BUIJLETII¢ OF

IV£ATHEMATICA]5 BIOLOGY

VOLUME 38, 1976

OXYGEN TENSION IN THE I N V I V O CORNEA

[] S. H. LIN Department of Chemical Engineering, University of Melbourne, Parkville, Victoria 3052, Australia

The s t eady s t a t e d i s t r i b u t i o n of oxygen t ens ion in t h e in vivo cornea is e s t i m a t e d in t he present s t u d y b y us ing a non l inea r oxygen c o n s u m p t i o n r a t e e q u a t i o n of t he Michae l i s - Menten type . Such a r a t e express ion is more accu ra t e t h a n t he p rev ious s implif ied vers ions in p red ic t ing t h e oxygen c o n s u m p t i o n ra te . I t is f o u n d t h a t for a n open eye w i t h or wi thou t con t ac t lens, t h e oxygen tens ions p r e d i c t e d p rev ious ly are in good ag reemen t wi th these p r ed i c t ed in t h e p r e s e n t work. However , for a closed eye w i t h or w i t h o u t contact lens, t h e p rev ious p red ic t ions u n d e r e s t i m a t e t he oxygen tens ion .

1. Introduction. The mammalian cornea, although rather complex in structure, can be considered as a composite tissue consisting of three layers, viz endo- thelium, stroma and epithelium. The endothelium is about 0.001 cm thick, the stroma 0.045 cm, and the epithelium 0.004 cm (Takahashi et al., 1967). Due to their thinness, it is extremely difficult to measure experimentally the oxygen tension in the corneal tissue. Fortunately, the physical parameters that govern the oxygen diffusion are known with enough certainty so tha t the steady state distribution of oxygen tension can be estimated fairly accurately.

Takahashi et al. (1967), Fa t t and Bieber (1968), Fa t t (1968), and F a t t e t al.

(1974) calculated the steady state oxygen tension using a simplified oxygen consumption rate expression. The physical data required for the prediction of oxygen tension include the diffusion coefficient, solubilities of oxygen in the corneal tissue, oxygen consumption rate and oxygen tensions on the anterior and posterior surfaces of the in vivo cornea. Takahashi et al. (1967), Fa t t and Bieber (1968) and Fa t t (1968) estimated the oxygen tension in the cornea of an open or closed eye by assuming a constant oxygen consumption rate.

269

270 S. I t . LIN

However, as evidenced by several investigators (Amberson, 1928; Blum, 1960; Presser and Brown, 1962; Takahashi et al., 1968), the oxygen consumption rate h~s been found to be a function of oxygen tension. This fact leads Fa t t (1968) and Fat te t al. (1974) to consider a sectioned oxygen consumption rate expression in predicting the oxygen tension in an eye covered by contact lens. They assumed that the oxygen consumption rate takes a constant value when the oxygen tension is above a certain critical value (20 mm Hg for this specific case) and is a linear function of the oxygen tension if it is below that critical value. Although such as oxygen consumption rate is better than a constant oxygen consumption rate in predicting the oxygen tension in the corneal tissue, it still can not display the true characteristics of the oxygen consumption rate. The oxygen consumption in the in vivo cornea, like that in many other bio- logical systems, exhibits nonlinear characteristics (Amberson, 1928; Blum, 1960; Presser and Brown, 1962). The purpose of this work is to a t tempt to use a more rigorous oxygen consumption rate to estimate the oxygen tension in the corneal tissue. This rate expression of Michaelis-Menten type predicts fairly well the oxygen consumption in cells and tissue of many biological systems (Amberson: 1928; Walshe, 1948 ; Presser and Brown, 1962). Hence, it is believed that a more accurate estimate of oxygen tension can be obtained.

2. Theoretical Formulation. The oxygen consumption due to the metabolic reactions in the corneal tissue is assumed to be represented by the following Michaelis-Menten equation

Oxygen consumption = FP/(P + kin), (I)

where F is the maximum reaction rate, P the oxygen tension, and km the Michaelis-Menten constant. Equation (1) with /~m = 4.834 mm Hg has been shown (Lin, 1975) to predict fairly welt the experimental observations of Amberson (1928) and was adopted in the present work as the oxygen con- sumption rate equation.

The physical parameters in each layer of the corneal tissue have been found to be different (Takahashi et al., 1967), hence, separate diffusion equations must be written for describing the oxygen tension in each layer. The steady state diffusion equation of oxygen can be represented by

d2Pi VIPi D ~ k ~ - - - - = 0 . (2)

dx 2 Pi + lemi

The parameters with subscript i = 1, 2, 3, respectively, represent the properties of the endothelium, stroma and epithelium. The above equation represents the oxygen tension in the cornea for an open or closed eye without contact lens

OXYGEN TEI~SION IN THE IN VIVO CORNEA 271

I f the cornea is covered by an oxygen-permeable contac t lens, an addit ional equation is needed for represent ing the oxygen diffusion in the contact lens

d2P4 D4/c4 ~ = 0. (3)

dx 2

Equat ion (3) does not have oxygen consumpt ion t e rm since no oxygen is consumed in the contact lens. The bounda ry conditions for the above equat ions are given by

P1 = Pa, for x = 0 (4)

Pz = P2, D1/Cl dP1 _ D2k2 dP2 dx "-~-x ' for x = xz (5)

P2 = Pa, D~k2 dP2 = Dak3 dPa d--~ --~--, for x -- x2 (6)

P3 = Pb, for x = x3. (,7)

For an open or closed eye wi th contac t lens, (7) is to be replaced by

dP4 P3 = Pa, D3ka dP-~xa = Daka -~x ' for x = xa (8)

P4 = Pb, for x = Xd. (9)

In the above equations, x is the distance measured f rom the interface between the aqueous humor and the endothel ium. Equat ions (5), (6) and (8) represent the cont inui ty of the oxygen tension at the interface between two layers. The physical parameters , D~k~, Pa, Pb, and x~, were given b y F a t t and Bieber (1968) and have recent ly been upda ted by F a t t e t al. (1974). The Michaelis- Menten constants in each layer are assumed to be the same, i.e. ]~ml = kin2

= km3 = ]Cm because of lack of exper imenta l data. The m ax im u m oxygen consumption rates, Vi, can be de termined f rom the m ax im u m oxygen con- sumption rates, Qi. I t can be no ted t h a t the oxygen consumption ra te as given by (1), is more critical to the m a x i m u m oxygen consumption rates, V,, t h a n to the Michaelis-Menten constants, kmi, because the oxygen tension P is general ly greater t ha n km for all the cases under consideration and var ia t ion of km does not exer t as much influence on Q as does the var ia t ion of V~. Therefore, the est imated lcm value is much less accurate than the V~ values. All the physical parameters are listed in Table I.

I t is no ted t ha t several cases were considered b y F a t t and Bieber (1968), Fa t t (1968) and F a t t et al. (1974). They es t imated the oxygen tensions for open and closed eyes wi th and wi thout contac t lens. These different si tuations determine the numerical values of Po in (7) or (9). F o r example, Pb assumes a

272 S. II. LIN

value of 155 m m Hg for an open eye and 55 m m Hg for a closed eye. All these cases are considered here and the results are compared with previous predictions using simplified oxygen consumpt ion ra te equation. I t is fur ther no ted tha t the oxygen tension at the interface between the aqueous humor and the endothel ium might be affected by the contac t lens. However , due to lack of experimental evidence, it was assumed to be constant (Fat t , 1968 ; F a t t et al., 1974) and the same assumption was made in the present work.

T A B L E I

Numerica l Values of Physica l Pa ramete r s

Symbol Explanation Numerical Value

Endothclium Stroma Epithdium

Dk oxygen permeability 0.53 x 10 -20 3.0 x 10 -lo 1.88 x 10 -10 ml(O2)-cm2/ml-mm Hg- s e c

x thickness, cm 0.001 0.045 0.004

Q oxygen consumption rate 11.0 X 10 -4 1.5 x 10-Tp rain Hg 2.1 x 10 -4 ml(O2)/ml-sec

Km Miehaelis-Menten 4.834 4.834 4.834 constant ram I-Ig

Because of the nonl inear i ty of the oxygen consumpt ion term, (2) is hot amenable to analyt ical solution. An i tera t ive R u n g e - K u t t a m e th o d (Lapidus, 1962) was employed to in tegrate numer ica l ly the equation. According this i terat ive scheme, an a rb i t r a ry dP1/dx at x = 0 is assumed and (2) integrated by the R u n g e - K u t t a method. I f (7) or (9) is satisfied, the assumed dP1/dx is correct. Otherwise, an improved value can be obta ined b y the reguli-fald method (Lapidus, 1962). This procedure is repea ted unt i l P a a t x = xa (or P4 at x = x4) deviates f rom Pb less t han a prescribed tolerance (10 -4 for the present work). Depending on the initial guess of dP1/dx, five to eight iterations are sufficient to yield ve ry sat isfactory results for all the cases under con- sideration.

3. Results. The oxygen tension dis t r ibut ion of an open eye to the air is shown in Figure 1. Also shown in this figure is the predic ted oxygen tension using a simplified oxygen consumpt ion ra te equat ion (Fa t t et al., 1974). Al though Fat t et al. (1974) predicted a sl ightly lower oxygen tension t h an the present one, the

O X Y G E N T E N S I O N I N T E E I X V I V O C O R N E A 273

difference between them is less than 5 per cent. For this particular case, the oxygen tension is fairly high, being between 55 mm Hg and 155 mm Hg, and for this high oxygen tension, a simplified oxygen consumption rate equation yield a good approximation to the nonlinear one as given by (1). As noted in the governing equation (2), the simplified oxygen consumption rate expression significantly simplifies the problem and analytical solution can be obtained (Fall, 1968; F a t t e t al., 1974).

I I I l

140

120

E I00 E

= 8 0

Endothelium

Slroma

Epi|h(slium ~

4ekP--.~_ (b)

Figure 1.

L I ..

o J I r I I o o . , o . a o . 3 o . a 0 .5

Distance from aqueous humor, mm

O x y g e n t e n s i o n p rof i l e s in (a) a n o p e n e y e a n d (b) c l o sed eye . - - P r e s e n t p r e d i c t i o n . . . . . p r e d i c t e d b y F a t t et al. (1974)

The two lower curves in Figure 1 display the oxygen tension distributions for a closed eye. As seen in this figure, the agreement between the present and previous predictions is very good for x < 0.02 cm; however, for x > 0.02 cm, the difference becomes rather appreciable. A maximum difference of 8.35 per cent is reached at the interface between the stroma and epithelium. This marked difference between the two predictions is not unexpected because as the oxygen tension becomes less than 55 mm Hg, the difference of oxygen consumptions predicted by the previous and present rate equations increases.

The oxygen tension distributions for open and closed eyes with oxygen- permeable contact lens are demonstrated in Figures 2 and 3, respectively. The

274 S .H. LIN

two upper curves in Figure 2 show the oxygen tension profiles a t a high oxygen permeabi l i ty of contac t lens of 500 × 10 -11 ml(02)-cm2/ml-mmI-Ig-sec, while the two lower ones show these a t a lower oxygen pe rmeab i l i t y of 13.1 × 10-11

1 6 o I I I I . - -

140 Stroma / t

,20 ~ r / ' ~ 7 I 7 y'/Y f

I00

E - 8 0 i

- - 6 0

$ 4.0 Contact I I J --

lens

Endathe l lum Ep~the l i~ l l [

o I P f I i o o.~ 0.2 0.3 o.4 0.5 o.6

Distance f r o m ~ u e ~ J s humor, mm

Figure 2. Oxygen t en s i on profiles in open eye w i t h c o n t a c t lens. The oxygen p e r m e a b i l i t y of t he c o n t a c t lens is (a) 500 x 10-11 a n d (b) 13.1 x 10 -11

ml (02)-em2/ml ( t i s sue) -mm Hg-sec.

5 0 S t r o m a

i ,o ,o, _ _ . . S - ¢

} I " ~ Contact , , ~' I

3o \ i , . , ~ / / d

~ I I// [

I 0 I .,~

O" I 0 - 2 @ 3 0 " 4 0 " 5 O'S

Distance from aqueous humor, mm

:Figure 3. Oxygen t en s i on profiles in a closed eye w i t h con t ac t lens. The oxygen p e r m e a b i l i t y of t he con t ac t lens is (a) 500 x 10 -11 a n d (b) 13.1 x 10 -11

ml (O2)-cm2/ml ( t i s sue) -mm Hg-see.

OXYGEN TENSION IN THE I1V VIVO CORNEA 275

ml(02)-cm2/ml-mm Hg-sec. I t is seen t h a t a t high oxygen pe rmeab i l i ty of

contact lens, the two predict ions do not show m u c h difference in oxygen tension; however , a t lower oxygen permeabi l i ty , the difference be tween t h e m becomes ra the r significant. A t the interface be tween the contac t lens and epithelium, F a t t et al. (1974) predic ted a lower oxygen tension of 16.2 per cent when compared to the present predict ion.

Figure 3 shows the compar ison of oxygen tensions pred ic ted b y F a t t et al.

(1974) and the present work for a closed eye wi th contac t lens. A t a high oxygen permeabi l i ty of contac t lens, a m a x i m u m difference of 7.4 per cent be tween these two predict ions is reached a t the in terface be tween the s t r oma and epi thel ium. As the oxygen pe rmeab i l i t y of con tac t lens is reduced to 13.1 × 10 -11 ml(02)- cm2/m]-mm Hg-see, the oxygen tension of the present work is more t h a n twice tha t predic ted b y F a t t e t al. (1974) a t a dis tance of 0.047 cm f rom the aqueous humor. Fo r this par t i cu la r case, the oxygen tension in the cornea is fa i r ly low. Such a lower oxygen tension generates a significant difference in oxygen consumptions calcula ted f rom the ra te equat ions used b y F a t t e t al. (1974) and the present work. This in t u rn influences s t rongly the oxygen tension predictions, as ant ic ipated.

The au thor is indeb ted to Dr. I . F a t t of the Univers i ty of California (Berkeley) and Dr. J . J . B lum of D u k e Unive r s i ty for thei r cons t ruc t ive suggestions.

L I T E R A T U R E

Amberson, W. 1%. 1928. "The Influence of 02 Tension on the Respiration of Unicellular Organisms." Biol. Bull., 55, 79-85.

Blum, J. J. 1960. "Concentration Profiles in and Around Capillaries." Am. J. Physiol., 198, 991-998.

Fa~t, I. and M. I. Bieber. 1968. "The Steady State Distribution of Oxygen and Carbon Dioxide in the in vivo Cornea. I. The Open Eye in Air and the Closed Eye." Exptl. Eye Res., 7, 103-112.

Fatt, I. 1968. "The Steady State Distribution of Oxygen and Carbon Dioxide in the in vivo Cornea. II . The Open Eye in Nitrogen and the Covered Eye." Exptl. Eye Res., 7, 413-430.

Fatt, I., 1~. D. Freeman and D. Lin. 1974. "Oxygen Tension Distributions in the Cornea: a 1%e-examination." Exptl. Eye Res., 18, 375-365.

Lapidus, L. ]962. Digital Computation for Chemical Engineers. New York: McGraw-Hill. Lin, S. H. 1975. "Oxygen Diffusion in a Spherical Cell with Nonlinear Oxygen Uptake

Kinetics." J. Theor. Biol., in press. Prosser, C. L. and 1% A. Brown. 1962. Comparative Animal Physiology. :Philadelphia:

Saunders. Takahashi, G. H., I. Fatt and T. K. Goldstick. 1967. "Oxygen Consumption 1%ate of

Tissue Measurement by a Mieropolarographie Method. '° J. gen. Physiol., 50, 317-335. Walshe, B. M. 1948. "O~ 1%equirements of Chironmid Larvae." J. Exptl. Biol., 25, 35-44.

I~ECEIVED 2-3-75 REVISED 10-20-75