) is the oxygen consumption as a function of the partial pressure, which is absent in the contact lens region and follows a Monod kinetics form in the corneal system:

In Eq. (A), the solubility (k) and diffusion coefficient (D) are considered function of the position, taking constant values across each of the two regions (contact lens and cornea) in the system. Using the above approach we could obtain the complete pressure profile, provided the continuity of the pressure is satisfied in the lens–cornea interface. This is automatically satisfied within our numerical scheme.

We choose standard Dirichlet boundary conditions in the spatial coordinate:

where Pair is the open eye pressure, corresponding to the atmospheric pressure, and Pac is the oxygen pressure in the anterior chamber (aqueous humor).

As for the initial condition, in order to reproduce the evolution of the pressure profile from the closed eye condition, we need to feed the stationary pressure profile in Eq. (A). This stationary closed eye profile can be obtained by solving the steady-state equation:

which is obtained from Eq. (A), by removing the temporal evolution. Eq. (D) is subject to the boundary conditions:
where PPC is the contact-lens/palpebral conjunctiva oxygen pressure, equivalent to 61.5mmHg, similar to data used by Chhabra et al.

We then use the solution to Eqs. (D) and (E) to define:

as the last boundary condition for Eq. (A).

The system of Eqs. (D) and (E) and (A)–(C) and (F) are solved using FiPy (Python Software Foundation), a finite volume PDE solver using Python. Table 1 shows the different values for the parameters used in the numerical solution of the equations. We use a spatial grid with 103 points in all computations, and time steps of 10−1s for the time-dependent equations.

Eqs. (D) and (E) are solved numerically, and the resulting profile is used as initial condition for Eqs. (A–C) and (F). An iterative procedure was used due to the nonlinear nature of the transport Eqs. (A)–(F), by “sweeping” the solutions over few iterations (see FiPy manual for details). Convergence was reached after the residual was below a predefined value (10−11 in our case). We checked both, grid size and time step parameters, so that further decrease in size did not result in any improvement. All the computations were performed in a personal computer with an Intel Core i7-3770K under Debian Linux. FiPy version 3.0 was used in all computations.

Multidimensional parameter optimization subject to bounds was done through the “fmin_tnc” function in the Scipy package, which uses a Newton conjugate-gradient method. We used this optimization procedure to determine optimized values of the Qc,max and Km parameters, for a predefined set of the remaining parameters in the model.

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Current address: Unidad Multidiscliplinaria de Docencia e Investigación-Juriquilla, Facultad de Ciencias, Universidad Nacional Autónoma de México (UNAM), CP 76230, Juriquilla, Querétaro, México.

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