Methods of mathematical differentiation in tonography

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  • Albrecht v. Graefes Arch. klin. exp. Ophthal. 195, 175--186 (1975) 9 by Springer-Verlag 1975

    Methods of Mathematical Differentiation in Tonography

    M. Dutescu

    Abteilung ffir Augenkrankheiten der Medizinischen Fakultat an der Rhein.-Westf. Technischen Hochschule, Aachen (Vorstand: Prof. Dr. M. Reim)

    Received January 3, 1975

    Summary. Our bulbar-compressure isotonographical method was applied to 28 normal and 40 glaucomatous eyes. This method was carried out twice on all patients using two pressure gradients: Pt = Po -- 10 and Pt = Po d- 30. Also, the outflow facility (C) was deter- mined by Grant's tonography. Thus, 3 C-values were obtained with 3 varying pressure differ- ences (Pt--Po = AP). With this differential tonography the mathematical determination and the graphic representation of a real C-value (Cdiff) by A P = 0 are possible. With normal eyes the average C-vMues were indirectly proportional to the pressure differences. This relationship was found to be directly proportional on glaucomatous eyes.

    The tonographical coefficients Po/Co_4, Pa/CLa_7, Po -- 10/C0 4, and Po/Cintegral were calcu- lated on 104 normal and 312 glaucomatous eyes. The results show small differences between the average values of the various coefficients for normal and glaucomatous eyes. The per- centage of the pathological values on glaucomatous eyes varies from 57.4% to 67.3%.

    The diagnostic efficiency of the tonography is increased by the determination of Cdi ff (percentage of positive va lues - 70 %) and the calculation of Po/Cdiff value (95 % ).

    Zusammen/assung. Bei 28 normalen und 40 glaukomkranken Augen wurde die eigene iso- tonographische Kompressionsmethode benutzt. Sic wurde an allen Patienten mit zwei unter- sehiedliehen Druckgradienten (Pt = Do -t- 10 und Pt ~ Po ~ 30) durchgefiihrt. Zus~itzlich wurde der Abflu$1eichtigkeitskoeffizient (C) bestimmt. So erhiMt man drei C-Werte fiir die ver- schiedenen Druckgradienten (Pt -- Po = P). 1Viit dieser Differentialtonographie ist die Berech- nung und die graphische Darstellung des tatsi~chlichen C-Wertes (Cdiff) fiir AP = 0 m6glich. Bei normalen Augen verhielt sich der mittlere C-Weft indirekt proportional zur Druckdiffe- renz (~ P). Direktproportional war diese Relation bei glaukomkranken Augen (Abb. 1).

    Bei 104 normalen und 312 glaukomkranken Augen wurden die tonographischen Koeffi- zienten: Po/Co_4 und Pa/CL3 ~ nach Leydhecker, Po-- IO/C nach Stepanik und Po/Cintegral nach Mc Ewan berechnet. Die Resultate weisen nur geringe Unterschiede innerhalb der Durch- schnittswerte dieser Koeffizienten, ffir normMe (Tabelle 3) und glaukomat6se (Tabelle 4) Augen auf. Der prozentuale Anteil pathologischer Werte an glaukomkranken Augen reicht von 57,4% bis 67,3%. Die diagnostische Effizienz der Tonographie steigt mit der Cdiff-Be- stimmung (der Prozentsatz der positiven Werte : 70 %) und mit der Berechnung des Po/Cdiff- Wertes (95 %).

    The value of tonography lies in the fact that it can show different diminishing degrees of the outflow facility of the aqueous humour in eyes affected by un- treated simple chronic glaucoma. This decrease of aqueous outflow generally varies with the value of the intraocular pressure and the malignity of clinical signs, thus representing the most incipient sign of glaucoma.

    Among methods for the determination of the outflow capacity of the aqueous humor at the level of the angle of the anterior chamber, tonography is the only one to have been acknowledged in the g laucoma practice. It is unfortunate, though, that tonography with all its modifications and betterments, is still an


  • 176 M. Duteseu

    unsure method in the study of ocular hydrodynamics, especially in the differentia- tion of normal from glaucomat~)us eyes.

    In spite of the demonstrated inaccurancy of the theoretical basis of tono- graphy and its value in determining the parameters of the rheological equation, the results it supplied are in agreement with those obtained with other proce- dures, e.g. by fluorometric determination of the ciliar secretion flow. These results validate the method and call for acceptance of its principle.

    Since the method was described by Grant in 1950, the following proposals have been made with the aim of increasing the diagno~ic influence of tonography: (1) modification of the technique--most important would probably be t~)nography with cor~stant tonometric pressure; (2) combination of tonography with the provocation tests; arid (3) use of methods of mathematicM differentiation. The aim of the present paper is to determine the value of various methods of mathe- matieM differentiation in the early diagnosis of glaucoma.

    Since the determination of the outflow facility coefficient (C) or that of the resistance to the outflow (R) of the aqueous humor has proved not absolutely reliable for the differentiation of normal from glaucomatous eyes, the so-called coefficients of mathematical differentiation were proposed.

    Leydhecker (1956) extends the duration of tonography to 7 mill and calculates two C-coefficients for the 1st and the last, 4 min. He relates the vMue of the intra- ocular pressure (-Po) to that of the two coefficients (Co- 4 and Ca-~) then shows tha,t the _Pc~Ca- ~ ratio gives better results than Pc~Co-4.

    Stepanik (1961, 1974)uses the (1~--10) : C-coefficient. Table 1 shows normal, probable pathological, and definite pathological values

    of these coefficients. Prijot (1960) calculates the logarithmic value of the outflow resistance coeffi-

    cient (R), while Weekers (1966) proposes the utilization of a chart on whose ordinate the P0 values and abscissa the logarithmic values of the R-factor are

    Table 1. The normal, probable pathological, and definite pathological values of the tonographic coefficients

    Coefficien~ Normal Proba.ble Definite va.lues pathological pathological

    Pc]Co-4 Leydhecker (1956) < 100 t00-140 > 140 Leydhecker (1968) < 114 1t4-160 >160

    Leydhecker (1956) < 120 120-165 >165 Leydhecker (1968) < 142 142-213 > 213

    P0-10/Co-4 Step~nik (t961) < 27 27.5-34 >34.5 Hrachovina (1967) and

  • Methods of Mathematical Differentiation in Tonography 177

    represented. The author believes that by employing the Gaussian curve, a better distinction between normal and glaucomatous eyes is obtained.

    In recent years two mathematical differentiation methods were proposed that are applied in the calculation of the tonogram. The first one considers the tonography as a linear decrease, while the second one regards it as an exponential decrease of the intraocular pressure. For the first method the integral calculation of Friedenwald's equation is used (Me Ewan et al., 1969). Woodhouse (1969) com- putes the exponential coefficient of the pressional decrease as defined by Gold- mann (1959).


    A proper applano-tonographic method with a constant tonometric pressure was used. This is identical in principle and, with respect to formal mathematics similar to the isotonographie methods described by Stepanik (1966) and Vancea et al. (1967).

    Stepanik creates a 5-rain digital compression, while the applanotonometer in the meantime checks a value Pt constant and equal to P0+l l mm ttg. Vancea utilizes the ophthalmodynamometer to perform a Pt = P0+ 10 compression of the globe, which he maintains constant for 4 rain.

    We applied a suction cup 13 mm in diameter to the temporal eyeball, which aids in producting a pressure increase equal to P0+10. This value is controlled by the applanotonometer and kept constant during a 4 min period of time through the successive increase of the negative pressure within the suction cup. 4 min later the C-value is computed after Grant's formula.

    The great advantage of the isotonographic method is that it allows the choice of various gradients of pressure (Dutescu, 1971). Based on this finding, we carried out two gpplanotonographies with two Pt-constant values of P0+ 10 and P0+30. With this procedure the determination of a real C-differential (Cdiff) is possible, at a zero value of the pressure gradient (APt =0), that is for an eye not influenced by any instrument (tonometer).


    For the calculation of the C 1 (when dPt~10) and C 2 (when APt~30) values we precede from Grant's equation:

    CAP=A V/t (1)

    whereas A P = Pt P0--A P~. For the coefficient C1, A P--8.75, because Pt-- Po ~- 10 and AP~=1.25 (Linner, 1955).

    For the C 2 value AP is 28.75. The value d V is the result of the addition of corneal (Vc) and scleral (Vs)

    sinking volumes. Using our method, the value V~ is practically zero since the applanation surface has a 3.06 mm diameter, and therefore no volumetric dis- placement appears to be related to the corneal sinking. That means, according to Grant (1950):

    1 Ptl V = V~ = ~- log p, ~ (2)

  • 178 M. Dutescu


    030Z, 0.291

    I I I I I I I I I

    C'I 0.117 . . . . . . . . . . . . . . . . . . . . . + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

    0.0850101 ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

    0.071 : I i I

    0.046 " I i I i1 I

    I I I J I I

    1012.1815 20 s 30 11.6

    Fig. 1. The relationship between C-vMue stud pressure difference by the three tonographies on normal (at the top) and glaucomatous (bottom) eyes

    Thus, for a testing t ime of 4 min, C 1 and C 2 calculated from Eq. (1) amount to :


    logPt~/Pt2 logptl/Pt~ C1- AP1 .E - - 35E ' (3a)

    log Pt ~lPt2 log Pt 1tP+2 C2-- AP 2-E - - l15E (3b)

    The Cdifr value can be computed if all of the C-values are considered to be on the straight line :

    C =n- f -mAP t.

    Within a Cartesian