Oxygen tension in the in vivo cornea

BUIJLETII¢ O F IV£ATHEMATICA]5 B I O L O G Y

VOLUME

38, 1976

OXYGEN TENSION IN T H E I N

V I V O CORNEA

[] S. H. LIN

Department of Chemical Engineering, University of Melbourne, Parkville, Victoria 3052, Australia

The s t e a d y s t a t e d i s t r i b u t i o n of o x y g e n t e n s i o n i n t h e i n vivo c o r n e a is e s t i m a t e d i n t h e present s t u d y b y u s i n g a n o n l i n e a r o x y g e n 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 h e M i c h a e l i s M e n t e n t y p e . S u c h a r a t e e x p r e s s i o n is m o r e a c c u r a t e t h a n t h e p r e v i o u s simplified versions in p r e d i c t i n g t h e o x y g e n c o n s u m p t i o n r a t e . I t is f o u n d t h a t for a n o p e n eye w i t h or w i t h o u t c o n t a c t lens, t h e o x y g e n t e n s i o n s p r e d i c t e d p r e v i o u s l y are i n good a g r e e m e n t w i t h t h e s e p r e d i c t e d i n t h e p r e s e n t work. H o w e v e r , for a closed eye w i t h or w i t h o u t contact lens, t h e p r e v i o u s p r e d i c t i o n s u n d e r e s t i m a t e t h e o x y g e n t e n s i o n .

The mammalian cornea, although rather complex in structure, can be considered as a composite tissue consisting of three layers, viz endothelium, 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 t h a t govern the oxygen diffusion are known with enough certainty so t h a t the steady state distribution of oxygen tension can be estimated fairly accurately. Takahashi et al. (1967), F a t t and Bieber (1968), F a 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), F a t t and Bieber (1968) and F a t t (1968) estimated the oxygen tension in the cornea of an open or closed eye by assuming a constant oxygen consumption rate. 1. Introduction.

269

270

S. It. L I N

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 F a t t (1968) and Fattet al. (1974) to consider a sectioned oxygen consumption rate expression in predicting the oxygen tension in an eye covered b y 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 biological systems, exhibits nonlinear characteristics (Amberson, 1928; Blum, 1960; Presser and Brown, 1962). The purpose of this work is to attempt 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 consumption 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 b y

d2Pi D~k~-dx 2

VIPi --=0. Pi + lemi

(2)

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 b y an oxygen-permeable contact lens, an additional equation is needed for representing the o x y g e n diffusion in the contact lens d2P4 D4/c4 ~ = 0. dx 2

(3)

E q u a t i o n (3) does not have o x y g e n consumption t e r m since no o x y g e n is consumed in the contact lens. The b o u n d a r y conditions for the above equations are given b y

P1 = Pa,

for x = 0

P z = P2,

D1/Cl dP1 _ D2k2 dP2 dx "-~-x '

for x = xz

(5)

P2 = Pa,

D~k2 dP2 dPa d--~ = Dak3 --~--,

for x -- x2

(6)

P 3 = Pb,

for x = x3.

(4)

(,7)

For an open or closed eye w i t h contact lens, (7) is to be replaced b y

P3

Pa,

D3ka dP-~xa

P4 = Pb,

for x = Xd.

=

=

dP4

Daka -~x '

for x

=

xa

(8) (9)

In the above equations, x is the distance measured from the interface between the aqueous h u m o r and the endothelium. E q u a t i o n s (5), (6) and (8) represent the c o n t i n u i t y of the o x y g e n 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 h a v e r e c e n t l y been u p d a t e d b y F a t t e t al. (1974). The MichaelisMenten constants in each layer are assumed to be the same, i.e. ]~ml = kin2 = km3 = ]Cm because of lack of e x p e r i m e n t a l data. The m a x i m u m o x y g e n consumption rates, Vi, can be d e t e r m i n e d from the m a x i m u m oxygen consumption rates, Qi. I t can be n o t e d t h a t the o x y g e n consumption rate 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 o x y g e n tension P is generally greater t h a n km for all the cases u n d e r consideration and variation of km does not exert as m u c h influence on Q as does the v a r i a t i o n of V~. Therefore, the estimated lcm value is m u c h less accurate t h a n the V~ values. All the physical parameters are listed in Table I. I t is n o t e d t h a t several cases were considered b y F a t t and Bieber (1968), F a t t (1968) and F a t t et al. (1974). T h e y e s t i m a t e d the o x y g e n tensions for open and closed eyes with and w i t h o u t contact lens. These different situations 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 H g for an open eye and 55 m m H g for a closed eye. All these cases are considered here and the results are c o m p a r e d with previous predictions using simplified o x y g e n consumption r a t e equation. I t is f u r t h e r n o t e d t h a t the oxygen tension at the interface between the aqueous h u m o r and the endothelium might be affected b y the contact lens. However, due to lack of experimental evidence, it was assumed to be constant (Fatt, 1968 ; F a t t et al., 1974) and the same assumption was m a d e in the present work.

TABLE I N u m e r i c a l Values of Physical P a r a m e t e r s Numerical Value Symbol

Explanation Endothclium

Dk

oxygen permeability ml(O2)-cm2/ml-mm Hg-

Stroma

Epithdium

0.53 x 10-20

3.0 x 10-lo

1.88 x 10-10

0.001

0.045

0.004

1.5 x 10-Tp rain Hg

2.1 x 10-4

4.834

4.834

sec

x

thickness, cm

Q

oxygen consumption rate ml(O2)/ml-sec

Km

Miehaelis-Menten constant ram I-Ig

11.0 X 10-4 4.834

Because of the n o n l i n e a r i t y of the o x y g e n c o n s u m p t i o n term, (2) is hot amenable to analytical solution. An iterative R u n g e - K u t t a m e t h o d (Lapidus, 1962) was e m p l o y e d to integrate n u m e r i c a l l y the equation. According this iterative scheme, an a r b i t r a r y dP1/dx at x = 0 is assumed and (2) integrated b y 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 i m p r o v e d value can be obtained b y t h e reguli-fald m e t h o d (Lapidus, 1962). This p r o c e d u r e is r e p e a t e d until P a a t x = xa (or P4 at x = x4) deviates from P b less t h a n 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 v e r y satisfactory results for all the cases under consideration.

3. Results.

The o x y g e n tension distribution of an open eye to the air is shown in Figure 1. Also shown in this figure is the predicted o x y g e n tension using a simplified o x y g e n c o n s u m p t i o n rate e q u a t i o n ( F a t t et al., 1974). Although Fatt et al. (1974) predicted a slightly lower o x y g e n tension t h a n the present one, the

OXYGEN TENSION IN TEE IX VIVO CORNEA

273

difference between them is less t h a n 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 Slroma 120

E E

I00

=

80

Endothelium

Epi|h(slium~

4ekP--.~_

(b)

I

L..

o J o

I o.,

r o.a

I o.3

I o.a

0.5

Distance fromaqueoushumor, mm

Figure 1. - -

O x y g e n t e n s i o n p r o f i l e s i n (a) a n o p e n e y e a n d (b) c l o s e d e y e . Present prediction ..... 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 oxygenpermeable contact lens are demonstrated in Figures 2 and 3, respectively. The

274

S.H. LIN

two u p p e r curves in Figure 2 show the o x y g e n tension profiles a t a high o x y g e n permeability of contact lens of 500 × 10 -11 ml(02)-cm2/ml-mmI-Ig-sec, while the two lower ones show these a t a lower o x y g e n p e r m e a b i l i t y of 13.1 × 10-11 16o

I

I

140

I

.--

I

Stroma

/ t

,20

~r/'~

y'/Y

7I

7

f

I00 E -

80

--

i

60

$ 4.0

Contact

I

--

J

lens Ep~theli~ll

Endathellum

o

I o

P

o.~

0.2 Distance

f

I

0.3

o.4

[

i 0.5

o.6

humor, mm

from~ue~Js

F i g u r e 2. O x y g e n t e n s i o n profiles i n o p e n eye w i t h c o n t a c t lens. T h e oxygen p e r m e a b i l i t y of t h e c o n t a c t lens is (a) 500 x 10-11 a n d (b) 13.1 x 10 -11 m l (02)-em2/ml ( t i s s u e ) - m m Hg-sec.

50

i

Stroma

,o

"

,o,

__..S-¢

~ 3o

Contact

\

}

,

i,.,~//

~

I

,

I I//

I0 I

~'

I

d

[

.,~

O" I

0-2

@3

0"4

0"5

O'S

Distance from aqueous humor, mm

:Figure 3. O x y g e n t e n s i o n profiles i n a closed eye w i t h c o n t a c t lens. T h e o x y g e n p e r m e a b i l i t y of t h e c o n t a c t lens is (a) 500 x 10 -11 a n d (b) 13.1 x 10 -11 m l (O2)-cm2/ml ( t i s s u e ) - m m 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 at high o x y g e n p e r m e a b i l i t y of contact lens, the t w o predictions do n o t show m u c h difference in o x y g e n tension; however, at lower o x y g e n p e r m e a b i l i t y , the difference b e t w e e n t h e m becomes r a t h e r significant. A t the interface b e t w e e n the c o n t a c t lens a n d epithelium, F a t t et al. (1974) p r e d i c t e d a lower o x y g e n tension of 16.2 per cent when c o m p a r e d to the p r e s e n t prediction. Figure 3 shows the c o m p a r i s o n of o x y g e n tensions p r e d i c t e d b y F a t t et al. (1974) a n d the p r e s e n t w o r k for a closed eye w i t h c o n t a c t lens. A t a high o x y g e n p e r m e a b i l i t y of c o n t a c t lens, a m a x i m u m difference of 7.4 per cent b e t w e e n these two predictions is r e a c h e d a t the interface b e t w e e n t h e s t r o m a a n d epithelium. As the o x y g e n p e r m e a b i l i t y of c o n t a c t lens is reduced to 13.1 × 10 -11 ml(02)cm2/m]-mm Hg-see, t h e o x y g e n tension of the p r e s e n t w o r k is m o r e t h a n twice t h a t p r e d i c t e d b y F a t t e t al. (1974) a t a distance of 0.047 cm f r o m the aqueous humor. F o r this p a r t i c u l a r case, the o x y g e n tension in the cornea is fairly low. Such a lower o x y g e n tension generates a significant difference in o x y g e n consumptions calculated f r o m t h e r a t e equations used b y F a t t e t al. (1974) and the p r e s e n t work. This in t u r n influences s t r o n g l y the o x y g e n tension predictions, as anticipated. The a u t h o r is i n d e b t e d to Dr. I. F a t t of t h e U n i v e r s i t y of California (Berkeley) and Dr. J. J. B l u m of D u k e U n i v e r s i t y for their c o n s t r u c t i v e suggestions.

LITERATURE 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