ACTA O P H T H A L M O L O G I C A 64 (1986) 26-32

Hypotony and retinal detachment

Tron Solberg? Tor Ytrehusl and Amund Ringvold

Division of Applied Mechanics, NTHl (Head: Henry Oiann) and Department of Ophthalmology (Head: Amund Ringvold),

University of Trondheim, Norway

Abstract. The present work has been based on the assumption that the decreased intraocular pressure in eyes with retinal tletachment is due to an abnormal posterior aqueous outflow, in addition to the normal anterior pathways. Subretinal fluid may leak out either through the peri-optic connective tissue or across the retinal pigment epithelium. In order to test the validity of these alternatives itgainst clinical observations, a fluid dynamic model ha\ been developed to calculate intra- ocular pressure-drop versus detached area in both cases. In contrast to the choroidal alternative, the numerical results from the peri-optic route exhibit qualitative agreement with clinical observations. It is concluded that both clinical and tht.oretica1 findings lend support to the hypothesis that the intraocular pressure-drop in eyes with retinal detachment is due to drainage of subretinal fluid via the peri-optic tissue.

Key words: hypotony - intraocular pressure - retinal detachment - outflow facility.

Ad

A, C

Fl"

FA

FN

Fs

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List of symbols

detached area, (m') total fundua area, i.e. area of circular disc, (mZ) facility of outflow through

m3 the trabecular meshwork, ( - )

P a . S

m3 formation r.tte of aqueous, ( -)

trabecular outflow rate in an eye

with retinal detachment, (-)

trabecular outflow rate in a normal eye, (-)

subretinal aqueous outflow rate, ( -)

m3

m3 S

m3

Fw aqueous outflow rate from a detached area m3

through the corresponding eye-wall, ( -)

thickness of subretinal interstitial matrix, (m) thickness ofthe eye-wall, (m)

+ k Cartesian unit vectors

permeability of aqueous flow in subretinal interstitial matrix, (m') permeability of aqueous movement in eye-wall tissue, (mZ) length, (m)

unit normal vector pressure, (Pa) background pressure, i.e. choriocapillary pressure, (Pa) intraocular pressure in an eye with retinal detachment, (Pa) episcleral vein pressure, (Pa) intraocular pressure in a normal eye, (Pa) background pressure, i.e. pressure around optic nerve, (Pa) detached radius, (m) inner radius of the circular disc, (m) outer radius of the circular disc, (m)

+ m v velocity vector, ( -)

x, y. Cartesian coordinates, (m)

V = 0 normalized pressure 0 d

p

+ + + i 3/3x + j 3/3y + k 3/32 del operator, [m-l]

normalized intraocular pressure in an eye with retinal detachment dynamic viscosity of aqueous, (Pa.s) non-dimensional physical parameter in the subretinal outflow model non-dimensional physical parameter in the choroidal outflow model

q

The pathogenesis of the hypotony following retinal detachment has challenged ophthalmologists for more than one century, and the results of the numerous efforts to explain it may be grouped in the following way: 1) The retinal detachment causes decreased aqueous humour production for some unknown reason (Becker 1963; Dobbie 1963; Regan & Rousseau 1966; Syrdalen 1970). 2) The aqueous secretion is normal. However, the detach- ment introduces an aberrant posterior pathway for fluid exit through the pathologic vitreous humour and the retinal hole into the detached subretinal space (Leber 1916; Kleiner 1933). From here on the abnormal outflow may principally proceed either through the non-detached subretinal space to the optic disc, the peri-optic tissue and the extrabulbar compartment (Ringvold 1980), or across the retinal pigment epithelium to choroidal vessels (Moseley et al. 1984). This additional post- erior drainage makes the intraocular pressure (IOP) stabilize at a lower level.

There is still no agreement as to which descrip- tion is the most correct one. Experimental studies have shown that subretinal fluid floats freely into the peri-optic tissue without any obstades at the optic disc (Flage & Ringvold 1980), so this pathway is indeed possible. On the other hand, there is also some evidence that subretinal fluid is actively trans- ported across the retinal pigment epithelium by a mechanism independent of the IOP (Frambach & Marmor 1982; Pederson 1982). In the peri-optic drainage alternative the magnitude of the pressure drop obviously depends on the shortest detach- ment-disc distance, whereas in the choroidal model is it related to the size of the detached area. Accordingly, it seems reasonable that a graph showing IOP-drop versus size of the detached area must be quite different for these two alternatives. These graphs may be determined either from clinical observations or on a theoretical basis.

The clinical curve is already available (Ringvold 1980) based on 3 separate studies (Kleiner 1933; Dobbie 1963; Syrdalen 1970). It should be stressed that none of them were performed for this parti- cular purpose, so they were undoubtedly non- ,

biased. It is therefore taken as more than coin- cidence that they all independently showed essen- tially the same result.

The present investigation has been undertaken in order to elucidate the problem from a theore- tical point of view. The graph showing IOP-drop versus deached area will be calculated in a fluid

dynamic model both for the peri-optic and the choroidal draining alternative. Subsequently it will be possible to determine which one of these curves matches the clinical observations best, i.e. which drainage route is the most likely one to occur in vivo.

Physical model applied on peri-optic drainage The IOP is determined by conservation of aqueous in- and outflow. The aqueous, here taken as a Newtonian fluid, is formed in the ciliary body at a constant rate (FI"), independent of the amount of retinal detachment, so the flow rate is assumed to be the same in both normal and detached eye. The uveoscleral outflow, which is unaltered by IOP- changes, is not included in the calculations. The trabecular outflow (FN) in normal eyes is con- sidered to be dependent on the difference between IOP (pi) and the episcleral pressure (p,).

Regarding the retinal detachment, the trabecular outflow (FA) is appointed by the pressure dif- ference pd-pe, where pd is the detachment IOP. As soon as a detachment occurs, a new outflow path- way through the subretinal space to the peri-optic tissue is formed. The subretinal outflow (Fs) is modelled by a creeping flow in a porous medium, assuming that the viscous force is in balance with the driving pressure force. Resistance of aqueous

L

t

Q Fig. 1.

Physical model for subretinal aqueous outflow: aqueous flow rate Fs through a subretinal route from the detached boundary to the perioptic connective tissue due to the pressure difference pd-Po. rs = 30.75 mm; r, =

0.75 mm; hs = 50 pm.

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flow in the porous subretinal material is described by the permeability (kJ. The pressure gradient is related to the difference pd-po, where po is the background pressure around the optic nerve. The pathophysiologic- condition prescribes pi > pd >po. The pressure po, which is unknown, is later absorbed by normalization. Still, the relation be- tween po and p. is decisive for the kind of trabe- cular flow in the model, i.e. po > pe ensures trabe- cular outflow, whereas po I pe implies no trabe- cular flow at all, or even inflow.

To describe the flow of aqueous in the subretinal material, the extension of the substance will be modelled as a circular disc with a constant thick- ness, Fig. 1. Here, h, is the thickness of the space between the external limiting membrane and the pigment epithelium filled up with rods, cones, and interstitial matrix, rs is the meridian distance from the centre of the optic disc to the ora serrata and ro is the radius of the optic nerve. The detachment then occurs as part of an expanding circle with instantaneous radius (rd) centered at an arbitrary point (Q) on the outer boundary of the disc.

As a retinal detachment starts to develop, the subretinal material is almost impermeable to aqueous flow due to the low permeability, and the major part of the outflow rate is through the anterior pathway. The IOP is therefore little disturbed by the detachment. As the detachment advances towards the optic disc, the pressure gradient within the subretinal material becomes steeper, the subretinal rate of aqueous outflow increases, and the IOP decreases. When the de- tachment has reached the optic disc, the posterior outflow will not increase any more, and the IOP will be appointed by the background pressure of the peri-optic connective tissue itself.

Governing equations Aqueous is formed in the ciliary body at a constant flow rate,

F1,l = cor1st. ( 1 )

The rate of trabecular aqueous outflow in a normal eye is given in analogy with Poiseulle’s law as

FN = C(pj-pe) 9 (2)

where C is the facility of flow. Conservation of mass in the cavity of the eye gives for

Eq. (1)

Fln = C(pi-pe) (3)

The rate of trabecular aqueous outflow in an eye with retinal detachment is described through a similar expres-

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-r

Fig. 2. Boundary conditions: computational domain Q, boun- daries @, r, P with prescribed pressure and velocity

values.

sion as Eq. (2), but with an IOP pd, which is generally different from the pressure pi of the normal case;

FA = C(pd-pe) (4)

To determine the subretinal outflow rate, the creeping flow of a viscous fluid in a porous medium is considered. The migration, or flow, of a tracer fluid into a biological structure consisting of some solid matrix with interstitial voids, are in many cases described by Darcy’s law for flow in a porous medium (Dullien 1980).

( 5 )

The Darcy’s law in this form expresses the fundamental assumption that the viscous force is in balance with the driving pressure force in a steady, non-accelerated flow.

Conservation of mass in the subretinal tissue is expres- sed by the continuity equation for an incompressible fluid.

0.3 = o (6)

Combining Eqs. ( 5 ) and (6), assuming constant p and ks yields the Laplace equation for the pressure distribution in the subretinal tissue 8,’Fig. 2.

w p = o in Q (7)

The pressure is therefore determined by finding the solution of Laplace’s equation subject to the boundary conditions introduced in Fig. 2: the pressure around the optic nerve is equal to the background pressure, i.e.

P = Po on @ (8 )

The pressure along the detached boundary is the un- known intraocular pressure, i.e.

P = Pd on X (9)

The zero-flow condition of the outer subretinal boundary yields

?.d = 0 on r (1")

Combining Eqs. (5 ) and 10) gives for the zero-flow condition

5.10

5 =loo Ad 0.8 1 .o A i

_ - aP - 0 an on

The totdl aqueous flow rate into the subretinal tissue is now obtained by integrating the flux of the velocity vector? over the surface X.

Fs = - l?.i?hsdl (12) x Here the minus sign is introduced since fluid outflow from the cavity of the eye is taken positive. Combining Eqs. ( 5 ) and (12) yields

Conservation of mass in the cavity ofthe eye gives

FI" = FA + Fs (14)

1.0

0.8

0.6

0.4

0.2

0

//i

Introducing Eqs. (3), (4) and (13) into Eq. (14) yields the expression

As normalized pressure we introduce

@ = p-po pi-po

The Laplace equation for the pressure, Eq. (7) still reads

wo = 0 in 51 (17)

The bounddry conditions, Eqs. (8), (9) and (1 1) yield

o=o on @ (18)

a@ - " - - an on r

Here the normalized intraocukar pressure o d varies from unity to zero as, respectively, no retinal detachment occurs, and the optic disc is reached. The continuity equation of aqueous, Eq. (15) yields

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kshs . P

Here = Ci- IS a physical parameter for the problem at hand.

The numerical problem is now to solve Eqs. (17)-(20) for a given parameter E, for a guessed value of the intraocular pressure @d, until Eq. (21) is satisfied.

Numerical results and discussion The potential problem Eq. (17) with its mixed boundary conditions Eqs. (18)-(20) discussed before was solved numerically by means of the Boundary Element Method with linear variation along the elements. The theory and simple com- puter programs are discussed in detail in Brebbia (1980). The intraocular pressure @d was iterated by the Secant Method to satisfy the continuity equation, Eq. (21).

The only physical parameter in the model, kbs

CL namely 5. = C / - is not known nominally, be-

cause no permeaility data (k,) on aqueous flow in subretinal tissue were found. Numerical calcula- tions were therefore performed for three different values of the parameter 5, namely 5 = 1, 10 and 100 to include the effect of the permeability k,.

In Fig. 3 the intraocular pressure drop 1-@d is

plotted versus normalized detached area - along

with clinical findings (see Ringvold 1980). The curves A and B are based on mean values from Syrdalen (1970). Curve A is based on the total material, and curve B on data derived from 1 - 14 days old detachments.

According 1 0 the 3 clinical studies taken together, maximum hypotony is present in the range from below 50% to 66% detachment, which corresponds to the interval in which the detach- ment reaches the optic disc. In our 2-dimensional model this limiting situation occurs at exactly 37.5 ’% detachment. The discrepancy between the clinical findings and the computational model as far as this limit is concerned may partly be due to the abnormal planar geometry in the latter, and it may also indicate that the expanding detachment rim does not exactly imitate the circular form.

By comparison with the clinical findings, and with the computational results for 5 = 10 and 5 =

100, the case = 1 in Fig. 3 exhibits some contra- dictory features. This may suggest that the perme- ability k, for the interstitial matrix is overestimated in this case. Realistic values for the parameter 5

Ad Ai

therefore appears to lie in the range 10-100 for the problem at hand.

A different aspect is that the interstitial matrix between the external limiting membrane and’ the pigment epithelium is soluble in saline (Berman 1969). The physical properties of this substances may therefore be modified through the flow. This would tend to increase the permeability and, hence, reduces the slope in the experimental pressure-area relation in Fig. 3.

The clinical curves have 2 characteristic features in common (Ringvold 1980): 1) There is a bend in the first part of the curves from where they raise steeply to their maxima. 2) The maximum point happends to occur roughly around 50% detached area. Both of these features are principally imitated in the graph for calculated peri-optic drainage.

Regarding the second half of the clinical curves, the results are 1ess.consistant. It seems that this to some extent may refer to differences in the disease duration combined with possible reparative proces- ses. The calculated curve is, on the ohter hand, by definition running horizontally at maximum level from 37.5 to 100% detachment.

Comparison with choroidal outflow We next consider the possibility that aqueous leaves the subretinal space through the pigment epithe- lium of the detached area. The model of retinal detachment in Fig. 1 will be kept, but the abnormal aqueous outflow is now in the z-direction vertical to the ‘retina’. The thickness of the eye-wall (h,) is estimated to 0.1 mm, roughly corresponding to that of the retinal pigment epithelium and the Bruch‘s membrane, assuming’ the drainage is mainly performed through the choriocapillaris. The background pressure is then the choriocapil- lary pressure (pc). As for the peri-optic drainage model the condition pc > pe ensures trabecular outflow.

The formation- and trabecular elimination rates of aqueous are as previously given by Eqs. (3) and (4). The governing equations for the abnormal aqueous outflow then read

Fw = vz . Ad (23)

Unlike the previous case, the flow rate now becomes proportional to the detached area. Integration of Eq. (22) combined with Eq. (23) yields

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-2 1-ed 4 rl .lo

1.0 c

06 -! I rl.10- /

0 0 0.2 0.4 0.6 0.8 1 .o

Fig. 4. Normalized pressure drop versus normalized detached area.

Conservation of mass in the cavity of the eye gives

Fin = FA + Fw (25)

By introducing Eqs. (3), (4) and (23) we obtain an expression for the intraocular pressure;

Here the normalized intraocular pressure is defined in analogy with Eq. (16) by

kwhw . and the parameter q = C/ IS similar to 5 previously CL

found for subretinal outflow.

The intraocular pressure drop 1 - @ J is plotted

versus normalized detached area- in Fig. 4. Ad Ai

It is seen that all the curves in Fig. 4 strongly contradict the character of the clinical findings shown in Fig. 3. In addition, maximum hypotony

- Ad

A i

occurred in our subretinal outflow model and in the clinical findings at 37.5 and roughly 50% detachment, respectively, whereas maximum hypo- tony in the trans-choroidal model occurs at 100% detachment.

The model of radial outflow through the eye- wall is therefore inconsistant with clinical findings. Accordingly, our main conclusion is that both clinical observations and numerical results lend support to the hypothesis that decreased IOP in eyes with retinal detachment is due to aqueous outflow through a subretinal route to the peri-optic connective tissue.

References

Becker B, in discussion of Smith J L (1963): Retinal detachment and glaucoma. Trans Am Acad Ophthal Otolaryngol67 : 73 1-732.

Berman E R ( 1969) : Mycopolysaccharides (Glycosamino- glycans) of the retina: identification, distribution and possible biological role. Mod Probl Ophthalmol 8: 5-31.

Brebbia C A (1980): The Boundary Element Method for Engineers. Pentech Press, Plymouth.

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Dobbie J G (1963): A study of the intraocular fluid dynamics in retinal detachment. Arch Ophthalmol69: 1.59- 164.

Dullien F A L (1980): Porous Media, Fluid Transport and Pore Structure. Academic Press, N.Y.

Flage T 8c Ringvold A (1980): Demonstration of a diffusional pathway between the subretinal space and the juxtapapillary iissue. An in vitro experiment using horseradish peroxidase as a tracer. Acta Ophthahol (Copenh) 38: 899- 907.

Frambach D A & hlarmor M F (1982): The rate and route of. fluid resorption from the subretinal space of the rabbit. Invest Ophthalmol Vis Sci 22: 292-302.

Kleirier L (1933): Dcr intraokulare Druck bei Netzhaut- abliisung. Graefes .\rch Ophthalmol 129: 485-506.

Leber T ( 1916): In: Graefe-Saemisch-Hess: Handbuch der gesamten Augenhelkunde. Bd VIII2: 1416- 1428. Wilhelm Engelmaiin Verlag, Leipzig.

Moseley H, Foulds W S, Allan D & Kyle P M (1984): Routes of clearance of radioactive water from the rabbit vitreous. Br.1 Ophthalmol68: 145- 151.

Pederson J E (1982): Experimental retinal detachment. 1V. Aqueous humor dynamics in rhegmatogenous detachments. Arch Ophthalmol 100: 1814- 1816.

Regan C D J & Rousseau A P (1966): The intraocular dynamics of eyes with retinal detachment. Am J Oph- thalmol61: 696-702.

Ringvold A (1980): Evidence that hypotony in retinal detachment is due to subretinal juxtapapillary fluid drainage. Acta Ophthalmol (Copenh) 58: 652-658.

Syrdalen P (1970): Intraocular pressure and ocular rigidity in patients with retinal detachment. I . Pre- operative study. Acta Ophthalmol (Copenh) 48: 1024- 1035.

Received on March 8th, 1985.

Author's address:

Dr. Amund Ringvold, M.D., Dpt. of Ophthalmology, University of Trondheim, N-7000, Trondheim, Norway.

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