A mathematical formulation of intraocular pressure as dependent on secretion, ultrafiltration, bulk outflow, and osmotic reabsorption of fluid

A mathematical formulation of intraocular pressure as dependent on secretion, ultrafiltration, bulk outflow, and osmotic reabsorption of fluid Ernst H. Bdrdny An equation has been derived under simplified assumptions showing how the steady-state value of intraocular pressure depends on rate of secretion of aqueous, episcleral venous pressure, outflow facility at the chamber angle, colloid-osmotic pressure of the blood, systemic arterial blood pressure, pressure distribution over the vascular tree of the eye, and filtration properties of the vasculature in different intraocular regions. It is shown that intraocular fluid formation will be reduced by increased intraocular pressure in such a manner that the presence of a facility is simulated. In the conscious rabbit, this pseudofacility is estimated to account for about 10 per cent of the measured tonographic facility. Its absolute magnitude can be expected to rise with arteriolar vasodilatation (as might be caused by miotics) and to drop with arteriolar vasoconstriction. A method is outlined which should allow pseudofacility to be measured in the glaucomatous human eye. If intraocular pressure is not excessively high, changes in arteriolar tone will tend to cause parallel changes in pressure and pseudofacility simulating homeostatic adjustment of facility to pressure. -y- JLhe possibility that filtration from the vascular tree can contribute to formation of the intraocular fluid and that colloid osmotic forces can contribute to its removal has been recognized a long time. Equations governing an eye where these factors are the only important ones were published in 1946. 1 In later years, interest has mainly been focused on the other factors in intraocular pressure, which almost certainly are From the Department of Pharmacology, University of Uppsala, Sweden. This study was supported by Research Grant B 3060 from the Institute of Neurological Diseases and Blindness, United States Public Health Service, Bethesda, Md. This paper was read in part at the Glaucoma Research Conference, Hot Springs, a., May- June 1963. 584 of major importance in most cases: secretion, episcleral venous pressure, and outflow facility. 2 The following is an attempt to deal simultaneously with all these factors determining the steady state pressure of the eye. The treatment is only a first approximation and undoubtedly will have to be refined. Even at this stage, however, it helps to clarify the interactions between the many different factors. The mathematics employed are the simplest possible. To keep them so, it has been necessary to introduce a few unfamiliar concepts. A list of symbols appears at the end of the paper. I. The concept "pressure index" Fig. 1 is a schematic drawing of the vascular system of the eye. The main feature distinguishing it from other vascular

Mathematical formulation of intraocular pressure 5S5 n- -v territories is that the tissue pressure or intraocular pressure, p i( under the conditions dealt with here determines and is equal to the lateral pressure, p v, of the veins at their point of exit from the ocular cavity. (Note that p v here is pressure justinside the globe, not just outside.) The arterial pressure, p n, is the mean lateral pressure in the arterial tree outside the eye. The exact point where it is measured is immaterial, as long as it is central enough not to be appreciably influenced by the amount of blood flowing through the eye. The ophthalmic artery may qualify. As a start, consider p a and p v as constant. Take a certain pressure p x between p,, and p v ; with the exception of values of p x very close to p,, or p v, there will as a rule be a great many vessels on which there is one point (or rather cross section) in which the lateral pressure is p*. These points which have p x in common will still have a common pressure if p,, or p v changes a little, because their similarity in pressure depends on their position on the scale of resistance between vein and artery. This position can be expressed by a "pressure index" x which is defined as - P* " P v (1) The formula implies that for instance all points with x = 0.333 are one-third of the way between vein and artery, pressurewise. Moderate changes in p,, or p v will not alter this fact, which depends on the distribution of flow resistance throughout the tree. As long as the calibers of the vascular tree are unchanged, a certain pressure index identifies a certain group of points on a number of vessels. The same group will, however, acquire a different pressure index if for instance arteriolar vasodilatation redistributes and changes the resistance in the vascular tree. It is evident that the range of variation of pressure index is between 0 and 1, the lower value characterizing the veins at their point of exit, the upper one the arterial branch in which p., is measured. At intermediate points, the pressure in- Fig. 1. Schematic diagram of the vascular tree of the eye, to illustrate the concept "pressure index." essels with low pressure index are close to the veins on the pressure scale and mainly affected by venous pressure, since p.< =.vp«+ (1 " x)-p T. dex decides in which proportions p,, and p v contribute to p. v (Fig. 1). From now on, "point x" on a vessel means the point where pressure index is x. II. The forces involved in filtration Rearrangement of equation (1) gives the filtering hydrostatic pressure head acting from die lumen of the vessel at point x Px - pi = p* - p,- = x(p. - pv) (2) Osmotic forces will be disregarded except for the colloid-osmotic pressure, p o,,n. This is assumed to be constant, and fully active throughout the vascular territory. Thus, Pressure head for filtration = x(p n - p v ) - peon In assuming constancy of p c0 n one assumes that the colloid-osmotic pressure of the aqueous is negligible and in disregarding all other osmotic forces one assumes that the permeability of the blood-aqueous barrier is large enough to minimize the influence of crystalloid-osmotic differences. Where this is not true, for instance in die ciliary processes, the mobility of water under crystalloid-osmotic pressure is not impressive. 3 Anyhow, the assumption of one and only one p co ii is an approximation. III. The concept "conductivity coefficient for filtration" A conventional measure of conductivity for filtration across a membrane would be (3)

586 Bdrdny Investigator2 Ophthalmology December 1963 t Pi=Pv Y Conductivity coefficient for filtration t Fig. 2. Conductivity coefficient for filtration of the vascular wall as a function of pressure index. The black dot indicates the center of gravity of the area A under the curve. Its pressure index is x c. milliliters per minute per unit area and millimeters of mercury pressure difference. Since position along the vessel is not measured in length but in pressure index, a special measure for conductivity has to be used in the present treatment. Consider the part of the vascular tree located between pressure index x and pressure index x + Ax. It consists of a number of short segments of vessels, all of which carry the same lateral pressure. Some of the vessels may have very permeable walls, others may be thick walled. Some of them will be located superficially in the iris, others may be deeply embedded between double layers of epithelium in the ciliary processes. Some are retinal or choroidal, even if their contribution to the filtrate must be negligible. Some of the vessel segments are shorter than the others, because they are narrow and the pressure drops fast along them. In all of them the pressure drops by the amount Ax- (p a - p v ) from one end to the other. The contribution of filtrate per unit time from this aggregate is AF. It is proportional to the pressure head for filtration (determined by x), to the lengths of the individual segments (determined by Ax) and to a composite proportionality factor Y, which will be termed "conductivity coefficient for filtration." This coefficient Y embraces all the remaining local factors, such as number of vessels, their circumference, wall properties, location, etc. The conductivity coefficient so defined is evidently not constant all the way from artery to vein but is a function of x, Y(x). Thus we have AF = Y(x)-Ax- pressure head for filtration (4) The detailed distribution of Y(x) over the vascular tree is not known. It is evident, however, that Y is very close to zero when x is zero, at the point of exit of the large veins, and when x is close to 1, in the thickwalled arteries outside the eye. If filtration occurred here it would not be counted anyhow. Fig. 2 shows a diagram of Y as a function of pressure index x. It is possible but not proved that the maximum corresponds to the anatomic capillaries. The shape of the curve between x = 0 and 1 is pure guesswork and not essential to the argument. The total area below the Y-curve will be called A and represents the total conductivity for filtration or for colloid-osmotic reabsorption of the vascular tree. Its dimension is volume per unit time and unit pressure difference, similar to facility. asomotor changes in the eye, which affect pressure distribution over the vascular tree (without shutting off circulation in certain regions) will change the shape of the Y- curve but not affect A to any large extent. I. Filtration as a function of vascular and intraocular pressures Inserting the value for filtering pressure head from (3) into (4) one obtains AF = Y(x)-Ax-[x(p. - pv) - pcull] (5) Passing to the limit and rearranging gives df _ Total filtration rate F is obtained by integration from x = 0 to x = 1, h h +A

Mathematical formulation of intraocular pressure 587 -A Jl F = (p. - Pv)Jx-Y(x)dx - p coll JY(X) (7) The secoiid integral is the total area under the Y-curve, A. The first integral is the moment of the area under the Y-curve around the Y-axis. Hence it is equal to the area A times the x-coordinate x c of the center of gravity of the area (Fig. 2). Thus, filtration rate F can be expressed as F = (p. - pv)-a-xc - pco,,-a (8) Here, the first term represents outward filtration from the vessels as it would appear in the absence of colloid-osmotic pressure and the second term expresses total colloid-osmotic reabsorption to the vascular tree.. The pressure at the center of gravity of the area under the Y-curve: A measure of capillary pressure The pressure index x c of the center of gravity of the area under the Y-curve is hard to visualize. It can be translated, albeit somewhat loosely, into more common physiologic concepts, however. Making use of equations (1) and (8) one obtains F = A(po - Pv - pen,,) (9) Here, p 0 is the absolute pressure corresponding to the center of gravity. What is its physiologic meaning? Consider the statement: No net transfer of fluid into or out of the vascular tree will occur by hydrostatic forces if the excess of capillary pressure over intraocular pressure equals the colloid-osmotic pressure of the blood. This statement, which is a reformulation of Starling's hypothesis, is loose insofar as capillary pressure is ill defined; capillary pressure is a pressure region, not a pressure. However, if capillary pressure is defined as p c, the pressure at the center of gravity of the Y-curve, the statement becomes exact, as equation 9 shows: Whe = peon, F = 0 Thus, a possible interpretation of x c is x,. = the pressure index of the capillaries A word of caution is necessary. The Y- curve embraces all vessels contributing to intraocular fluid production or removal. It is quite possible that the several capillary beds present in the eye occupy different pressure regions. The idealized capillary with pressure p c may not exist at all and be only a mathematical construct, an "equivalent capillary" summarizing and averaging the properties of all the real capillaries. I. Pseudofacility Since, according to the assumptions, p v = p i; equation (8) shows that if intraocular and venous pressures are increased, for instance as a result of a tonometer being put onto the eye, F decreases. The rate of change of F with pressure is dp v dpi dpi Since p., has been defined as the pressure in a vessel where blood flow through the eye has no influence, changes in intraocular pressure will also leave p., unaffected. Therefore, dp,, -r±- = 0 and equation 10 simplifies into df (ID dp, Ax c thus represents a rate of decrease in filtration with increasing intraocular pressure. Its dimension is volume per unit time and unit pressure rise, exactly as facility. Ax c can be termed "pseudofacility" if one considers secretion to be pressure independent as is done in tonography or it can be termed "ease of suppression of formation" or "ease of increase in reabsorption." The possible existence of such a pressureproportional decrease in fluid formation was discussed by Grant 4 in his exposition of tonography at the Third Macy Glaucoma Conference. It is interesting to note that pseudofacility increases with x c, the pressure index

588 Bdrdny Investigative Ophthalmology December 1963 of the capillaries. Since miotics are vasodilators, they can be expected to increase x c. Moreover, it is conceivable that unfolding of the iris somewhat increases total conductivity for filtration, A. It is therefore possible that one component in the facilityincreasing action of miotics is their effect on pseudofacility. Conversely, vasoconstrictor catecholamines should cause a drop in pseudofacility. Another special case is that of the patient with arterial hypertension. It is well known that his intraocular pressure is normal. Because of the diffuse increase in arteriolar resistance he can be expected to have a low x c and (assuming a normal A) a low pseudofacility. It is evidently important to know how large pseudofacility is under various conditions. Direct experiments are still lacking. An estimate for the rabbit eye is made in Section III. II. Intraocular pressure as a function of secretion, filtration, outflow facility, and episcleral venous pressure It is not known whether secretion from the ciliary processes is markedly affected by the hydrostatic pressure difference across the epithelium. Certainly secretion of fluid can go on without such a pressure difference. 5 If there is a dependence, it would formally behave as filtration and be taken care of by the composite conductivity coefficient Y. Hence, in the following, true secretion will be considered pressure independent and its rate designated S. The symbol for episcleral venous pressure will be p c and refers to vessels so far downstream that their pressure is independent of aqueous flow. 0 Conventional outflow facility through the chamber angle and associated veins is C. Filtration rate F can be positive or negative, depending on whether net production of aqueous or net reabsorption occurs. With formula (8) a positive sign means net production of aqueous and one has P. = P. (12) Inserting F from equation (8) into (12) and rearranging gives equation (13) at the bottom of the page. This equation summarizes the influence of all the factors taken into account in the present treatment. It is seen that if the contribution of the vascular tree is neglected, by putting A =0, the classical relation of Goldmann 2 between p^ p e, C, and S results. III. Estimation of the size of pseudofacility As a first approximation, one can consider episcleral venous pressure p 0 to be independent of arterial pressure p,,. Secretion S and colloid-osmotic pressure p, :o n are also independent of this variable. Hence, a change in arterial blood pressure will act on intraocular pressure only through the third term of equation (13) and iel = (14) Simultaneous measurements in the conscious rabbit of changes in ophthalmic artery pressure (by an indirect method) and in intraocular pressure (by calibrated tonometry) following unilateral carotid ligation are available. 1-7 The change in steady state intraocular pressure was about 4 mm. Hg for a change in ophthalmic artery blood pressure of 40 mm. Hg. Hence 0.1 (15) C + Axe Available tonographic measurements of C embrace Ax c and have given a value of 0.33 in the conscious rabbit. 8 This then is the denominator of expression (15) and pseudofacility Ax c in the rabbit under physiologic conditions is of the order of h y- *y P< = Axc C + Axc Axc (13)

Mathematical formulation of intraocular pressure 589 Lx. 0.03 unit. Arteriolar dilatation could well increase pseudofacility a few times, but it is unlikely to become dominant as long as the normal outflow channels at the angle are patent. No corresponding figures are available for man. A study of the steady state effect of an increase in episcleral venous pressure on intraocular pressure should be possible and would allow an estimate of pseudofacility to be made with the aid of the second term of equation (13) which gives: Ap, _ Ap0 ~~ C (16) If the rise in p c in this experiment is produced by a cuff around the neck or by pressure breathing, it will be accompanied by a rise in pressure in all the periocular veins. It will then be necessary to limit the change in venous pressure so that it keeps well below resting intraocular pressure. Too large a rise in venous pressure is transmitted to the intraocular veins and causes a rise in intraocular pressure no longer exclusively due to the rise in p c. Probably, sufficient precision will be easier to obtain in cases of glaucoma, where a considerable venous pressure rise can be allowed, than in normal subjects, in whom the permissible rise is only a few millimeters of mercury. IX. The pressure during angle-closure While the factors determining filtration (Pa, Pcoiij A, X C ) perhaps play only a minor role in an eye with normal C, they are very important indeed in an eye with blocked outflow at the chamber angle. If C is made equal to zero in equation 13 one obtains: S - A'Proll pi (with blocked angle) = p., + ^ (17) The assumptions behind the derivation of equation (13) certainly do not hold very well for the large pressure changes in angle closure, but equation (17) still should be of interest. It indicates (not unexpectedly) that even with a completely blocked angle, the intraocular pressure will level off below mean arterial blood pressure, provided that colloid-osmotic reabsorption Ap co u exceeds secretion S. If this is the case, the second term of equation (17) is negative and decreases with increasing pseudofacility Ax c. If in this situation a vasodilating miotic is given and does not succeed in freeing the angle, the increase in pseudofacility will reduce the negative term and increase pressure, making the situation worse. This seems to be in accord with clinical experience that strong miotics "add to the congestion." X. Pressure effects of changes in arteriolar tone The pressure-increasing effect of vasodilatation just discussed makes it interesting to enquire under which conditions the increase in pseudofacility Ax c brought about by vasodilatation outweighs the increased filtration of fluid. Does vasodilatation ever cause a decrease in pi? The factors entering into the situation become clearer on differentiation of pi in equation (13) with respect to x c : dp, _ - [C(p» - p.) + A-p,.,, - S] (18) (C With C in the normal range, about 0.3, and p a - p c of the order of 10% the positive terms inside the square brackets certainly dominate by a wide margin, the differential is positive, and intraocular pressure rises with arteriolar vasodilatation. Only if C becomes quite small and A-p,,,, is small enough, the possibility arises for S to dominate over the other terms. The expression then becomes negative and intraocular pressure drops with arteriolar vasodilatation. The conditions for this to happen are the same as make for a very high intraocular pressure and in fact it is easy to show that the sign of the square bracket changes from positive to negative only as pt reaches and exceeds p.,. Thus, changes in arteriolar tone occurring in the normal or moderately high range of intraocular pressures cause changes in pressure and in pseudofacility both in the same

590 Bdrdmj estigntioe Ophthalmology December 1983 direction. This simulates a homeostatic adaptation with an increase in facility tending to counteract a rise in pressure. Comments Despite the simplifying assumptions on which they are based, the relations derived are probably not very far from the truth. How important they are clinically is quite another question, the answer to which will have to wait until reliable estimates of A and x c have been obtained for the human eye under various conditions. REFERENCES 1. Barany, E. H.: The influence of local arterial blood pressure on aqueous humour and intraocular pressure. An experimental study of the mechanisms maintaining intraocular pressure. I. Intraocular pressure and local blood pressure from seconds to hours after unilateral carotid occlusion. A search for homeostatic reflexes in the undisturbed eye, Acta ophth. 24: 337, 1946. 2. Coldmann, H.: Abflussdruck, Minutenvolumen und Widerstand der Kammerwasserstromung des Menschen, Docum. ophth. ('s. Grav.) 5-6: 278, 1951. 3. Auricchio, C, and Barany, E. H.: On the role of osmotic water transport in the secretion of the aqueous humour, Acta physiol. scandinav. 45: 190, 1959. 4. Grant,. M.: In Newell, F. W., editor: Glaucoma, Tr. of the Third Conference, New York, 1959, Josiah Macy, Jr. Foundation, pp. 19-20. 5. Berggren, L.: Unpublished, quoted by Barany, E. H.: Pharmacology of aqueous humour formation, in Uvniis, B., general editor: Proc. First International Pharmacological Meeting, Stockholm, 1961, vol. 4, p. 109, Oxford, 1963, Pergamon Press, Inc. 6. Goldmann, H.: In Newell, F. W., editor: Glaucoma, Tr. of the Second Conference, New York, 1957, Josiah Macy, Jr. Foundation, pp. 207-210. 7. Barany, E. H.: The influence of local arterial blood pressure on aqueous humour and intraocular pressure. An experimental study of the mechanisms maintaining intraocular pressure. II. The recovery of intraocular pressure, arterial blood pressure, and heat dissipation by the external ear after unilateral carotid ligation, Acta ophth. 25: 81, 1947. 8. Becker, B., and Constant, M. A.: The facility of aqueous outflow, A. M. A. Arch. Ophth. 56: 305, 1956. Symbols poj mean blood pressure in ophthalmic artery or even more centrally pc, Pf.ni, pn, pi, pv, px, mean blood pressure corresponding to center of gravity of area under Y-curve. Loosely, capillary pressure colloid-osmotic pressure of blood episcleral venous pressure intraocular pressure pressure in intraocular veins at their point of exit from the ocular cavity a blood pressure between p,, and pv x, pressure index corresponding to p.* XK, pressure- index of center of gravity of area under Y-curve. Loosely, pressure index of capillaries Y,Y(x), conductivity coefficient for filtration of the vascular wall A, total conductivity for filtration of the vascular tree C, facility of outflow through the chamber angle and its associated veins F, filtration rate S, secretion rate -