Integrals Formula

Integral solution of the formula for facility of outflow W. K. McEwen, Catherine S. Lyon, Marvin D. Shepherd, and Richard R. Hibbard Integral solution of the classical formula for facility of outflow is in essential agreement with the current tables except in the normal and higher ranges. Polynomial equations equivalent to Friedenwald's equations have been generated. The integral solution is also the solution of an electrical model of tonography. This indicates that a true model or analogue can be fashioned for a nonlinear elastic concept of the eye. Key words: aqueous humor outflow, mathematical analysis. .he determination of the facility of outflow (C) from a tonographic trace is currently approximated by the formula: (1) c = 4 (Prav. - (Po + 1.25) ). The development of this formula and the related tables was a superb achievement of Grant1 and Friedenwald2 and was continued by Moses and Becker.3 With the advent of computers, it is now possible to approximate more closely the facility of outflow by integrating the underlying differential equation. This can be achieved with a digital or analogue computer. A Vr In this paper we are concerned only with the solution of the formula for C, and not with the validity of the formulas for P and VT. For P and VT we will use the classical relations of Friedenwald2 because they are in current use and not because they are necessarily correct.5"7 Methods 1. Classical method for solving the formula with Friedenwald's equations. When Friedenwald's2 equations are substituted in the formula for facility of outflow, the formula can be solved for different values of RO and R4. The values are given in Tables I and II for eyes of average ocular rigidity. An asterisk in these tables indicates a value different from that given by Moses and Becker3 in their Tables 1 and 2. An underlined value also indicates a value different from that given by Moses and Becker, but in this case the difference may be explained by the fact that Moses and Becker on the one hand, and the Fortran computer program on the other, used different methods of rounding off. In any case, the difference is no greater than ±0.01, and the accuracy of the tables can be said to be confirmed. No value is given for RO = 3, JR4 = 7 in Table II because the R4 value is lower than Po + 1.25. From the Francis I. Proctor Foundation for Research in Ophthalmology and the Department of Ophthalmology, University of CaliforniaSan Francisco Medical Center, San Francisco, Calif. Supported in part by the National Institutes of Health, Grant NB 05451. Computer time supported by the National Institutes of Health Grant FR 00122 to the Electronic Computer Training and Research Facility of this Medical Center. 206 Volume 8 Number 2 Integral solution for facility of outflow 207 2. Integral method using Friedenwald's equations. Formula (1) above is an approximation of the underlying equation: f dt dVs PT - (Po + 1.25) dVc PT - (Po + 1.25) By appropriate substitutions on the right hand side of this equation, these integrals can be solved by a computer using any standard numerical integration technique. To substitute for dVs and dVc, Friedenwald's relations of volume to pressure2' 3 have to be differentiated. A Vs — log Vc = dPr a(b-log PT) to give: dVc = -fl(10)"» where a = 2.016 for the 5.5 Gm. weight and 2.174 for the 7.5 Gm. weight, b = 2.029 for the 5.5 Gm. weight and 2.092 for the 7.5 Gm. weight, but log Po = log PT - 0.0215Vc is used undifferentiated. Hence, for the standard four-minute interval, the facility of outflow equation becomes: PT at R4 x-i Making these substitutions causes the equation for C to become quite complicated but it is easily managed by a computer. There are several standard methods of integration available to computer users. We have selected the technique of Gaussian quadrature because of its simplicity and accuracy. Gaussian quadrature subroutines of varying degrees of precision are available as elements of the IBM system/360 scientific subroutine package (360ACM-03X) version II. The program shown in Table III is an adaptation of the 9-point Gaussian quadrature subroutine. This program will calculate a facility of outflow given initial and final R values and time for the 5.5 Gm. weight. By making appropriate numerical changes, the program is applicable to the 7.5 Gm. weight. Results will have an accuracy to three decimal places and the computer integration values given in Tables I and II may be considered exact. It will be noted that the only large differences are in the higher C values (to the right of the staggered line). The differences probably have little clinical significance but may be of interest to those doing precise tonographic work. 3. Classical and integral methods for solving the formula with polynomial functions. As an alternative to computer integration it is possible to substitute approximating polynomials for the formulas used in the classical method. The formulas are generated from the standard tables.3 Vc5.5 = 0.06751 - 10.7676 (1/P) + 11,925.6 (1/P 2 ) VC7.5 = 1.7272 - 219.19 (1/P) + 24,572.9 (1/P 2 ) PT5.5 = 49.1726 - 4.8841 R + 0.3006 R2 - 0.007925 R* Pi-7.5 = 68.1334 - 7.3559 R + 0.5651 R2 - 0.025221 R3 + 0.000475 R" P00.5 + 1.25 = 40.5703 - 6.0240 R + 0.3701 R2 - 0.008977 R3 POT.5 + 1.25 = 57.9348 - 8.4884 R + _ J 20.20 dPT PT (PT - (Po + 1.25) ) PT at RO Pr at R4 r (a + l) ( P j , _ ( p Q PT at RO + 12 5) 0.5507 R2 - 0.018651 R3 + 0.000272 R* These functions are essentially the same as the classical formulas within the limits 3 < R < 12. When substituted for the classical formulas, the C values are the same as those obtained with method No. 1 above within ±0.004. When these approximations are used in the integral method by substituting into equation (2) above and integrating explicitly, the resulting C values are within ±0.01 of the exact integral values. Fox4 used an approximation of Friedenwald's formula for volume of corneal indentation, for the 5.5 Gm. weight only, in order to show When integrating by computer it is more convenient to express the above integrals in terms of R and integrate directly from RO to R4. This is achieved simply by noting from Friedenwald's equation that: PT = _ weight (0.0138) dR gives dPr — ~ 0.107 + 0.0138 R (0.107 + 0.0138 fl)a where weight = plunger load, 5.5 or 7.5 weight 208 McEtoen et al. Investigative Ophthalmology April 1969 Table I. Facility of outflow. Average ocular rigidity, 5.5 Gm. weight. Classical values (top) compared with integral values (bot.) AR R 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 0.50 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.03 0.04 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 1.00 0.09 0.09 0.09 0.09 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.07 0.07 0.07° 0.07 0.07* 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 1.50 0.14 0.15 0.14 0.14 0.13 0.14 0.13 0.13 0.13 0.13 0.12 0.13 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.11" 0.12 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 2.00 0.20 0.22 0.20 0.21 0.19 0.20 0.18 0.19 0.18 0.19 0.18 0.18 0.17 0.18 0.17 0.17 0.17 0.17 0.16* 0.17 0.16 0.16 0.16 0.16 0.16* 0.16 0.16* 0.16 0.15 0.16 0.15 0.16 0.15 0.15 2.50 0.27 0.30 0.26 0.29 0.25 0.27 0.25 0.26 0.24 0.25 0.23* 0.24 0.23 0.24 0.22* 0.23 0.22 0.23 0.21* 0.22 0.21 0.22 0.21 0.21 0.20 0.21 0.20 0.21 0.20 0.20 0.20 0.20 0.20 0.20 3.00 0.35 0.42 0.34* 0.39 0.32 0.36 0.31 0.35 0.30 0.33 0.29* 0.32 0.29 0.31 0.28 0.30 0.27 0.29 0.27 0.28 0.26 0.28 0.26 0.27 0.25 0.27 0.25 0.26 0.25 0.26 0.25* 0.25 0.24 0.25 3.50 0.44 0.57 0.42 0.52 0.40 0.48 0.39* 0.45 0.37 0.42 0.36 0.40 0.35 0.39 0.34 0.37 0.33 0.36 0.33 0.35 0.32 0.34 0.31* 0.33 0.31 0.33 0.30* 0.32 0.30 0.31 0.30 0.31 0.29 0.31 4.00 0.54 0.82 0.51 0.71 0.49 0.63 0.47 0.58 0.45 0.54 0.43 0.51 0.42 0.48 0.41 0.46 0.40 0.44 0.39 0.43 0.38 0.41 0.37* 0.40 0.37 0.39 0.36* 0.39 0.36 0.38 0.35* 0.37 0.35 0.37 °See text for explanation of asterisk and underline. Volume 8 Number 2 Integral solution for facility of outflow 209 Table II. Facility of outflow. Average ocular rigidity, 7.5 Gm. weight. Classical values (top) compared with integral values (bot.) AR R 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 0.50 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.03 1.00 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.06 0.07 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06*' 0.06 0.05 0.05 0.05 0.05 1.50 0.12 0.12 0.11* 0.12 0.11 0.11 0.11 0.11 0.10 0.11 0.10 0.10 0.10 0.10 0.10 0.10 0.09* 0.10 0.09* 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.08 0.09 2.00 0.17 0.19 0.16 0.18 0.16 0.17 0.15* 0.16 0.15 0.15 0.14* 0.15 0.14 0.14 0.14 0.14 0.13 0.14 0.13 0.13 0.13 0.13 0.13 0.13 0.12 0.13 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 2.50 0.24 028 0.22 0.25 0.21 0.24 0.20 0.22 0.20 0.21 0.19 0.20 0.18 0.20 0.18 0.19 0.17 0.18 0.17 0.18 0.17* 0.17 0.16 0.17 0.16 0.17 0.16 0.16 0.16* 0.16 0.15 0.16 0.15 0.16 3.00 0.31 0.41 0.29 0.36 0.28 0.33 0.26 0.31 0.25 0.29 0.24* 0.27 0.23 0.26 0.23 0.25 0.22 0.24 0.22 0.23 0.21 0.22 0.21 0.22 0.20 0.21 0.20 0.21 0.20* 0.20 0.19 0.20 0.19 0.20 3.50 0.40 0.65 0.37 0.53 0.35 0.46 0.33 0.42 0.32 0.38 0.30 0.35 0.29 0.33 0.28 0.32 0.27 0.30 0.26* 0.29 0.26 0.28 0.25* 0.27 0.25 0.26 0.24* 0.26 0.24 0.25 0.23* 0.25 0.23 0.24 4.00 0.46 0.90 0.43 0.68 0.41 0.58 0.39 0.51 0.37 0.46 0.35 0.43 0.34 0.40 0.33 0.38 0.32 0.36 0.31 0.34 0.30* 0.33 0.29* 0.32 0.29 0.31 0.28 0.30 0.28 0.30 0.27 0.29 °See text for explanation of asterisk and underline. 210 McElVen et al. • Investigative Ophthalmology April 1969 Table III. Program for integration of classical formula C ~Z C ~Z C ~Z T H I S PROGRAM C O M P U T E S FACILITY OF OUTFLOW BY INTEGRATING F R 1 E D E N W A L 6 EQUATIONS FOR A 5.5 GRAM W E I G H T . THREE OATA CARDS OF FORMAT (FIO.5) ARE REQUIRED GIVING THE FOLLOWING DATA: C A R D 1, R INITIAL; C A R O 2, R FINAL; CARD 3, E L A P S E D TIME IN MINUTES BETWEEN R INITIAL AND R F I N A L . QATA IS TO APPEAR IN THE FIRST TEN COLUMNS ON EACH CARO. EXTERNAL FNVStFNVC R E A D (5,1) RTl " R E A D (5,1) RT2 R E A D (5,1) DEL T 1 FORMAT (F1Q.5) CC = 0.0 CALL QG9 ( R T 1 , R T 2 , F N V S , C C ) cvs = CC/DEL T CALL QG9 (RTl,RT2,FNVC,CC) CVC = CC/DEL T C = CVS+CVC W R I T E (6,2) 2 FORMAT (1HL,83H FACILITY OF OUTFLOW VALUE FROM INTEGRATED FRIEDENW 1ALD E Q U A T I O N F O R 5.5 G R A M WEIGHT) R = ,F1O.5/IX, WRITE ( 6 , 3 ) RT1,RT2,DEL T , C 3 FORMAT ( 1 H 0 . 1 3 H I N I T I A L R = , F 1 0 . 5 / L X , 1 1 H F I N A L 116H ELAPSED TIME = , F 1 0 . 5 / l X , 5 H C = , F 1 O . 5 > TTOP" END C _C FUNCTION FNVS (RO,R) THIS FUNCTION IS THE INTEGRAND CONTRIBUTION TO FACILITY OF OUTFLOW FROM THE SCLERA DISPLACEMENT. PT = 5.5/(0.107*0.O138*R) PT0=5.5/(0.107»0.0138»RO) DPT = -0.0138/(0.107*0.0138*RI*PT DVS ~ ~ 2 0 ' 2 0 / P T * D P T VCO*10.**(Z.016*Z.029I/PTO**2.016 FNVS - DVS/(PT-(PO*1.25I) RETURN FUNCTION FNVC(RO,R) THIS FUNCTION IS THE INTEGRAND CONTRIBUTION TO FACILITY OF OUTFLOW FROM CORNEAL INDENTATION. PT = 5.5/<0.107*0.0l38*R» — — PT0=5.5/(0.107*0.0138*R0) DPT = -U.01iB/(0.10/*0.01JB»K)»PT VC = 10.**(2.016*2.029)/PT**2.016 VCO= 10. •»(-. 016*2. UW) /Ml 0**2.016 DVC=-2.016/PT*VC*DPT PO = PTO/10.**(U.02lb*VCO) FNVC = DVC/(PT-(PO«-1 .25) ) RETURN " END SUBROUTINE QG9(XL,XU.FCT,Y ) INTEGRATION OF A GIVEN FUNCTION BY 9" ~C C C T H I S bUBKUUIlNb KtKhUKMS _C POINT GAUSS Q U A D R A T U R E . A = .5*(XU+XL J B=XU-XL = .0A063 7I9*(FCT(XL,A»CI»FCT(XL,A-C> ) C=.4180156*B Y = Y».09032408*(FCT(XL,A»C)»FCT(XL,A-C) C= Y = Y*.l303053*(FCT(XL,A*C)*FCT(XL,A-C) ) .1561735*(FCT(XL,A*C)*FCT(XL,A-C) Y=B*lY*.1651l97*FCT(XL,A)I RETURN Volume 8 Number 2 Integral solution for facility of outflow 211 TO X,Y RECORDER OUT CAP PL Fig. 1. An electrical analogue of the eye during tonography. Rs and Ps are the resistance and pressure of secretion, respectively. PT is the initial rise (16 msec, contact) upon tonometer placement. CAPOR is the derivative of the reciprocal of nonlinear ocular rigidity (dVs/dP) as given by Friedenwald. CAPPL is the nonlinear plunger effect in derivative form (dV0/dP). ROUT is the resistance to outflow (1/C). Pv is episcleral venous pressure. A is a log converter to change R values to pressure values (cf. ref. 5). T T TT Fig. 2. Nonlinear capacitance network. mathematically that the integral solution differed from the classical solution, but she offered no quantitative estimation of the difference. 4. Analogue computer method. Shepherd and co-workers5 fashioned an electrical analogue of tonography by means of linear models of both an elastic eye and a viscoelastic eye. We now wish to develop an electronic analogue of a nonlinear elastic model like that shown in Fig. 1. Both the capacitor representing ocular rigidity (CAPUR) and the capacity representing indentation by the plunger (CAPPL) are nonlinear acand relations given dP dP above. They may be added together to give a single nonlinear capacitance. The electronic netcording to the dVs work required to produce this particular nonlinear capacitance is given in Fig. 2. When this network is substituted for CAPon and CAPVL in Fig. 1, the circuit can be used as an electronic analogue of a nonlinear elastic eye. In determining C values with various values of RO and R4, this analogue computer gives the same values, within its limits of error (±10 per cent of the Roar value), as those obtained by the integral method. It should be noted that the mathematical analysis of the decay of the circuit in Fig. 1 gives the same formula as the integral formula (2) above. Hence, agreement between the analogue simulation and the integral method confirms that the electronic analogue is a true analogue model of the Grant-Friedenwald concept of tonography. 212 McEioen et al. Investigative Ophthalmology April 1969 REFERENCES 1. Grant, W. M.: Tonographic method for measuring facility and rate of aqueous flow in human eyes, Arch. Ophth. 44: 204, 1950. 2. Friedenwald, J. S.: Tonometer calibration; an attempt to remove discrepancies found in the 1954 calibration scale for Schi0tz tonometers, Tr. Acad. Ophth. & Otol. 61: 108, 1957. 3. Moses, R. A., and Becker, B.: Clinical tonography: The scleral rigidity correction, Am. J. Ophth. 45: 196, 1958. 4. Fox, M. C : Continuous derivation of the pressure-flow relationship and outflow resistance for living human eyes from tonographic, mano- metric and pressure cup pressure-decay curves, Exper. Eye Res. 6: 243-260, 1967. 5. Shepherd, M., McBain, E. H., and McEwen, W. K.: An electrical model of the human eye. II. The model and the eye during tonography, INVEST. OPHTH. 6: 160, 1967. 6. McEwen, W. K., and St. Helen, R.: Rheology of the human sclera. Unifying formulation of ocular rigidity, Ophthalmologica 150: 321, 1965. 7. McEwen, W. K.: Difficulties in measuring intraocular pressure and ocular rigidity, in Glaucoma Symp., Tutzing Castle, 1966, Basel, 1967, S. Karger, AG, pp. 97-125.