An electrical model of the human eye

Transcription

1 An electrical model of the human eye I. The basic model W. K. McEwen, Marvin Shepherd, and Earle H. McBain An electrical model which simulates the action of the human eye is developed from experimental data and current concepts. Three fundamental parameters of the eye (volume, pressure, and time) are directly translated into analogous electrical elements. All other parameters are derived from these fundamental ones. The classical elastic Friedemoald model is compared to the viscoelastic model. M.any physical systems, hydrodynamic, mechanical, acoustical, and electrical, are analogous. 1 This article is concerned with the development of an electrical model which is analogous to the hydrodynamic and mechanical properties of the human eye. In general, changes in these properties of the eye, occasioned by any test, procedure, or altered physiology should be matched or reproduced by this electrical model within the limits of its error. If there is no correspondence, either our present concepts of the mechanism of the action of the eye or the experimental data regarding its fundamental properties are faulty. This model then allows us to test the validity of concepts and data by comparing the model's action with the action of the eye during clinical or experimental tests. Because the model performs in a time continum 60 times faster than the eye, and From the Department of Ophthalmology and the Francis I. Proctor Foundation of the University of California-San Francisco Medical Center, San Francisco, Calif. Supported in part by National Institutes of Health Grant No. NB also because supplemental electronic circuitry (to be given in later papers) can substitute for many manual procedures, reiterative matching or reproducing techniques are facilitated. Analogous quantities and units In the system of the eye there are three fundamental quantities, volume, pressure and time, from which all other quantities are derived. These quantities are translated into the electrical system by changing the units without changing the symbols. Volume (V) in microliters (fil) of the eye system is translated into the electrical system as microcoulombs (ju-q). Similarly, pressure (P) in mm. Hg becomes volts, and time (t) in minutes becomes seconds. This last translation allows the model to act sixty times faster than the eye. For instance, a 4 min. tonogram of the eye will be followed by the model in 4 sec. Other parameters of the eye are easily derived from the above quantities and translated into electrical units. Flow (F) in /xl per minute translates into juq per second, which by definition is microamperes (jiamp), and facility of outflow (C) is the ratio of flow to pressure (/JL per minute per mm. Hg) which becomes ^amp.

2 156 McEioen, Shepherd, and McBain Investigative Ophthalmology April 1967 per volts. By Ohm's law, the ratio of microamperes to volts is the reciprocal of resistance in megohms. The outflow resistance in megohms (abbreviated R ou t) is thus seen as the reciprocal of facility of outflow. Other terms will be developed later, and all symbols will be found in Table I. The model and the eye under steady state conditions The eye maintains a steady pressure because the rate of inflow (secretion) is equal to the rate of outflow. The rate of inflow is probably determined mostly by the active process of secretion of aqueous by the ciliary body. The inflow rate is probably not influenced by pressures within the physiological range, but is considered to be constant under steady state conditions. The electrical analogue of constant secretion is constant current. This is obtained by a source of electrical energy with a high internal resistance which tends to maintain a constant current independent of external circuit changes. The actual component in the model is a high-voltage source (battery) in series with a large resistance. The series resistance is kept at least ten times greater than the value of any external resistance. This means that fluctuations of current, due to the external resistance changes, will be less than 10 per cent. The rate of outflow is related to the pressure drop from inside the eye to the episcleral venous pressure (P o - P v ) and the resistance to flow (reciprocal of facility). The episcleral venous pressure has been measured 2 ' 3 and found to be constant within a few mm. Hg. This indicates that the episcleral venous plexus represents a constant pressure pool uninfluenced to any large extent by the outflow or the resistance to outflow. The electrical component analogous in this situation is a voltage source (battery) which is variable within small limits. The resistance to aqueous outflow from the eye is represented electrically by a resistance to current flow, which is a resistor. The resistance from inside the eye to the episcleral venous system is probably not uniform that is, the resistance may be greatest across the inner wall of Schlemm's canal and less in the corneoscleral trabecular mesh work or in the intrascleral plexus. Since we are not yet in a position to measure the individual resistance at intermediate points, we will call the total resistance to outflow (1/C) and represent it with a single electrical resistance (R O ut)- The constant current device may now be associated in a circuit with the outflow resistance and the voltage source representing episcleral venous pressure. A diagram of the steady state condition of the Table I. Conversion of units between eye and model Quantity Fundamental Volume Time Derived Flow Facility of outflow Resistance to outflow Ocular rigidity dp/dv Reciprocal of ocular rigidity dv/dp Symbol V P T mm. Hg min. Eye Units nq volts sec. Electrical F /tl/min. /iq/sec. = /xamp C /tl/min./mm. Hg Ma/volts Rout mm. Hg/ il/min. volts/juamp = megohms K /d ( M L) d (volts )/d(mq) CAP d(/il)/d(mm. Hg) d(fiq)/d (volts) = /xfarads Note that Rout is the reciprocal of C, and CAP the reciprocal of K.

3 Volume 6 Number 2 An electrical model of the human eye. I 157 eye is compared with the analogous electrical circuit in Fig. 1. It will be noticed that the well-known formula 4 for the eye under steady state conditions (F = C [P o - P v ] ) describes both the eye and the electrical model when the appropriate units are used. The model of the eye under dynamic conditions Any change in pressure in an eye must be caused by a change in the stress in the walls of the ocular coats. Thus, to represent dynamic conditions with changing pressure we must take into account the stressstrain relation of the ocular coats. The stress-strain relation may be divided into the immediate stress-strain relation 5 commonly known as ocular rigidity and the time-dependent stress-strain relation known as stress-relaxation or change of ocular rigidity with time. 0 The immediate stress-strain relation or ocular rigidity has been determined by many investigators, culminating in a unifying formulation. 5 In developing our model, we are using an average of the data of McBain (0.023P ) and Prijot (0.025P ) on enucleated eyes. Both authors used modern rapid methods for determining the immediate stress-strain relation, thus avoiding complications of the time-dependent variation, and the data of both were in close agreement. We will use, also, Friedenwald's data for comparative purposes; the data of other investigators may be used by substituting those particular numerical data in the translation from eye quantities to electrical quantities. The stress-strain relation for the eye is dp per dv. We will use 5 finite quantities rather than infinitesimals: dp/dv ~ AP/AV = P Friedemvald = P average of McBain and Prijot From Table I it can be seen that ocular rigidity is K in units of To convert K to capacitance (CAP), we need to take the reciprocal, i.e., and express the value in microfarads, i.e., d (volts) This results in the following: 1 CAP = Friedenwald 0.050P CONSTANT CURRENT DEVICE I 1 F I J INTRAOCULAR OUTER COATS C(Po-Pv) Po R or Fig. 1. Comparison of eye and model under steady state conditions. average of 0.024P McBain and Prijot. It may be noticed that the capacitance value changes with pressure. Since variable capacitors in the microfarad range are difficult to obtain, we have employed the device of using a fixed capacitor over a pressure range which would be covered by allowing an error of ± 20 per cent in the capacitance value. The capacitance values obtained from the above equations and the pressure ranges used are given in Table II. The actual value used will therefore depend upon: (1) which investigator's values are used, (2) the pressure range over which the model is operating, and

4 158 McEwen, Shepherd, and McBain Investigative Ophthalmology April 1967 Table II McBain-Prijot Capacitance value range Friedenwald Capacitance value range (3) changes which might be made to match the ocular rigidity of a specific eye if it is known that that eye has an ocular rigidity much greater or less than the average value. The time dependent stress-relation has been determined quantitatively by St. Helen and McEwen, 0 and Itoi and associates, 7 but we shall use the former data. St. Helen and McEwen stressed human scleral segments and analyzed the resultant decay of the stress (stress relaxation) into three components: an equilibrium elastic component, and two pressure decay components with half-lives of 0.09 and 1.7 min. The decay components are composed of an elastic part and a loss part. Heretofore we have considered ocular rigidity as a single elastic component. The above authors' analysis equates ocular rigidity to the sum of three separate parts: the equilibrium elastic part (K E Q), and the two elastic parts (K Q * and K s ) of the decay components. The values for K EQ, K Q, and K s were given as 1.5, 0.5, and 0.9 respectively, yielding a total K of 2.9. It will be noticed that this total K of 2.9 when converted in CAP, by taking the reciprocal (1/2.9 = 0.345), is 3V2 times smaller than the value previously given for McBain and Prijot (1.21) in the pressure range of 20 to 30 mm. Hg. This discrepancy caused by the geometry used in stressing scleral segments: the periphery of the segments clamped, rather than free to move. To correct this, we divide each individual K by 3.5, and take the reciprocal, obtaining "The original notation was Kr, "f" standing for "fast", but because of confusion of "f" with flow, the subscript "Q" for "quick" is used. Table III ranges CAPc CAP KQ CAP* the CAP values (i.e. = CAP EQ = 2.3; l.o ± = CAP Q = 7.0; = CAP S - 3.9) and the total equals 1.21, since capacitances in series add as reciprocals (i.e. + + = ). Therefore the ocu lar rigidity from McBain and Prijot may be considered as the sum of three separate capacitances in series, and have values in the different ranges given in Table III. Similar calculations may be made for any other ocular rigidity data. Because the capacitances remain in a fixed ratio, the designation of CAP Q, for example, as 7.0 indicates the values for the older capacitances. We are now in a position to convert the loss elements of the decay components into resistances. The decay components are recorded as half-life values, the time necessary to decay to V2 its value in minutes. This is a time constant which is composed of an elastic element K or that is combined with a loss component

5 Volume 6 Number 2 An electrical model of the human eye. I 159 C or min. x mm. Hg in order to give units in reciprocal minutes ( /*L (d (j"l) min. x mm. Hg min. / and half-life in minutes ) STRESS-RELAXA TION : FRIEDENWALD : ROUT = 3.6 :O28 A capacitor in parallel to a resistance also gives a time constant ( d (nq) volts x sec. x = sec. 1 d (volts) nq in units of seconds, but this time constant is the time necessary for it to decay to l/e th of its value, which is 1.47 times longer than a half-life. Thus, we may establish the following relation: R x CAP (in sec.) = 1.47 x half-life (in min.). Since we know the half-life and the capacitance for each decay component, we can solve for R: and R«= 1.47 x 0.09 = megohms," 1.47 x 1.7 R s = - = 0.64 megohms. These values are in the ratio of approximately 1:35. The complete basic electrical model of the eye We have translated data obtained from laboratory conditions into analogous electrical components and have calculated average values for these components from two sets of data: (1) the classical data of Friedenwald, and (2) the newer data of an average of McBain-Prijot for immediate ocular rigidity, combined with that of St. Helen and McEwen for time dependent ocular rigidity. The two models are shown in Fig. 2. It will be noticed that "Usually recorded as 19 kilohms =F C (P O-Pv) 0.28(15-10) IS "OUT = 3.6 C = ROUT = 0.28 P V=1O T Fig. 2. Viscoelastic and Friedenwald electrical models of the eye compared. Note that the capacitors have the value for the range of pressures from 10 to 20 mm. Hg. under steady state conditions the models are indistinguishable and both obey the formula: F = C (P o - P v ). In a subsequent paper it will be specifically shown that tonography can distinguish between the two models and that the electrical model incorporating stress relaxation is a more faithful analog of the action of the eye. REFERENCES 1. Olson, Harry F.: Dynamical analogies, ed. 2, New York, 1958, van Nostrand Company, Inc. 2. Goldmann, H.: Der Druck in Schlaemmschen Kanal bei Normalen und bei glaucoma simplex, Experientia 6: 110, Linner, E.: Further studies of the episcleral venous pressure in glaucoma, Am. J. Ophth. 41: 646, Becker, B.: Glaucoma ( ), Arch. Ophth. 56: 898, McEwen, W. K., and St. Helen, R.: Rheology of the human sclera. II. Unifying formulation of ocular rigidity, Ophthalmologica 150: 321, St. Helen, R., and McEwen, W. K.: Rheology of the human sclera. I. Anelastic behavior, Am. J. Ophth. 52: 539, Itoi, M., Kaneko, H., and Sugimachi, T.: Anelasticity of eyeball and errors in tonometry, Jap. J. Ophth. 9: 61, 1965.