Stanford Univeraity School of Medicine, Stanford, California 94305, US.A. (Received 18 ..... (19 decimal digits) on a main frame computer. Using this method ...
J. Phyaiol. (1979), 292, pp. 107-134 With 10 text-figuree Printed in Great Britain
107
NUMERICAL SOLUTION OF COUPLED TRANSPORT EQUATIONS APPLIED TO CORNEAL HYDRATION DYNAMICS
BY S. D. KLYCE AND S. R. RUSSELL
From the Diviimon of Ophthalmology, Department of Surgery, Stanford Univeraity School of Medicine, Stanford, California 94305, US.A.
(Received 18 September 1978) SUMMARY
1. A quantitative basis for the currently accepted theory on the regulation of corneal hydration was derived using the technique of finite element analysis to integrate a set of coupled flow equations. The model was based on non-equilibrium thermodynamics and incorporated the transport and permeability properties of the corneal epithelium and endothelium as well as the gel properties of the central connective tissue layer. 2. Considerable errors were introduced in the prediction of corneal hydration dynamics (unsteady-state behaviour) unless allowance was made for the development of trans-stromal gradients in pressure and solute concentration. 3. Thicknesses of in vitro rabbit corneal epithelium and stroma were measured with an automatic specular microscope during responses to changes in the osmolarity of the tear-side bathing medium. The time course of these experiments was fitted with the mathematical model to obtain a set of membrane phenomenological coefficients and transport rates. 4. The model with the redetermined membrane parameters was tested by predicting the influence of other variations in boundary conditions with excellent match to several well-documented experimental observations, including an explanation for the slight stromal swelling observed in hibernating mammals. 5. The regulation of corneal stromal hydration can be explained accurately by a balance between the dissipative flows across the serial array of corneal layers and the active HCO3 transport by the endothelium, supporting the earlier 'pump-leak' hypothesis. 6. It was found that stromal retardation of fluid flow, as well as gradients in solute concentration, significantly influences the dynamics of corneal stroma hydration. Tissue gel properties may be a more important factor in coupled transport across cell layers than generally appreciated. INTRODUCTION
The transparency of the mammalian cornea is dependent upon the preservation of epithelial and endothelial membrane integrity. Disruption of either of these layers permits the stroma to swell through the passive imbibition of fluid. The precise qualities of the limiting cell layers are still being explored, since no comprehensive model for corneal hydration dynamics has yet been detailed. 0022-3751/79/3720-0739 801.50 © 1979 The Physiological Society
S. D. KLYCE AND S. R. RUSSELL Several mechanisms for the regulation of stromal hydration have been proposed. Early impressions that the epithelium and endothelium might be impermeable to salt led Cogan & Kinsey (1942) and von Bahr (1948) to speculate that slight ambient hypertonicity produced homeostasis in stromal thickness. With the recognition (Maurice, 1953) that these membranes had a finite solute permeability, other external driving forces were explored. It was felt that the concentrative effect of evaporation from the precorneal tear film would be a sufficient force to dehydrate the cornea (Yasuda & Stone, 1967; Friedman, 1972a, b, 1973), but these theories overlooked the fact that normal hydration could be maintained for long periods of time with no transcorneal solute gradient (e.g. under closed lids or in vitro). The most widely accepted theory for the regulation of corneal hydration is based on the observation (Davson, 1955; Harris & Nordquist, 1955) that the stromal swelling in the refrigerated, enucleated eye could be reversed by rewarming the cornea to body temperature (the temperature reversal phenomenon). It is currently felt that this metabolism-dependent process is primarily the consequence of endothelial active transport of fluid into the aqueous humor (Mishima & Kudo, 1967; Dikstein & Maurice, 1972). A probable driving force for this fluid pump has been tentatively identified as the active endothelial transport of HCO3 (Hodson & Miller, 1976; Hull, Green, Boyd & Wynn, 1977). This new information supports Maurice's (1953) hypothesis that active ion transport provides the necessary osmotic gradient across the limiting membranes to counter stromal swelling.
108
There are, however, deficiencies in the 'pump-leak' theory. First, there exists no quantitative description which incorporates all the well known corneal properties and which can then be used to explain corneal reactions to various experimental procedures (cf. Maurice, 1969). Secondly, there appear to be considerably different corneal hydration characteristics in vivo as compared to in vitro (from which situation most current data is obtained). For example, the cornea of the refrigerated excised eye swells markedly overnight; yet corneas of rabbits (Mishima & Maurice, 1961b), amphibians (Bito & Saraf, 1973), and hibernating mammals (Bito, Roberts & Saraf, 1973) apparently do not exhibit a similar behaviour at low temperatures in vivo. Hence, the idea that stromal hydration is maintained by membrane solute pumps in vivo needs clarification. In this paper a mathematical model is developed incorporating the well known transport characteristics of all corneal layers. The resulting set of coupled transport equations was integrated numerically to redetermine membrane phenomenological coefficients and active solute transport rates. The model successfully simulated the dynamics of corneal hydration for a variety of boundary conditions. A preliminary report of this study has appeared in abstract form (Klyce, 1978). THEORY
The objective of this section is to write a complete set of equations describing coupled flows across a system with the physical properties of a polyelectrolyte gel, but whose boundaries have the properties of ion transporting epithelia. In cornea the groundwork for this has been laid by previous experiments that have documented stromal gel behaviour and limiting membrane-transport processes and by previous
COUPLED TRANSPORT IN CORNEA 109 theoretical work which provides a first principle-based formalism with which to quantitate coupled flows. The solution must of necessity be numerical, since it arises from a set of simultaneous non-linear partial differential equations (stroma) whose boundary conditions are described by simultaneous quasi-linear equations (epithelium and endothelium whose boundary conditions are in turn constant (bathing medium). List of symbols A C
(cm2) effective osmotic solute concentration (mole/cm3) D solute diffusion coefficient (cm2/sec) d corneal stromal thickness (cm) H stromal hydration (mass H20/dry mass) i boundary index J.b active solute flow (mole/cm2. sec) Jd diffusive solute flow (mole/cm2. sec) total solute flow (mole/cm2 sec) volume flow (cm3/cm2. see) compartment index Lp hydraulic conductivity (cm3/dyne sec) m solute molarity (mole/cm3) number of stromal compartments n Up hydrostatic pressure (dyne/cm) R gas constant (dyne cm/mole 'K) S stromal swelling pressure (dyne/cm2) area
.
.
T
temperature (0K)
t
time (sec) empirical constant, dS/d(e-H)
y
(dyne/cm2)
a e
i1 K v
6 iff p
oa T
0 O
thickness of stromal slice (cm) thickness of dry stroma (cm) specific volume of dry stroma (cm) empirical constant (cm) number of ions/molecule solute empirical constant relating stromal hydraulic conductivity to hydration osmotic pressure (osmole/cm3) empirical stromal constant, dL./dHg (cm4/dyne. sec) reflexion coefficient time interval (see) osmotic coefficient for NaCl solute permeability (cm/sec)
General equations and considerations Coupled flows across membranes can be described phenomenologically on the basis of non-equilibrium thermodynamics (Kedem & Katchalsky, 1958). For simple systems their linear differential equations may be integrated analytically (with some reservations) to obtain a kinetic analysis of corneal thickness as has been done previously (Mishima & Hedbys, 1967; Stanley, Mishima & Klyce, 1966). However, serious inaccuracies can arise when not allowing for the existence of pressure and concentration gradients across the stroma. Nevertheless, there are certain advantages in the determination of phenomenological coefficients by fitting data over an entire experimental time course (time sequence analysis) including the elimination of uncertainties stemming from zero time extrapolation for the estimation of phenomenological coefficients. The major disadvantage is that analytical solutions become rapidly incogitable for all except the simplest of systems, primarily due to macroscopic non-linear behaviour. The cornea is modelled as a series array of n + 1 thin, planar and homogeneous membranes bounding n concatenated stromal compartments (Fig. 1). This system is open (in the thermodynamic sense) on each outer surface to semi-infinite well-stirred, temperature controlled media characterized by adjustable solute concentration and hydrostatic pressure. While the model is illustrated with discrete compartments, it is apparent that when n is very large, the description of the stroma becomes continuous, analogous to the physical situation. The influence of unstirred layers in the bathing solutions on the outer corneal surfaces is not incorporated in this model for the following reasons. First, an adequately detailed description
Ito
S. D. KLYCE AND S. R. RUSSELL
of the surface properties of the bounding membranes is not available (i.e. roughness factor, surface tension, velocity profile of solution parallel to membrane, dimensions and physicochemical properties of adherent 'fuzzy coats' etc.). Secondly, volume flows across the membranes generated by experimental procedures in these studies were purposely made small to minimize Dainty's (1963) steady-state approximate correction for solute concentration next to the membrane. Finally, the influence of external unstirred layers on the determined coefficients is a characteristic of tile in vivo situation or of the membranes themselves or both, as long as zero time extrapolation procedures are not used in their estimation.
The limiting membranes (i = 0, n) are given the attributes of corneal epithelium and endothelium respectively. Each is characterized by a reflexion coefficient (C-), a hydraulic conductivity (Lp), a solute permeability (wRT), and an active solute pump, Ja. The solute pump is coupled to metabolism but is assumed to be independent of
Endothelium
Epithelium
Bathing solution
Stroma
Bathing solution J,
Jd
Compartment
1indices:ji= 01 Membrane indices: i=
0
1
jj4
3
2
1
2
3
n+
n
Fig. 1. Schematic representation of corneal model. Flows of volume, J,, and solute, both active, J., and passive, Jd, can be calculated across the series of n + 1 membranes where n is determined by stability criteria for a given calculation. The thicknesses of the stromal compartments, hence the relative positions of the membranes, are permitted to vary in time.
other solute flows. The total solute flow, J., across these membranes is the sum of passive solute flow (id) and J.. The incorporation of Ja is necessary both to match physical observations as well as to sustain the non-equilibrium status of the stroma. A metabolism-coupled volume pump as proposed by Hoshiko & Lindley (1967) for the more general theoretical case is intentionally not included, since there is no a priori or a posterior reason to do so. While it would be more physically descriptive to treat each of the cellular layers as a series-parallel array, doing so would treble the number of unknown coefficients for the simplest expansion. In the cornea it will be shown that coupling of fluid flow to solute flow does not require the presence of intercellular or intracellular compartments. The stroma can provide the gradients necessary for this coupling. The remaining membranes (i = 1, ..., n - 1) are characterized by the space
ill COUPLED TRANSPORT IN CORNEA average transport properties of the adjacent stromal compartments at any given point in time and space. Their evaluation is discussed below. In order to simplify the phenomenological description, the model has been initially developed to treat the flow of a single solute: NaCl is chosen since this is a fair approximation of extracellular fluid composition. The net flow of ionic current across each membrane is set to 0, simulating experiments performed at resting potential. Under these conditions the coupled flow equations for electrolyte/water reduce to those derived for a non-electrolyte. Finally, it is assumed that net radial flows are everywhere 0 so that the problem is spatially unidimensional. With the above considerations and with reference to Fig. 1, the following set of general equations can be written (for membranes, i = 0, ..., n; compartments, j= , ...,n+1; and times, t= 0, . . .,T):
ivi t = Lpi t (APi, t-aoRTACi, t), i
t =
(1- 0)i, tvi + ('i
(1 a) (l b)
tRTACt+Jai
in which
ACi, t A7i, t/RT voAmit, Ci, t ACi, t/1n(Cj, tl =i/C,+l, tl =i) =
(1 C)
=
=
~ (Ci'tlj+Cj+1'tlj=i)12
(
1
d)
when ACit < 10-6M. Eqns. (la-d) expand the discrete forms of the simultaneous partial differential equations derived by Kedem & Katchalsky (1958) into a set of 2(n + 1) equations at any instant in time. They are first order correct, that is, second and higher order derivatives have been ignored, and therefore a first order correct numerical integration procedure is developed below. They make most accurate predictions when AP and A\C are kept small. Note that the set is non-linear, as the coefficients are allowed to vary with time and space. In the calculation of osmotic pressure (eqn. 1 c) deviation from non-ideal behaviour is accounted for by including the average osmotic coefficient, 5, of NaCl for the range of concentrations encountered in the calculations below. Stromal relationships (i = 1,...,n-1; j = 1,...,n;t = 0 r) The corneal stroma forms the bulk of the cornea, and as a consequence of the organization and dimensions of its component collagen fibres (a necessary requirement for transparency (cf. Maurice, 1957; Goldman, Benedek, Dohlman & Kravitt, 1968)) there is little inherent structural stability in the thickness dimension. The stroma, sharing many properties common to loose polyelectrolyte gels, has an equilibrium swelling volume several times the in vivo volume. The driving forces for stromal swelling have been shown to result from long range electrostatic repulsion between the acidic groups of the mucopolysaccharide component of the ground substance with a similar contribution from the Donnan osmotic contribution of the macromolecules (Hedbys, 1963; cf. Katchalsky, 1954). A great deal of effort has been spent in the past characterizing the properties of the stroma; sufficient information is currently available to describe flows through this structure quantitatively.
S. D. KLYCE AND S. R. RUSSELL For a connective tissue, low molecular weight solute reflexion coefficients can be assumed to be close to 0. In this situation, eqn. (la) reduces to a restatement of Darcy's law for the flow of fluid within tissue. One important feature of fluid transport in gels is that the driving force for volume flow is the local gradient in hydrostatic pressure. A second important characteristic is that the local hydraulic conductivity of the tissue can be strongly dependent upon the local gel hydration. For these reasons relatively large gradients in tissue hydration can be sustained over short distances, which is the basis for corneal dellen (local corneal depressions below imperfections in the evaporation-preventing oily layer of the precorneal tear film). As noted above, the stromal transport coefficients can be calculated from the average properties of the two adjacent stromal elements. Hence, the local stromal solute permeability (wptRT) was calculated from the stromal solute diffusion coefficient DNaCl: (2a) j, t = 2D\aCl/RT(&i, t + 8j+l, t)l j=1, where values of 8 are thicknesses of the bounding slices at time t. Local stromal hydraulic conductivity can be calculated in a similar fashion, expanding an empirical relationship first found by Fatt & Goldstick (1965): 112
LPit p(H3j, t/&j,t + Hj+i,t/18j+i,t)/21 '= =
(2b)
in which p and f are emprical constants, and H is the tissue hydration (the ratio of tissue water weight to tissue dry weight). Eqn. (2b) is of sufficient accuracy for H ranging from 1*5 to 5. Hedbys & Mishima (1966) have demonstrated that, since changes in stromal volume are primarily confined to the thickness dimension, there is a linear dependence of H on d. This can be written as:
Hjlt = Knfl,,t
yd (2c) where q is the specific volume of dry stroma. The slope, K, of this relationship is not only species-dependent but varies according to animal age (body weight). Hence, for the purpose of accuracy, K is determined from eqn. (2c): K = (Ho + #)/doX (2d) where do and Ho are the in vivo closed eye steady-state values of total stromal thickness and the space-averaged stromal hydration respectively. The fluid pressure and solute concentration in each stromal element must also be evaluated. The hydrostatic pressure in a stroma below its equilibrium swelling volume is at a negative value with respect to the ambient solution pressure. The consequent pressure gradient is the primary force underlying stromal swelling. The relationship between the space average swelling pressure, 5, of corneal stroma and H has been measured (Hedbys & Dohlman, 1963), is similar in a variety of species, and is relatively independent of solute concentration and ambient fluid pressure. The data have been fitted with the empirical equation (Fatt & Goldstick, 1965):
Sit
=
-
ye-1ji,
(2e)
which is relatively accurate over a range of stromal hydration of 1 5 5.5. In the in vivo eye, there is an extracorneal hydrostatic pressure gradient (AP) due to the intra-ocular pressure. After the analysis of Friedman (1972a), it is assumed
113 COUPLED TRANSPORT IN CORNEA that the tension in the stromal collagen fibres caused by the intra-ocular pressure is taken up by the most anterior lamellae, so that the stromal hydrostatic pressure at any point approximates the simple relationship (Hedbys, Mishima & Maurice, 1963):
(2f)
Pi, t = Pn+1-Sj,t.
Clinical observations support this conclusion as well (Ytteborg & Dohlman, 1965). The thickness of each stromal element with time can be derived by summation of the volume flows occurring at each boundary of each element over the time interval, A, from some initial condition (t = 0):
1,T
=
aj, tO+ 1/A g
(Jvi,
at i -vi,
tAtIi=j+l)At*
(2g)
Similarly, the solute concentration of each stromal element may be evaluated as a function of time by summation of the solute fluxes:
Cj,
T
=
Jiatlsj-
t
(Cjt=o (4j, to-e/n) +
Here correction is made for the presence of non-solvent stromal volume in each element (e/n) where e is the dry thickness of the stroma. Total stromal thickness, d7, after any time interval will be the simple sum of the component element thicknesses: n
(2i)
d= X s, t
Equations (2e, f and h) define the space average hydrostatic pressure and solute concentration in each stromal slice as a function of time. From these equations, histograms can be constructed (C and P vs. depth in stroma) which in turn can be used to draw a continuous curve approximating the exact solution. Since stromal gradients are continuous, a good approximation of the gradients and flows across the stromal 'membranes' can be calculated without an excessively large number of stromal compartments. However, the gradients across the model epithelium and endothelium are for simplicity modelled as being discontinuous, and when stromal gradients are large the values of the space average concentration and pressure in the adjacent stromal compartments will not accurately approximate the concentrations and pressures next to the limiting membranes (Cm, Pm) without making n large. Hence, Cm and Pm were calculated by linear extrapolation from the respective space average quantities in the two adjacent stromal compartments (subscripts 1 and 2):
and
Pmj=o, n
=
Pj=n
-
C1 + 1 (Cl- C2)/(81 + &2) For example, flows across the epithelium were calculated with APi-O
Cmj=o,
n
(2j)
y exp {- [HI + 41(Hl - H2)/(&L + &2)]}
(2 k)
=
=
Po -Pmj=o
and ACiO = CO- CmjO. Stromal hydrations were extrapolated (eqn. (2j)) rather than fluid pressures, since the former are more linear in the stroma.
S. D. KLYCE AND S. R. RUSSELL
114
Numerical procedures The sets of equations (1) and (2) can be evaluated numerically. They are a first order finite difference approximation (cf. von Rosenberg, 1969), increasing in accuracy as n - so and At -O 0. A small time increment is chosen during which a quasi-steady state is assumed and for which the stromal coefficients in eqns. (1) are very nearly constant. A variable time step integrator (At) was used to maximize efficiency, stability and accuracy. Its minimum bound was chosen to reduce round-off error and its upper bound was set to reduce truncation errors. Within these limits the integrator was adjusted with a semi-empirical test designed to keep stromal element solute concentration changes small during any time step with respect to the total solute concentration in each element. All of these tests were sensitive to the value of n employed and were adjusted accordingly. For each time increment the discrete forms of eqns. (1) were solved to calculate solvent and solute flows across each membrane. These were used to adjust the thickness, solute concentration and hydrostatic pressure in each stromal element (eqns. 2 c-hk). Finally, the hydration- and thickness-dependent stromal coefficients were recalculated for use during the subsequent time increment (eqns. 2a, b). This procedure was reiterated until a specified time interval, T, was reached. In order to approximate a continuous system (n -> xo) for the stroma, n was increased stepwise until the time course of stromal thickness for a given set of boundary conditions and long time interval converged to a common relationship. The minimum value of n was sought for efficiency in time of calculation. Most of the numerical procedures were accomplished with a laboratory minicomputer in 6 decimal digit precision. Possible round-off errors were judged insignificant by comparison to results of sample calculations made in double precision (19 decimal digits) on a main frame computer. Using this method, preliminary predictions of well known corneal responses to various boundary conditions were not satisfactory, employing any set of previously published epithelial and endothelial phenomenological coefficients (cf. values in Table 2). Hence, all six of these coefficients as well as membrane ion transport rates were designated as unknowns ((To, on, LP,0, LPit, co0, (o)n Jao and Jan), and osmotic perturbation experiments were designed for their evaluation. Fits to experimental data were accomplished by successive approximation of these unknowns in time sequence analyses. Initial steady-state calculations Steady-state membrane pump rates and stromal solute and hydration gradients were calculated from in vivo rabbit cornea 'closed eye' conditions. These were (j = 1, ..., n): CO Cn+l; Pn+1 = normal intra-ocular pressure; Hi = Ho the space average stromal hydration in vivo; 6, = do/n and
Cj
=
(LpOAPO/RT + ooLpOCO + LPnAP,/RT + oLpC,+ )/(oLpo+ ,L P) (3a) the steady-state analytical solution (Jso = Jo,,, J,0 = Jvn) for the model
which is when n = 1. The latter equation uses values for the epithelial and endothelial phenomenological coefficients which are the current best guess. For whole corneal
COUPLED TRANSPORT IN CORNEA 115 experiments the endothelial solute pump rate was first approximated by the steady state single stromal compartment solution:
Jan
=
(o-n-o-0)(APOo- ORTACO)LpOCO + (wo0 + o)RTACO + Ja8.
(3b)
Subsequently J.. was adjusted (in general, the analytical approximation overestimated Ja, by about 5 %) to obtain a steady stromal thickness (defined as less than a 1 #sm change in thickness over a time interval of 10 h), which was equal to in vivo thickness. Steady-state stromal gradients of solute concentration and hydration were established for these 'closed eye' values by 'running' the model until flows across each corneal membrane were uniform (for the whole cornea calculation) or zero (for the endothelial surface blocked). Following the calculation of steady state for a given set of membrane coefficients, the influence of changes in boundary conditions (e.g. ACO) on stromal thickness could be calculated. Paramaterization The parameters used in the model are listed below and have been drawn from literature values derived from rabbit cornea where not determined in this paper. Ambient conditions A = 1 cm2 for the corneal surface area exposed in the chamber used (Klyce, 1972). 'TO nn+ = 300 m-osmolar (Brubaker & Kupfer, 1962), which is the osmolarity of 163 mM-NaCl (cf. eqn. (2c)). Oac1 = 0-92 (Robinson & Stokes, 1959). VN&C1 2. PO = 0 dyne/cm2 using the tear solution for the reference hydrostatic pressure. Pn+l = 2-67 x 104 dyne/cm2 (20 mmHg). This is the normal intra-ocular pressure for the eye of the unanaesthetized rabbit (Sears, 1960). T = 35 0C (Schwartz & Feller, 1962). Hence, RT = 2-58 x 1010 dyne cm/mole. Stromal constants DNsa01 = 9 x 106 cm2/sec. Maurice (1961) obtained a value which was two times less than that for free solution diffusion. The resistivity of corneal stroma is about 120 Q cm (Klyce, 1972), which is twice that of isotonic saline. Hence, the stromal solute diffusion coefficient is taken as half its free solution value at 35 'C. p = 8-63 x 10-15 cm2/dyne. see (Fatt & Goldstick, 1965). = 4 (Hedbys & Dohlman, 1963; Fatt & Goldstick, 1965). 1 = 0-62 for rabbit cornea (Hedbys & Mishima, 1966). do is the thickness of the in vivo rabbit stroma measured with the Maurice & Giardini (1951) pachometer. In the experiments reported below it ranged from 0*0310 to 0-0365 cm. Ho = 3.45 (Duane, 1949). The space average hydration for the in vivo corneal stroma has been measured since by many workers with generally good agreement. y= 2'41 x 106 dyne/cm2 (Fatt & Goldstick, 1965). e = 0'008 cm (Hedbys & Mishima, 1966).
S. D. KLYCE AND S. R. RUSSELL
116
METHODS Eyes were enucleated from 3-S5 kg New Zealand White rabbits following the administration of a lethal dose of sodium pentobarbitone. The corneas were then dissected and mounted in a lucite incubation chamber using an atraumatic procedure (Klyce, 1972). Temperature-controlling water jackets maintained the preparation at 35+0.5 'C. When the endothelial surface was exposed to Ringer solution it was perfused at the rate of 0-6 cm3/hr. The epithelial surface was perfused with Ringer solution at 10 cm3/hr during steady-state periods and at the rate of 20 cm3/hr when perfusate osmolarity was changed. With these perfusion rates laminar epithelial surface flow was observed by light microscopic observation of 0-25 jam latex spheres. Solution velocity about 2 above the epithelial surface was 50 ,um/sec. For comparison, non-laminar flow at the endothelial surface was observed, probably due to thermal convection. These observations suggest that the application of Dainty's (1963) derivations of external solution 'unstirred layer' effects would over-estimate the experimental effects. A gradient in hydrostatic pressure of 20 mmHg (2-67 x 104 dyne/cm2) was imposed across the tissue by elevation of the endothelial compartment outlet tube. The composition of the parent Ringer solution used in this study was 124-7 mM-NaCl, 25.4 mM-NaHCO3, 1-21 mM-KH2PO4, 7-26 mM-KCl, 0.61 mM-MgSO4, 0-65 mM-CaCl2, 0-16 mMglutathione, 4 75 mM-adenosine, 2-79 mM-glucose and gentamycin, 25 mg/l. Just prior to use, the solution was saturated with 9500 02/50 CO2 at 37 'C. This solution is similar to the one developed by Dikstein & Maurice (1972), the only changes being a slightly increased concentration of NaHCO3 and the presence of the antibiotic. Variations in the osmolarity of this solution were achieved by altering only the .NaCl content. The accuracy of these osmotic changes was confirmed by freezing point depression measurements. Experiments were designed to first measure the phenomenological coefficients for the epithelium and then to determine those of the endothelium. In the former experiments, the aqueous humor bathing medium was replaced with silicone oil to block flows across the endothelium. In the latter, the functional integrity of both epithelium and endotheliumswas maintained to determine the endothelial transport coefficients while using those derived for the epithelium from the initial series. In both series osmotic perturbations were confined to the epithelial perfusate. Fluid volume flows were determined by the measurement of the thickness of the stroma. In the case of the experiments in which the inner surface was blocked with silicone oil, stromal thickness measurements were directly converted to volume flowx across the epithelium. In the case of whole corneal experiments, thickness changes correlated directly with the difference in volume flows across the constituent membranes, all of which could be calculated from the formalism presented above. Thickness measurements were made with the Mauirice (1968) specular microscope in the centre of the isolated cornea. These were done^^-ith attachments recently described (Klyce & Maurice, 1976; Klyce & Russell, 1978), which automatically scan the tissue and perform data analysis to increase the accuracy and reliability of measurements over those made with the original specular microscope. With these improvements, scaled averages of epithelial and stromal thicknesses were plotted directly on a chart recorder. Thicknesses were readable to 0-2jam with an over-all accuracy better than 1 %.
jzm
RESULTS
Pressure and solute gradients in the unstirredd' strorna Preliminary calculations were made to determine the minimum number of stromal compartments, n, for the simulation of the experiments designed to evaluate the designated unknowns. Originally these were done with a current set of best-guess parameters for membrane phenomenological coefficients; however, these were repeated using the final set determined below for the data presented in the subsequent two Figures. Significant differences were found in the calculated response of stromal thickness to a hypotonic medium applied to the epithelial surface as the stroma was divided into more than a single compartment (Fig. 2). In this particular experiment
117 COUPLED TRANSPORT IN CORNEA the calculated response was essentially constant for n > 2. The response to the hypotonic medium and to the subsequent reversal of this perturbation was not symmetrical. In general for most other types of experiments calculated below, the value of n had to be adjusted upward to allow for steeper and more non-linear gradients in the stroma. Errors introduced by assuming that the stroma is a single well-stirred compartment in the evaluation of membrane coefficients was small only for very short times and for small perturbations from the in vivo steady state. 420
-
E400CA
C.
-'I *380-2 0
360-
5 4 3 Time (hr) Fig. 2. The calculated influence of a 30 m-osmolar NaCL hypotonic perturbation applied to the epithelial surface on stromal thickness as a function of time and the number of stromal compartments, n. For relatively mild perturbations such as shown here, convergence was obtained at n > 2. Hypotonicity was applied at ; and the normal tonicity reapplied at t. 1
2
When analysing the cause of these discrepancies with increasing values of n, their source was not entirely anticipated. In Fig. 3 stromal solute and fluid pressure profiles are plotted for the hypothetical experiment in Fig. 2 at t = 0 (steady state) and at two instants after changing the tonicity of the epithelial perfusate. In the steady state, the trans-stromal solute gradient was found to be linear and small (< 0-2 mosmole/l.). Likewise, the trans-stromal fluid pressure gradient was both linear and small (7 mmHg). Following the osmotic perturbation, the stromal solute concentration fell, and there was a trans-stromal gradient of more than 2 m-osmole/l. While the anterior stroma swelled as expected, there was actually a reduction in the hydration of the posterior lamellae which was linked to the fall in stromal solute concentration. The trans-stromal gradient of swelling pressure became noticeably non-linear during the experiment, reaching a peak value of 31 mmHg 15 min after the perturbation. Both solute and fluid pressure gradients can be fairly substantial across the corneal stroma even for mild perturbations away from the steady state. Hence, there can be considerable error in the prediction of hydration dynamics and in the determination of membrane coefficients when neglecting stromal gradients.
S. D. KLYCE AND S. R. RUSSELL
118
Determination of membrane phenomenological coefficients and transport rates An example of a time sequence analysis made to fit stromal thinning following the application of hypertonic medium (+ 30 m-osmole/l.) to the epithelial side is given in Fig. 4A. In this experiment the endothelial surface was blocked with oil so that 298-
~i2960
E 0
0
#A292
290
a:r
~0
t-=l0o(ste
state
-40
-50 I
.
400 300 100 200 Distance from anterior stromal surface (inm) Fig. 3. Stromal solute and fluid pressure gradients at steady state and at times after a calculated -30 m-osmolar perturbation. Note that the anterior stroma swelled as predicted, but that there was thinning in the posterior regions. 0
water and solute flows traversed only the outer surface. The rate of active epithelial solute transport was small and was estimated in individual experiments by fitting the slope of the thickness change during the initial steady state. The reflexion coefficient, o- was determined by fitting the amplitude of the response. The initial slope of the response was fitted by then adjusting the value of epithelial hydraulic conductivity LPo* Finally, the epithelial solute permeability, )ORT, was determined by fit to the return of thickness to normal. The response was nearly reversible, but generally the stroma began to swell after about 6 hr of incubation. That period was the anticipated
COUPLED TRANSPORT IN CORNEA 119 lifetime of the preparation when the normal corneal access to metabolic substrate was prevented by posterior silicone oil (Klyce, 1977). The average results of this and subsequent series are given in Table 1. A
-E 400
370 E 0
,,340 3
2
1
5
4
6
-
45
B ~~~~~~C
0
400
Ir' g 40 4I
A'eT%>
I35=
*5-350 E0
300 _I
1
2
3 Time (hr)
4
I
5
I
6
Fig. 4. Influence of a 30 m-osmolar hypertonic perturbation on stromal thickness. Osmolarity of the epithelial perfusate was increased at t and returned to normal at i.
The calculated fit (@) is superimposed on the experimental data. A, response with the endothelial surface scraped away and that surface blocked with oil. B, response perfusing both corneal surfaces with Ringer solution. The influence on epithelial thickness is shown as well primarily to illustrate the resolution of the automatic measuring system.
In Fig. 4B an example is given of the response of stromal and epithelial thicknesses to a hypertonic perturbation with both corneal surfaces perfused with Ringer medium. The epithelium in this experiment thinned 2-3 ,um following the hypertonicity and- returned to its original thickness during the reversal. However, the behaviour of the epithelium was not consistent from experiment to experiment. In some cases there was no apparent relationship between solution tonicity and epithelial thickness and in other cases there was an apparent inverse relationship. In order to determine the endothelial unknowns for the whole cornea experiments, the epithelial coefficients were given the values determined in the companion series
S. D. KLYCE AND S. R. RUSSELL 120 with the endothelial surface blocked. Each individual whole cornea experiment performed was not fitted numerically, as several days of minicomputer time were required for each calculation. Rather, all experiments from a given series of whole cornea experiments were graphed together to identify the one experiment best representing A E 360
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I I 4 5 3 6 Time (hr) Fig. 5. Influence of a -23-3 m-osmolar hypotonic perturbation on stromal thickness. The osmolarity was decreased at ; and returned to normal at t. The plotted points (*) are the calculated fit. A, response with the endothelial surface blocked. B, response of the whole cornea. 0
1
I
I
I
2
TABLE 1. Corneal phenomenological coefficients and solute transport rates (x 10-6
osmole/cm3) + 30.0
-23-3
ONaC1
Lp ( X 10-12
(0
a-,RT
cm3/dyne sec) ( x 10-6 cm/sec)
Epithelial values with endothelium blocked* 0-15 ± 0-01 5-8 + 0-7 0-78+± 0-07 6-4+ 1-5 0-23+0-02 0-80+0-06
J. (x 10-1o
mole/cm2. sec) - 0-120
0-064
+0-038+0-031
Endothelial values using whole corneat 53 (43-55) 80 (73-80) 4-8 (3-1-4-8) 0-45 (0-45-0-60) 32 (24-53) 80 (80-100) 4-6 (4-6-5-5) 0-45 (0-43-0.50) * Means + S.E. of means. Number of observations = 5. t Values fitting median response. In parentheses is given the range of values determined by fit to minimum and maximum response to osmotic perturbation. Number of observations = 6.
+ 30-0
- 23-3
the median response plus the two experiments representing the extremes. The unknowns were determined as follows: the endothelial solute pump rate was determined as that value necessary to maintain steady corneal thickness with a given set of endothelial phenomenological coefficients. The endothelial reflexion coefficient was
COUPLED TRANSPORT IN CORNEA 121 found by fitting the amplitude of the response 60-90 min from the stimulus. Lp was found by fitting the mid-section of the slope of the response rather than the initial portion, since in this type of experiment the initial changes were determined primarily by epithelial properties. Endothelial solute permeability was found by matching the experiment during the return to normal Ringer medium. No other combination of endothelial oC and Lp delivered a good match for the reversal. A major consideration in the design of the experiments reported here was to minimize influence of a perturbation on the coefficients to be determined. To test the validity of experimental design, the two series reported above were repeated using a hypotonic perturbation. The responses of stromal thickness to a 23-3 m-osmole/l. reduction in epithelial side perfusate concentration for a cornea with the endothelial surface blocked and for a whole cornea are shown in Fig. 5A and B. As before, the epithelial coefficients were determined with the endothelial surface blocked, and subsequently these values were used in the determination of endothelial coefficients from whole cornea experiments. These data are shown in Table 1 together with the earlier series. The minimal dependence of any of the measured coefficients on perturbation osmolarity over the range utilized supports the assumption made above that the membrane coefficients could be characterized as constants.
Prediction of well documented corneal experiments It is valid to ask whether the model and its parameters given here can be of any general use for the analysis of corneal hydration dynamics, especially since the model was parameterized by fitting the theory to a single type of experimental observation. Below are presented some of the observations that were fitted with the method. In these calculations membrane coefficients and pump rates were given the average values listed in Table 1 unless expressly stated otherwise. The 'temperature reversal' phenomenon. The original temperature reversal experiment (Davson, 1955) consisted of swelling the cornea overnight by placing the enucleated eye in the cold and measuring its subsequent return to normal thickness when rewarmed to body temperature the following day. A more controlled demonstration of the metabolic dependence of corneal hydration involved swelling the cornea from the bare anterior surface in vitro, blocking this surface with oil, and following stromal thinning back to a normal value (Mishima & Kudo, 1967; Dikstein & Maurice, 1972). An example of this type of 'temperature reversal' experiment is shown in Fig. 6. To match this type of experiment, the coefficients of the outer membrane of the model were initially given those of stroma: o-0 = 0, J. = 0; coo and LPO were extrapolated from their respective values in adjacent stromal slices. When the stroma had swelled to the maximum thickness indicated in the experiment, the outer membrane coefficients were re-assigned to simulate the presence of oil: o-O = 1 and Ja0 = co = LLp0 = 0. The result of this calculation (shown in Fig. 6) was a fair approximation of the experimental values. By attempting to fit this experiment exactly along the entire time course, it was determined that one or more of the endothelial coefficients or its pump rate was not independent of time with such a large change from normal thickness. However, the correspondence between the calculations and experimental data is excellent for the first hour and fairly good for the remainder of the time course.
S. D. KLYCE AND S. R. RUSSELL
122
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E 0
c
-Y
.
420
E
360
-
I
I
L
0
1
I
I
4
3 Time (hr)
2
5
6
Fig. 6. Fit to a 'temperature reversal' experiment. The continuous line is the calculated stromal thickness after swelling the bare stromal surface with Ringer solution and subsequently blocking it with oil. The experimental data points (@*) were reproduced from Klyce & Maurice (1976) with the permission of the Association for Research in Vision and Ophthalmology.
410
F-
Observed Predicted
390
F-
._
an= 0 (ouabain)
-Q
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I
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80 60 40 Time (min) Fig. 7. The calculated effect of ouabain inhibition on stromal thickness. Continuous line is the calculated response with the endothelial solute pump set to 0 at |. The experimentally observed rate determined by Mishima et al. (1969) is indicated by the dashed line. >0
20
123 COUPLED TRANSPORT IN CORNEA Inhibition of endothelial solute transport. It is well known that ouabain inhibits fluid pumping by the endothelium but without producing a measurable change in endothelial solute permeability (Trenberth & Mishima, 1968). Maximum inhibition occurs at concentrations of > 10- M-ouabain, at which Mishima, Kaye, Takahashi, Kudo & Trenberth (1969) report an average rate of initial stromal swelling of 45 ,um/hr. If it is assumed that the endothelial 'fluid pump' is in essence a secondary consequence of active endothelial solute transport (eqn. 2b) and that it is the solute pump which is inhibited by ouabain, then the model can be used to predict the experiment of Mishima et al. (1969). This was done as shown in Fig. 7 by setting Jan = 0 with excellent correspondence between the experimental and calculated slopes. Corneal thickness during hibernation. To examine the apparent discrepancy between in vitro and in vivo cold swollen corneas, the influence of cold on the model cornea was determined, matching the experimental conditions used by Bito et al. (1973) in the marmot studies. The calculation was made for corneas at 9 0C, the hibernation temperature used for marmots, after adjusting the passive and active coefficients in proportion to the observed in vitro hypothermic effects. Membrane solute permeabilities were reduced by 2 % per 0C (Robinson & Stokes, 1959). While solute permeabilities of many other membranes are thought to have Q10 values which are substantially more sensitive to temperature, the thermal sensitivity of Na and Cl permeability match free solution ratios for the corneal endothelium (Maurice, 1969) and epithelium (Klyce, 1975). The hydraulic conductivities of corneal membranes were reduced to half their respective values determined at 35 0C, as the viscosity of water will be more than doubled over this temperature range. The in vivo steady state (normal) stromal thickness was set at 300 1um. (In sum, the passive coefficients of the model were adjusted strictly on the basis of diffusion kinetics.) The active epithelial solute transport rate was set to 0, which is its approximate value in vitro unless the Cl transport is stimulated pharmacologically (Klyce, 1975). The Q10 for the endothelial HCO3 pump is unknown at present. Therefore, the influence of cold on stromal thickness for several endothelial transport rates was determined, and the results of the calculations are shown in Fig. 8. When the pump rate was assigned a Q10 of 1 2 (a value consistent with diffusion processes), stromal thickness did not change as a function of time, since a similar temperature sensitivity was assigned in that case to both dissipative and accretive flows. The Q1o of 02 consumption and lactate production was about 1 6 for the hibernating marmot cornea. Using this value for endothelial solute transport rate sensitivity over the 26 °C temperature reduction, a maximum stromal swelling of 50 /am was calculated at steady state. This 17 % increase over normal thickness was accompanied by a drop in stromal swelling pressure from the normal 57 to 7 mmHg. Unfortunately, Bito et al. (1973) did not measure thickness upon entering hibernation for individual marmots or for even the same population of animals. Consequently, the increased thickness calculated above could well be within the scatter of his data, which ranged up to 120,um for some experimental series. However, there was a reported increase in stromal hydration (which it should be noted is directly related to stromal swelling pressure from animal to animal, whereas thickness is not). Scaling the measured changes in stromal hydration appropriately, it is estimated that the swelling pressure of the stroma in hibernating mammals was 9 mmHg, similar to the calculated steady state swelling
124 S. D. KLYCE AND S. R. RUSSELL pressure with a pump Q10 of 16. The time course of the corneal swelling in this calculation was identical to that observed for the frog upon entering hibernation, in which species Bito & Saraf (1973) did measure the time course of individual animals. Frog corneas swelled 50 % above normothermic values as compared to the 17 % calculate for the marmot. 500
990C
-
Q_Qloo (Jan =0)
460
E
B
420
Cu
X7
380
r
0
340
300 0
6
12
18
_1 -o= 1-2
24
Time (hr)
Fig. 8. The calculated effect of hypothermia on stromal hydration for different endothelial solute pump Q10 values. The temperature of the steady-state model cornea was reduced from 35 to 9 "C at 1. The initial temporal response and the final steady-state values (t -+ o) were drawn.
When the solute transport pump in the endothelium was set to 0 in the above calculation, the stroma swelled 62 % over normal. In this case the only force gradient preventing further stromal edema is the transcorneal pressure gradient set up by the intra-ocular pressure. However, with a Q10 of 1-6 for the endothelial solute transport sensitivity, the presence or absence of normal intra-ocular pressure exerted a minimal influence on the resultant steady corneal thickness. Hence, the minimal swelling observed in the cornea of hibernating animals can be explained with the model using coefficients derived from in vitro experimentation. It should be pointed out that the model so parameterized does not predict the more rapid and extensive swelling observed in the corneas of enucleated eyes placed in a cold, moist chamber. However, this is not unexpected, since tissue function generally deteriorates more quickly when no attempt is made at its preservation. It would not be surprising if the isolated and perfused cornea under sterile conditions swelled considerably less in the cold than a cornea exposed to a stagnant pool of aqueous humor. Osmotic perturbation on endothelial surface. The experimental work presented in this paper dealt with osmotic perturbations on the epithelial surface alone, attempting to reduce the possible influence of external unstirred layers and experimentally induced changes in membrane coefficients. Fischbarg (1973) and McCarey & Maurice (personal communication) have performed a number of similar experiments, imposing osmotic increases at the endothelial surface with sucrose as the added solute, and it
125 COUPLED TRANSPORT IN CORNEA would be interesting to test the usefulness of the model by applying it to their experimental situation. For simplification, the model was first developed to deal with the flow of a single solute as outlined in the Theory section above. However, expansion to treat multiple solutes is straightforward if possible interactions between different solutes are ignored. For a number of solute species, k, eqn. (1 a) and (b) can be rewritten: r
ivi,I=LKin (APi t - RT I (ri, ACi, k, t r JSi,k, = k=1 ((1-
iO, k)Ci, k, tJvi,
+
(4a)
ji, k, tRTACi k,t).
(4b)
To simulate Fischbarg's experiment in which the bare anterior stromal surface was covered with oil, epithelial values were assigned as follows: -O = 1 and wo = LPO = JSO = 0. The stromal sucrose diffusion coefficient derived from Robinson & Stokes (1959) in a manner similar to that used for NaCl videe 8upra) was 3-2 x 106 cm2/sec. 320
-
._
E310 C 300
290 E
°
280
-
270
l 0
l
l
4
l
8 12 Time (min)
16
l l l 20406080
Fig. 9. Fit to a sucrose osmotic perturbation applied to the endothelial surface at t. The continuous calculated line was shifted temporally to match the onset of the thinning. The data points (@) are reproduced with the kind permission of Dr J. Fischbarg (Fischbarg, 1973); copyright by Academic Press, Inc. (London) Ltd.
The stromal sucrose reflexion coefficient was assumed to be 0. Endothelial Lp, Ja, and were assigned their average determined values (given later in Table 2). Simulation of this experiment converged with n > 7. The data points from one of Fischbarg's (1973) experiments are reproduced in Fig. 9. A good calculated fit was obtained by making adjustments in the endothelial sucrose reflexion coefficient and permeability. The initial slope of the thinning curve was fitted with endothelial OTsucrose =0*78, while fit to the latter portion of his experiment was made by setting endothelial LoseRT = 4-8 x 10-5 cm/sec. Calculation beyond the experimental time course predicted that the stromal thickness should return to its original steady state thickness due to the flow of sucrose into the stroma. Parenthetically, it is interesting to note that if the epithelium were left intact and perfused with Ringer, the model predicts that addition of sucrose to the aqueous humor bathing medium would cause steady-state stromal swelling. This behaviour has been observed experimentally by McCarey & Maurice (personal communication) and has been predicted theoretically by Friedman (1973).
o,~l,
W-acl
S. D. KLYCE AND S. R. RUSSELL
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8 12 16 20 60 Time (min) Fig. 10. Analysis of force gradients across the corneal endothelium during the osmotic perturbation of Fig. 9. Sucrose (suc.), NaCl, and pressure gradients are plotted in the same units for ease of comparison. Note the rapidity with which the imposed sucrose gradient is dissipated by its flow into the stroma. 0
4
For a clearer understanding of the component forces operative during this type of experiment, the time courses of the calculated transendothelial sucrose osmotic pressure, NaCl osmotic pressure, and hydrostatic pressure gradients are compared in Fig. 10. The initial changes in the solute gradients were rapid, with 90 % of the imposed sucrose gradient being dissipated within 20 min. (Since the influence of unstirred layers in the external solution bathing the endothelium was ignored in these calculations, the initial time courses calculated for the gradients were steeper to some extent than the physical situation; this error became less important after about 1 min from time 0 videe infra).) The NaCI gradient across the endothelium reacted to the volume flow caused by the sucrose gradient in a multiphasic fashion. In the first 3 min there was a net flow of NaCl from the stroma to the aqueous humor leading to a temporary reversal of the normal NaCl gradient. This was followed in time by a small amplitude oscillation about the steady-state level before the NaCl gradient returned to its original value. The fluid pressure gradient was biphasic with an initial rapid increase in magnitude. Note that the fluid pressure gradient approximately equalled the sum of the two solute gradients from 14 to 18 min. This produced the apparent steady thickness of the stroma observed during that interval.
COUPLED TRANSPORT IN CORNEA 127 Other predictions. There is some disagreement as to how the stromal oedema observed clinically in glaucomatous eyes arises. Bowman & Green (1976) reported that elevated intra-ocular pressure in the in vitro rabbit cornea diminished the rate at which the endothelium could thin the stroma, whereas Hodson (1974) found no significant influence of pressures up to 37 mmHg. This point was re-examined by determining the steady-state stromal thickness with the model cornea for several values of intra-ocular pressure ranging from 0 to 50 mmHg. Calculations showed the dependence to be linear and to have a slope of 0-145 ,um/mmHg. Hence, if intraocular pressure increased from 20 to 50 mmHg, the stroma should thin less than 5 jtm at the new steady state. This result implies that the clinical observation of stromal oedema in glaucomatous eyes is due to pathological changes in membrane properties and not to an intra-ocular pressure-induced fluid imbalance. Evaporation from the precorneal tear film in vivo produces an anteriorly directed fluid flow of about 3,ul./cm2.hr in the rabbit (Mishima & Maurice, 1961a), which is accompanied by a 5 % corneal thinning (Mishima & Maurice, 1961 b) compared to the 'closed eye' thickness. These effects are presumably caused by an increase in the tonicity of the tears, but the tear osmolarity in rabbits has not been measured with accuracy. However, Mishima & Maurice measured a 1-5 % increase in the osmolarity of the aqueous humor in the normal 'open eye' situation compared to the 'closed eye' conditions. With this information the model was used to calculate the probable effective tonicity of the tears for the open eye situation. With the tonicity of the aqueous humor increased by 4-5 m-osmole, the tonicity of the tears that would cause a 3 /sl./cm2. hr flow was found to be 16 m-osmolar (- 0.05 % wt./vol. NaCl). The stromal swelling pressure gradient (calculated to be 7 mmHg for the steady-state cornea) increased to 31 mmHg when there was evaporation from the tears. The increase in tear osmolarity predicted for the rabbit is similar to that measured in the human (Mastman, Baldas & Henderson, 1961). It should be noted that the force generated by normal evaporation is not sufficient to maintain normal stromal hydration as previously calculated by Friedman (1973). Applying the gradient produced by evaporation to the model with the endothelial solute pump set to 0 (a Friedman assumption), it was calculated that the stroma should swell considerably, reaching a steady stromal thickness 45 % greater than normal in about 30 hr. The corneal stroma swells rapidly in vivo after destruction of either of its limiting cell layers; however, the extent of swelling is considerably less than when only the epithelium is destroyed (Maurice & Giardini, 1951). The model predicted a similar phenomenon. Calculated swelling 1 hr after the removal of the epithelium produced a 25 % increase in stromal thickness, which is identical to that measured in vivo annd with a time course similar to that reported by Maurice & Giardini (1951). Unlike the average in vivo result, however, the calculated stromal thickness achieved a steady state at about 2 hr at increased thickness of 26-5 % over normal. In the above experiments stromas continued to swell beyond this time, but in individual cases some corneas were stable. When the effect of destroying the endothelium was calculated, the time course of the response in corneal thickness was close to that given by Maurice & Giardini. The fit was good for the first 30 min, after which the model stroma began to swell less rapidly than experimentally observed. It is noted that the equations used here to characterize stromal hydration dynamics are applicable over
S. D. KLYCE AND S. R. RUSSELL 128 a hydration range of 1-5-5'5. Hence, some discrepancy between observation and calculation is not unanticipated for large deviations away from normal steady state. Klyce, Neufeld & Zadunaisky (1973) have demonstrated that the rabbit corneal epithelium actively transports Cl from the stroma to the tears by a mechanism localized deep in the epithelium (Klyce & Wong, 1977). The rate of solute secretion by the epithelium in the presence of epithelial resting potential was found to be 0 in vitro unless pharmacological measures were taken to ensure stimulation of cyclicAMP production (Klyce, 1975). In the presence of theophylline, one such stimulator, the epithelium was found capable of thinning a previously swollen corneal stroma at the rate of 1-3 ,um/hr (bare inner surface blocked with silicone oil) averaging the thickness change over a 6 hr period (Klyce, 1977). The model predicts an identical rate (1 28 1m/hr) for this experiment over the same time period. Hence, the usefulness and validity of the model and its paramaterization should be apparent from the accurate predictions made for a wide variety of different experimental observations. DISCUSSION
Comparison to previous work Experinmental data. The time courses for changes in stromal thickness following changes in external solution osmolarity were considerably longer than anticipated from previous time sequence analyses (Mishima & Hedbys, 1967; Stanley et al. 1966). It had been thought that a new steady state in corneal thickness could have been achieved in 20 min or less, but this was not the case, as shown in Figs. 4, 5 and 9. A qualitatively similar conclusion was reached in Friedman's (1973) theoretical work. The apparent steady state, reached 10- 15 min after increasing the osmolarity of the endothelial bathing solution (epithelial surface blocked), is actually the thickness minimum in a longer transient whose steady state is the original stromal thickness (Fig. 9). Osmotic adjustments made on the epithelial surface also do not yield steadystate conditions in 20 min, although the rate of response does decrease substantially after this initial period of time. Representative results of previous determinations of corneal phenomenological coefficients and ion transport rates are compared with the current data in Table 2. The values listed for the current model are averages of the results presented in Table 1. The correspondence between endothelial coefficients measured in this study and previous determination is generally good, but there is a large difference in endothelial solute permeability and hydraulic conductivity compared to any previous determination. The rate of active solute transport by the epithelium was small, comparing favourably with previous tracer measurements (Klyce et al. 1973; Klyce, 1975). It is noted that there is no significant net solute flow across the in vitro rabbit corneal epithelium in the presence of resting potential unless active Cl transport is stimulated with theophylline or other agent (Klyce, 1975). Hodson & Miller (1976) measured a net endothelial HCO3 flux of 2-1 x 10-1o equiv/ cn2 . see from the stroma to the aqueous humor. Assuming that a cation accompanies the HCO3 flux and that the osmotic coefficient for the latter salt is similar to that of NaCI, thenr the HCO3 transport should be osmotically equivalent to a non-electrolyte
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