Methods of mathematical differentiation in tonography

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<ul><li><p>Albrecht v. Graefes Arch. klin. exp. Ophthal. 195, 175--186 (1975) 9 by Springer-Verlag 1975 </p><p>Methods of Mathematical Differentiation in Tonography </p><p>M. Dutescu </p><p>Abteilung ffir Augenkrankheiten der Medizinischen Fakultat an der Rhein.-Westf. Technischen Hochschule, Aachen (Vorstand: Prof. Dr. M. Reim) </p><p>Received January 3, 1975 </p><p>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. </p><p>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%. </p><p>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 % ). </p><p>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). </p><p>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 %). </p><p>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. </p><p>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 </p><p>13" </p></li><li><p>176 M. Duteseu </p><p>unsure method in the study of ocular hydrodynamics, especially in the differentia- tion of normal from glaucomat~)us eyes. </p><p>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. </p><p>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. </p><p>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. </p><p>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. </p><p>Stepanik (1961, 1974)uses the (1~--10) : C-coefficient. Table 1 shows normal, probable pathological, and definite pathological values </p><p>of these coefficients. Prijot (1960) calculates the logarithmic value of the outflow resistance coeffi- </p><p>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 </p><p>Table 1. The normal, probable pathological, and definite pathological values of the tonographic coefficients </p><p>Coefficien~ Normal Proba.ble Definite va.lues pathological pathological </p><p>Pc]Co-4 Leydhecker (1956) &lt; 100 t00-140 &gt; 140 Leydhecker (1968) &lt; 114 1t4-160 &gt;160 </p><p>Leydhecker (1956) &lt; 120 120-165 &gt;165 Leydhecker (1968) &lt; 142 142-213 &gt; 213 </p><p>P0-10/Co-4 Step~nik (t961) &lt; 27 27.5-34 &gt;34.5 Hrachovina (1967) and </p></li><li><p>Methods of Mathematical Differentiation in Tonography 177 </p><p>represented. The author believes that by employing the Gaussian curve, a better distinction between normal and glaucomatous eyes is obtained. </p><p>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). </p><p>Methods </p><p>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). </p><p>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. </p><p>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. </p><p>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). </p><p>Mathematical </p><p>For the calculation of the C 1 (when dPt~10) and C 2 (when APt~30) values we precede from Grant's equation: </p><p>CAP=A V/t (1) </p><p>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). </p><p>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) </p><p>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): </p><p>1 Ptl V = V~ = ~- log p, ~ (2) </p></li><li><p>178 M. Dutescu </p><p>C </p><p>030Z, 0.291 </p><p>I I I I I I I I I </p><p>C'I 0.117 . . . . . . . . . . . . . . . . . . . . . + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . </p><p>0.0850101 ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . </p><p>0.071 : I i I </p><p>0.046 " I i I i1 I </p><p>I I I J I I </p><p>1012.1815 20 s 30 11.6 </p><p>Fig. 1. The relationship between C-vMue stud pressure difference by the three tonographies on normal (at the top) and glaucomatous (bottom) eyes </p><p>Thus, for a testing t ime of 4 min, C 1 and C 2 calculated from Eq. (1) amount to : </p><p>and </p><p>logPt~/Pt2 logptl/Pt~ C1- AP1 .E - - 35E ' (3a) </p><p>log Pt ~lPt2 log Pt 1tP+2 C2-- AP 2-E - - l15E (3b) </p><p>The Cdifr value can be computed if all of the C-values are considered to be on the straight line : </p><p>C =n- f -mAP t. </p><p>Within a Cartesian system this line represents the ratio between the C and ~ Pt values (Fig. 1). Cdi f f is identical with n in this equation. So one can further state: </p><p>APt 2" Cdiff - - Pt2 log (Pt 1/Pt~h __-- A Pt l " Cd i f f - - Pt I log (Pt 1/Pt2)2 QI" E Q2" E </p><p>or : </p><p>Cdiff-- (APt2-- APtl) E Q~I l~ Q2 l~ (Ptl/Pt2)2 </p><p>whereas Q1 and Q2 are identical with A P-values. </p><p>Clinical </p><p>At Leydhecker 's proposal (1958), Grant 's 7-min extended tonography was performed on 104 normal eyes as well as on 312 glaueomatous ones with an open </p></li><li><p>Methods of Mathematical Differentiation in Tonography 179 </p><p>camerular angle without previous surgery and without treatment for at least 24 hrs beforehand. The following coefficients of mathematical differentiation were computed for all eyes examined: Po/Co_4, Po/C3_7, (P0--10): C and P0/Cintegral" The percentual distribution of "normal", "probable pathological" and "definite pathological" values was also determined based on the data in Table 1. </p><p>Differential tonography was performed on 40 glaucomatous eyes. The patients were subjected to tonography free from the influence of medication, i.e. therapy was interrupted at least 24 hrs before examination. All patients suffered from simple chronic glaucoma with an open camerular angle, and did not undergo previous surgery. Twelve of them have not yet been given any treatment. </p><p>Each patient underwent tonography through indentation and two applano- tonographies after the technique described, using the pressure gradients Pt = Po ~ 15 and Pt = P0 ~ 30. The choice of another value Pt for the glaucomatous eyes (P0 + 15) compared to (P0+10) for the normal eyes is explained by the fact that for high abnormal values of the intraoeular pressure, lower values/1 Pt than in the case of normal tensional values correspond to the tonography through indentation. Thus, for example, for a P0=14.5 mm tit , according to FriedenwMd (1957), a Pt of 29mmHg (/JP=14.4) corresponds, and for P0=29mmItg , Pt 40.9 (/JP=11.9). For subsequent calibrations (McBMn, 1957; Prijot, 1960; Francois, Vancea and Vandekerckhove, 1973) the AP t values are lower than in the Friedeu- wald calibration. Thus the choice of P0+15 instead of P0~-10 was necessary in order to obtain a better dispersion of values that can be easily represented graphically and determined mathematically. In this case the calculation formula changes and becomes: </p><p>log Pt ~/Pt~ C(~ p~=15) -- 55E (3c) </p><p>The measurements were performed ceteris paribus (same daily hours and, as far as possible, same P0 values) during 3 different days. Results obtained from the glaucomatous eyes were compared to those obtained for 28 normal eyes, published in a preliminary report (Dutescu, 1971). The average values and the statistical evaluation of the C-coefficient which were obtained by 3 tonographies on normal eyes (Cdp~=10 , C~pt=30 and CGrant) in terms of/JPt are shown in Table 2. </p><p>Table 2. The values of the C-coefficient at various gradients of pressure in normal eyes (n - 2s) </p><p>Po ~- 10 g Pt MeBain ~ Po -]- 30 </p><p>Pt 24.60 ~: 0.66 31.13 ~ 1.06 44.60 Pt 10.00 -- 12.18 -- 30.00 - - </p><p>C 0.28 -- 0.065 0.268 0.233 </p><p>Results </p><p>The average, minimum, and maximum values of various coefficients of mathematical differentiation computed on 104 normal eyes are shown in Table 3. On the right side of the table the percentage of normal, probable pathological, and definite pathological values computed on the basis of data in Table 1 is given. </p></li><li><p>180 M. Dutescu </p><p>Table 3. Calculation of different mathematical coefficients and the percentage of normal, probable pathological, and pathological values in normal eyes (n = 104) </p><p>Coefficient Aver- Mini- Maxi- ~ Percentage age mum mum </p><p>normal values </p><p>probable definite patho- patho- logical logical </p><p>Po/Co-~ Leydhecker (1956) 63 28 127 17.0 85.6% 14.4% Leydhecker (1968) . . . . 95.2% 4.8% - - </p><p>P~lq-7 Leydhecker (1956) 69 30 144 84.6% 15.4% - - Leydhecker (1968) . . . . 96.1% 3.9 % - - </p><p>Po - - 10/Co_ 4 Stepanik (1961) 23 9.3 31 80.8% 19.2% - - Hrachovina (1967) and . . . . 80.8% 19.2% - - Stepanik (1974) </p><p>Po/Cintegral 60 27 127 ~17.1 85.6% 14.4% - - </p><p>Po/Cdiff 50 31 82 ~17.3 100.0% - - - - </p><p>Only normal and probable pathological values were determined. The greatest percentage of probable pathological values is given by the (P0-- 10)/C-coefficient. None of the coefficients have shown definite pathological values for the normal eyes examined. </p><p>The lower part of Table 3 shows the computed results of P0/Cdiff, which were obtained on 28 normal eyes through differential tonography. The P0/Cdiff values did not exceed 82. Table 4 shows the values obtained on 312 glaueomatous eyes. The Leydheeker coefficient was posit ive (higher than 160) in 57.4%, while R/Ca v occurred in 61.2% of the cases. Po/Cintegral does not increase the percentage of posit ive values when compared to the other coefficients. The (P0--10)/C co- efficient has somehow given higher positive values (67.3 %). </p><p>The P0/Cdiff-values in the Table 4 were obtained from 40 different patients. The variat ion of the P0/Cdiff values is very large, however the percentual distr ibu- t ion in the three groups is clear-cut. There are only 2 cases in the group of probable pathological values. The minimal value is 137. </p><p>Table 5 shows the values of intraocular pressure (P0) and of those of the out- flow faci l ity coefficient (C) obtained as a result of the 3 tonographies p...</p></li></ul>


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