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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. D u t e s c u

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 determined by Grant's tonography. Thus, 3 C-values were obtained with 3 varying pressure differences (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 percentage 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 v a l u e s - 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 verschiedenen 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 glaucoma practice. It is unfortunate, though, that tonography with all its modifications and betterments, is still an

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176 M. Duteseu

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

In spite of the demonstrated inaccurancy of the theoretical basis of tonography 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 tha t 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 intraocular pressure (-Po) to tha t 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 <2 27 27.548 .",48.4 Stepanik (1974)

P~/Cintegra I < 100 t00--140 > t40

Po/Cdiff < tO0 100-160 > 160

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 tha t 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).

Methods

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 P 0 + 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 t ime 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 tha t 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), tha t is for an eye not influenced by any instrument (tonometer).

Mathematical

For the calculation of the C 1 (when d P t ~ 1 0 ) and C 2 (when APt~30) values we precede from Grant 's equation:

C A P = 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 A P 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

C

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

1 0 1 2 . 1 8 1 5 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 tes t ing t ime of 4 min, C 1 and C 2 ca lcula ted from Eq. (1) amoun t to :

and

logPt~/Pt2 logptl/Pt~ C 1 - A P 1 . E - - 3 5 E ' (3a)

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

The Cdifr value can be computed if all of the C-values are considered to be on the s t r a igh t line :

C = n - f - m A P t.

W i t h i n a Car tes ian sys tem this line represents the ra t io be tween the C and ~ Pt values (Fig. 1). C d i f f is ident ical wi th n in th is equat ion. So one can fur ther s t a t e :

APt 2" Cdiff - - Pt2 log (Pt 1/Pt~h __-- A P t l " C d i f f - - Pt I log (Pt 1/Pt2)2 QI" E Q2" E

o r :

Cdiff-- (APt2-- APtl) E Q~I l~ Q2 l~ (Ptl/Pt2)2

whereas Q1 and Q2 are ident ica l wi th A P-values .

Clinical

A t Leydhecker ' s proposal (1958), Gran t ' s 7-min ex tended t onog ra phy was per formed on 104 normal eyes as well as on 312 g laueomatous ones wi th an open


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.

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.

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 t i t , according to FriedenwMd (1957), a Pt of 2 9 m m H g (/JP=14.4) corresponds, and for P 0 = 2 9 m m I t g , 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:

l o g Pt ~/Pt~ C(~ p~=15) -- 55E (3c)

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.

Table 2. The values of the C-coefficient at various gradients of pressure in normal eyes (n - 2s)

Po ~- 10 g Pt MeBain ~ Po -]- 30

Pt 24.60 ~: 0.66 31.13 ~ 1.06 44.60 • Pt 10.00 - - 12.18 - - 30.00 - -

C 0.28 -- 0.065 0.268 • 0.233 •

Results

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.

180 M. Dutescu

Table 3. Calculation of different mathematical coefficients and the percentage of normal, probable pathological, and pathological values in normal eyes (n = 104)

Coefficient Aver- Mini- Maxi- ~ Percentage age mum mum

normal values

probable definite patho- patho- logical logical

Po/Co-~ Leydhecker (1956) 63 28 127 • 17.0 85.6% 14.4% Leydhecker (1968) . . . . 95.2% 4.8% - -

P~lq-7 Leydhecker (1956) 69 30 144 • 84.6% 15.4% - - Leydhecker (1968) . . . . 96.1% 3.9 % - -

Po - - 10/Co_ 4 Stepanik (1961) 23 9.3 31 • 80.8% 19.2% - - Hrachovina (1967) and . . . . 80.8% 19.2% - - Stepanik (1974)

Po/Cintegral 60 27 127 ~17.1 85.6% 14.4% - -

Po/Cdiff 50 31 82 ~17.3 100.0% - - - -

Only normal and probable pathologica l values were de termined. The grea tes t percentage of p robable pathologica l values is given b y the (P0-- 10)/C-coefficient. None of the coefficients have shown defini te pathological values for the normal

eyes examined.

The lower pa r t of Table 3 shows the compu ted resul ts of P0/Cdiff, which were ob ta ined on 28 normal eyes th rough different ial tonography . The P0/Cdiff values d id not exceed 82. Table 4 shows the values ob ta ined on 312 g laueomatous eyes. The Leydheeker coefficient was posi t ive (higher t han 160) in 57.4%, while R/Ca v occurred in 61.2% of the cases. Po/Cintegral does no t increase the percentage of posi t ive values when compared to the other coefficients. The (P0--10)/C coefficient has somehow given higher posi t ive values (67.3 %).

The P0/Cdiff-values in the Table 4 were ob ta ined from 40 different pa t ients . The var ia t ion of the P0/Cdiff values is ve ry large, however t he percentua l d is t r ibu- t ion in the three groups is clear-cut. There are only 2 cases in the group of p robable pathologica l values. The min imal value is 137.

Table 5 shows the values of in t raocular pressure (P0) and of those of the outflow faci l i ty coefficient (C) ob ta ined as a resul t of the 3 tonographies per formed a t var ious pressure gradients . I t is found t h a t the difference be tween the C-values de te rmined b y var ious methods and for Pt closed values (d P t = 1 5 and zl P t=11 .6) are re la t ive ly smal ler and do no t exceed 0.07 in the S t a n d a r d method. I t also resul ts in, as in the case of no rma l eyes, the existence of a re la t ionship be tween the va lue of the C-coefficient and the pressure difference (APt) dur ing the tonography. Yet though this re la t ionship is inverse ly propor t iona l for normal eyes (Table 2), in g laucomatous eyes i t is a d i rec t ly p ropor t iona l re la t ionship. The average of the C-coefficient and its d i rec t ly p ropor t iona l re la t ionship to A P t


Table 4. Calculation of different mathematical coefficients and the percentage of normal, probable pathological, and pathological values in glaueomatous eyes (n =: 312)

Coefficient Aver- Mini- Maxi- g Percentage age mum mum

normal v a l u e s

probable definite patho- patho- logical logical

&lCo_~ Leydhecker (1956) 220 80 1358 ~264 6.6% 29.7% 63.7% Leydhecker (1968) . . . . 11.8% 30.7% 57.4%

p~/c~_ ~ Leydheeker (1956) 308 97 1564 ~287 6.4% 27.3% 66.3% Leydhecker (1968) . . . . . . . 9.9% 28.8% 61.2%

Po- lO/Co_~ 8tepanik (196t) 182 20 927 • 6.4% 26.3% 67.3% ttrachovina (1967) and . . . . . . 6.4% 35.6% 58.0% Stepanik (1974)

Po/Cintegral 220 80 1358 • 264 12.5 % 30.1% 57.4 %

Po/Cdiff 1083 137 1 5 0 0 0 • 0 . 0 % 5 . 0 % 95.0%

values are highlighted in Table 6 where APt---- 11.6 represents the average of the calculating parameters (P0~=31.06 and Prim=42.67) determined through Frie- denwMd's calibration. The data obtained are transposed on a chart where the C-values are represented on the ordinate and the ~d Pt values are on the abscissa. I t is then seen that a just approximation is possible through a straight line. This straight line determined by the relationship: C = n @ m A P t has, in the ease of glaucomatous eyes, a trajeet slightly descendent toward the ordinate (Fig. 1, bottom). The m-value is 0.0024 in normal eyes and 0.00106 for glaucomatous eyes. Since C-values obtained for various pressure gradients are not found to be directly on a line in the chart in Fig. 1, we preferred to use the C-values for the calculation of Caiff determined by the same method, namely through applano- tonography (A' and C'), while B ' was obtained by indentation tonography. Through the extrapolation of line A'C' a Cdiff=0.0851 is obtained. The greatest error represented in the chart at the crossing of lines C'B' (n'1=0.0713) and A'B ' (n'i'==0.0466) with the ordinate is relatively smalt, yet for the glaucoma eases where C is generally small, it is of practical importance.

If we consider 0.10 to be the highest limit of the pathological value of the outflow facility coefficient (Stepanik, 1961) we may conclude that all normal eases have a Cdiff higher than this value. For the glaueomatous eyes 28 of them pre- sented a pathological Cdiff coefficient (70%). The percentage of pathological values of the differential coefficient, Cdi f f , is greater than that of the other C-coefficients (CGrant=62% , Cappls=50%, Ca~ps0=35% ).

The coefficient Po/Caiff was also computed for aI1 eyes examined. The average, minimum, and maximum results, as well as the percentage of normM, probable pathological, and definite pathologiea,1 values are shown in Table 3 (normal eyes)

182 M. Dutescu

Table 5. Results of the research wi th 40 glaucomatous eyes, at which 3 tonographies wi th different pressure gradients were made

Gran t ' s Applanat ion tonography A Cap p Cdiff tonography

Pt = Po +15 Pt= Po + 30

/De CGrant Po C15 A C Po Ca0 A C

1. 33.9 0.076 30 0.119 +0 .043 31 0.143 +0.067 +0 .024 0.095 2. 31.6 0.020 33 0.019 - 0 . 0 0 1 29 0.035 +0 .015 +0 .016 0.003 3. 30.3 0.070 30 0.061 - 0 . 0 0 9 30 0.055 - 0 . 0 1 5 - 0 . 0 0 6 0.067 4. 32.5 0.150 28 0.144 - 0 . 0 0 6 26 0.143 - 0 . 0 0 7 - 0 . 0 0 1 0.145 5. 29.0 0.104 27 0.099 - 0 . 0 0 5 30 0.113 +0 .009 +0 .014 0.085 6. 27.8 0.020 33 0.050 +0 .030 31 0.060 - 0 . 0 2 0 +0 .010 0.040 7. 31.6 0.072 27 0.103 +0.031 30 0.112 +0 .040 +0 .009 0.094 8. 30.3 0.127 29 0.091 - 0 . 0 3 6 29 0.108 - 0 . 0 1 9 +0.017 0.064 9. 24.3 0.088 26 0.114 +0 .026 27 0.096 +0 .008 - 0 . 0 1 8 0.132

10. 34.5 0.109 28 0.136 +0 .027 26 0.088 - 0 . 0 2 1 - 0 . 0 4 8 0.184 11. 31.6 0.155 28 0.212 +0.057 31 0.220 +0 .065 +0 .008 0.204 12. 30.3 0.064 33 0.077 +0 .013 30 0.079 +0 .015 +0 .002 0.075 13. 36.2 0.084 36 0.103 +0 .019 35 0.127 +0 .043 +0 .024 0.075 14. 33.0 0.068 32 0.091 +0 .023 30 0.081 +0.013 --0.010 0.101 15. 31.6 0.144 28 0.098 --0.046 28 0.118 --0.026 +0 .020 0.078 16. 33.0 0.110 34 0.088 --0.022 33 0.105 --0.005 +0 .017 0.071 17. 27.8 0.059 27 0.100 +0.041 29 0.120 +0.061 +0 .020 0.080 18. 26.6 0.056 29 0.066 --0.010 30 0.082 --0.004 +0 .016 0.050 19. 27.8 0.072 26 0.095 +0 .023 29 0.066 --0.006 --0.029 0.124 20. 30.3 0.081 28 0.115 --0.034 26 0.169 +0 .038 +0 .054 0.061 21. 37.8 0.091 33 0.115 +0 .024 29 0.110 +0 .009 --0.005 0.120 22. 31.6 0.148 31 0.194 +0 .046 31 0.233 +0 .085 +0 .039 0.155 23. 29.0 0.118 29 0.108 --0.010 26 0.186 +0 .068 +0 .078 0.030 24. 33.0 0.050 29 0.115 +0 .065 27 0.096 +0 .046 0.019 0.134 25. - - - - 36 0.115 - - 33 0.091 - - --0.024 0.139 26. - - - - 30 0.072 - - 30 0.142 - - +0 .070 0.002 27. - - - - 28 0.080 - - 28 0.100 - - +0 .020 0.060 28. - - - - 32 0.083 - - 34 0.072 - - +0.011 0.094 29. - - - - 28 0.120 - - 30 0.154 - - +0 .036 0.084 30. - - - - 28 0.109 - - 27 0.120 - - +0.011 0.098 31. - - - - 30 0.072 - - 26 0.126 - - +0 .054 0.018 32. - - - - 31 0.069 - - 28 0.108 - - +0 .039 0.030 33. - - - - 26 0.133 - - 26 0.133 - - 0.0 0.133 34. - - - - 36 0.080 - - 31 0.135 - - +0 .055 0.025 35. - - - - 31 0.093 - - 32 0.101 - - +0 .008 0.085 36. - - 34 0.103 - - 30 0.153 - - +0 .050 0.053 37. - - - - 27 0.098 - - 27 0.087 - - --0.011 0.109 38. - - - - 29 0.066 - - 26 0.090 - - +0 .024 0.042 39. - - - - 27 0.134 - - 30 0.191 - - +0 .057 0.077 40. - - - - 33 0.114 - - 30 0.i40 - - +0 .026 0.088

m 31.06 0.091 30,0 0.101 +0 .010 29.4 0.117 +0 .026 +0 .016 0.0851

a n d T a b l e 4 ( g l a u e o m a t o u s eyes ) a n d c o m p a r e d t o t h e v a l u e s of o t h e r coe f f i c i en t s

o b t a i n e d fo r 104 n o r m a l a n d 312 g l a u c o m a t o u s eyes . T h e l i m i t s of n o r m a l , p r o b -

a b l e p a t h o l o g i c a l , a n d d e f i n i t e p a t h o l o g i c a l v a l u e s f o r t h e P0/Cdiff c o e f f i c i e n t

a r e i d e n t i c a l t o t h o s e of P0/CLeydhecker ( T a b l e 1).

Methods of ~[athematical Differentiation in Tonography 183

Table 6. The va, lues of the C-coefficient a t various gradients of pressure in glaucomat, ous eyes (~=40)

Pt Friedenwald ~i P0 § 15 g P0 A- 30

Pt 42.67 ~ 1.26 45.0 • 0.88 59.4 i 0.86 3 Pt 11.60 - - 15.0 - - 30.0 - - C 0.091 ~ 0.035 0.101 • 0.034 0.117 • 0.046

Table 7. Calculation of the average values of the C and Po/C coefficients determined through 3 measurements in normal and glaucomatous eyes

A. The C-coefficients

Eyes n Poapp C~rant s C+]:,p +~ C~tiff

normal 28 14.6 0.268 0.060 0.280 0.065 0.304 0.078 glaucomatouses 40 30.0 0.091 0.035 0.101 0.034 0.085 0.045

B. The Po/C-coefficients

Eyes n P0app Po/CGran~ 8 P0/Capp 8 Po/C4iff

normal 28 14.6 73.4 20.4 54.5 11.8 50.6 12.3 glaucomatouses 40 30.0 400 276 352 247 1083 2824

Po/Cdif~ has an average value for normal eyes of 50.6 (5= 12.3) and in no case has exceeded the t00 value. The average value of Po/Cdiff coefficient is 1,083 for glaucomatous eyes (Table 4).

Table 7 shows the average values of the C determined through 3 measurements, as well as of the Po/C coefficient obtained for the 28 normal and 40 glaucomatous eyes examined with statistical evaluation. The average value of the coefficient Po/Cdiff is lower for the normal eyes and significantly higher for the glaucomatous eyes than the other Po/C coefficients.

This indicates that better differentiation is achieved between normal and anormal eyes by applying the Po/Cdifr. The standard deviation, however, is very high, especially for glaucomatous eyes (~: 2,824). In our opinion, this should not be considered as disadvantageous, since the deviation of the greatest P0/C~iff values from the average value only runs in one direction, namely that of pathological values (P0/Cdi f f>160, with a spread between 137 and 15000). From the statistical point of view, the standard deviation gives no definite information because it is more than twice as high as the arithmetic average value (~ = ~ 2,824; m~=1,083). That is why the median value was taken, which is situated exactly in the middle of the assorted values. I t amounts to 354.

Discussion

Extended statistics show only 36% of the glaucomat~)us eyes to be pathological, relative to the outflow facility coefficient. When the coefficient, Po/C is

184 M. Dutescu

Fig. 2. Nomogram for the determination of the Cdiff value

employed the percentage increases up to 71% (Beeker and Christensen, 1956). The coefficients employed by us showed values between 57.4% and 67.3 % with 41% among low P0 values and 95% for higher initial values of the intraocular pressure. Leydhecker (1968) found a far greater dispersion of the pathological values for the coefficient P~/C3_ 7. The lower dispersion of the values in our eases is probably due to the steady intraocular pressure which was always at about the upper limit of the norm (22-26 mm Hg).

If one compares the originally found normal and the pathological limits of the coefficients (Leydheeker, 1956; Stepanik, 1961) with the values suggested later (Leydheeker, 1968; Hrachovina, 1967; Stepanik, 1974) the result for the part of definite pathological values with glaneomatous patients only deviates slightly (Table 4).

The advantage of the method proposed in this paper is tha t the tonographieal values are determined by a triple tonography each with different pressure gradient. All of the values measured are supposed to be placed on a line in the nomogram. Greater deviations indicate a mistake in measuring.

As the method is based on Grant 's equation it also incorporates many of the theoretical errors of classical tonography. Some of these mistakes can be corrected by double or triple tonography and by calculation of the real C-value (with ~Pt =0) .

With normal eyes the average C-value was indirectly proportional to the pressure difference (3 P). This relationship was found to be directly proportional in the average glaueomatous eye (Fig. 1). 28 of the glaucomatous eyes (70%) showed this direct proportional relationship. In these 28 eyes the Cdiff value has

Methods of Mathematical I)ifferentiation in Tonography 185

to be regarded as defini te pa thologica l (less t han 0.10). The rest of this group was e i ther in the p r o b a b l y pa thologica l (10 eyes) or in the normal (2 eyes) field.

The poss ib i l i ty of d i f ferent ia t ing normal values from pathologica l ones increases when the Po/Cdiff coefficient is employed. I n 95 % of all cases the coefficient was pa thologica l (higher then 160) and a m o u n t e d to be tween 137 and 15000.

F o r a r ap id de t e rmina t ion of the Cdiff coefficient we propose the nomogram in Fig. 2.

As a result , i t can be concluded t h a t the de te rmina t ion of the C-vMue is no t sufficient for the d i f ferent ia t ion of the normal from the g laucomatous eyes. Fo r this, the calcula t ion of tonographica l coefficients is necessary. None of the coefficients descr ibed up to now offers a rel iable diagnosis a t the onset of g laucoma. Only dif ferent ia l t o n o g r a p h y is a t least theore t ica l ly more exac t b y the de termina- t i on of the outf low faci l i ty coefficient wi th A P0 = 0.

Diagnos t ic efficiency is increased b y mul t ip le repe t i t ion of tonogTaphy wi th different pressure gradients . F r o m a prac t ica l po in t of view the value P0/Cdiff raises the percentage of the posi t ive values for g laucomatous eyes up to 95%, while the Cdiff coefficient allows a rel iable diagnosis in only 70 % of the glaucoma- tons eyes.

References

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Dr. med. Mircea Dutescu D-5100 Aachen/Richterich Josef-Ponten- Stral~e 15 Federal Republic of Germany