ACTA OPHTHALMOLOGICA 65 (1987) 147-152

Determination of the corneal thickness profile by optical pachometry

Carsten Edrnund

Department of Ophthalmology (Head: H.-W. Larsen), Gentofte Hospital, University of Copenhagen, Denmark

Abstract. A clinical method of measuring central- peripheral corresponding values of corneal thickness and measurement positions in the horizontal meridian is described. Employing a fitting procedure it is demon- strated that the relative increase in the corneal thickness from apex to limbus is proportional to the square of the chord distance. The proportionality factor is defined as the coefficient of thickness variation (TV). T, is de- fined as the minimal corneal thickness. In relation to the visual axis the chord distance x, denote the tem- poral or nasal position of T,. TV, T, and x, are characteristic constants of a particular eye. Measuring the corneal thickness and the chord distance in mm pooled values of 80 eyes demonstrates (k+ SD) T, to be 0.575 k 0.027, and TV to be 0.0088 k 0.0020. With the normal interval in parenthes these results imply an enlargement of corneal thickness 0.9% (0.4- 1.3%) 1 mm and 22% (12-32%) 5 mm from the visual axis. With respect to the position of T, 21 eyes (26%) showed a median temporal displacement at 0.4 mm, 4 eyes (5%) showed a median nasal displacement at 0.3 mm and 55 eyes (69%) showed no significant displace- ment. This corresponds to the ususal angle kappa value, which clinically expresses the often slight nasal decentration of the visual axis relative to the optic axis.

Key words: pachometry - corneal thickness variations - topographic pachometry - descriptiv model.

Measurement of the central and peripheral cor- neal thickness has been increasingly used to obtain information about physiological and clinical con- ditions of the cornea (Martola & Baum 1968; Mandell & Poke 1969; Tomlinson 1972; El Hage & Beaulne 1973, 1975; Binder et al. 1977; Hirji & Larke 1978).

The different estimates of the peripheral cor- neal thickness obtained earlier are, however, dif- ficult to compare due to different measurement positions and lack of descriptive models.

The purpose of the present study is to describe a method for the measurement of the central- peripheral corneal thickness. Edmund & SjGntoft (1985) previously demonstrated the parabola as a useful descriptive model for the central-peri- pheral radius variation. Because the corneal stress distribution depends on the proportion between the radius- and thickness variation, it is further the purpose to evaluate the parabola as a descrip- tive model for the corneal thickness profile. Finally it is the purpose to examine the relation of the visual axis to the location of the minimal corneal thickness.

Materials and Methods

Ascertainment of the central-peripheral corneal thickness variations involves the measurement of the corneal thickness in various locations and the calculation of the measurement position.

Modalities to measure the corneal thickness are described by Mishima (1968) and Ehlers & Kruse Hansen (197 1). In the present study a modifica- tion of the Haag-Streit pachometer is used which ensures a perpendicular profile to be measured (Ehlers & Sperling 1977).

To represent a true value, the recorded corneal thickness must be corrected as well with respect to

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the thickness observed as to the radius of the corneal curvature. For the central zone of normal individuals this correction may be ignored. In the periphery of the cornea, however, the correction is important. Here, a significant increase is do- cumented in both thickness and radius of cur- vature. A table of Pachometer correction follows the Haag-Streit pachometer. Employing the table, a lot of interpolations must be performed. A multiple linear regression procedure of the pa- chometer correction table given leads to the fol- lowing practically useful formula for the pacho- meter correction (c).

~ = 0 . 1 0 8 - 0.005. R - 0.484. T + 0.0 15 . R . T + 0.46 1 * T2

where R is the actual radius of the curvature and T is the measured thickness. Eq (1) is valid for variations of T between 0.36 mm and 1.00 mmm and of R between 5.5 mm and 11.0 mm. In the present study R is detemined form photokerato-

1)

scopic data as described by Edmund & Sj@ntoft (1985). To ensure a rapid and non-biased regi- stration, the pachometer is equipped with an atuomatic recording device which transforms the mechanical movement of the pachometer during a measurement to an electric impulse recorded by a printer. The automatic recording system is ac- tivated by a foot switch and on the third decimal calibrated to the linear scale of the Haag-Streit pachometer.

Fig. 1 is a sketch of the set-up for a peripheral corneal thickness measurement. The fixation de- vice F rotates around the same axis as the slit-lamp S, and can be adjusted in a given angle (i) to the light beam of the slit-lamp, thus allowing meas- urement of the pripheral cornea. The common axis is the corneal point M where the light of the slit-lamp is brought into focus and projected per- pendicular to the corneal surface. The corneal thickness is then recorded in the point M. For each position of the fixation device the chord

F /

D \ \

Fag. 1. The set-up for a peripheral corneal thickness measurement. Z-axis indicates the visual axis. F is the fixation light. i is the angle of rotation of the fixation light. S is the slit-lamp the light of which is perpendicular to the cornea in the measurement position M with the coordinates (z, x). T(x) is the corneal thickness in the chord distane x from the visual axis. To is the corneal thickness in the visual axis. Tm is the minemal corneal thickness. xm denote the chord distance between Tm and the visual axis. 45, D, G, V and u are dimensions used in the appendix for the calculation

of x.

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4

0.7.

0.6.

I X x .

- 4 - 2 0 2 4

Fig. 2. The distribution in a single eye of a serial of measure- ments of the central-peripheral corneal thickness in the horizontal meridian. The ordinate indicate the thick- ness T(x) in mm. The abscissa indicates the chord distance x from the visual axis. Negative values indi- cates nasal positions, positive values temporal positions. x, denote the dislocation of the minimal corneal thick- ness in relation to the visual axis. The parabola drawn (T(x) = 0.568 . (1 + 0.0088 . (x-0.84)2) is by curve- linear regression calculated to give the best description

of the measured thicknesses.

distance x form visual axis is calculated as des- cribed in the appendix. The visual axis is assumed to pass through the corneal apex.

Fig. 2 plots the corneal thickness against cal- culated chord distances. The proposed parabola, which is decentrated a little in the temporal direc- tion, seems suitable to describe the corneal thick- ness variation. In mathematical terms a parabola can be described as follows

T(x) = T, . ( 1 + TV . (X - xm)’) 2 )

where T(x) is the corneal thickness in a chord distance x from the visual axis. Tm denote the minimal corneal thickness. In relation to the visual axis the chord distance xm denote the temporal (positive) or nasal (negative) position of Tm. TV is defined as the thickness variation coefficient. It appears from eq (2) that

(T(x) - T m ) / T m . 100% =

100 * TV . (X-xm)’ 3)

The product T V . (X-xm)’ multiplied with 100 thus denotes the percentual change in corneal thickness in chord distance (x-xm) from the vi- sual axis in proportion to the minimal corneal thickness.

Topographic pachometric as well as photokera- toscopic recordings were obtained from both eyes of 40 persons, 19 men and 21 women (median age 31 year, range 17-66). All persons were in good health and presented no history of generalized diseases or eye anomalies apart from refractive errors below rt 3 diopters. The photokeratoscopic procedure has previously been described (Ed- mund & Sj@ntoft 1985). For every 10 degrees of rotation of the fixation light, the peripheral cor- neal thickness was measured with the topographic pachometric method described. This equals to 5 -6 measurement locations in the nasal as well as the temporal direction. The procedure involved double measurements in every peripheral and 4 measurements in the central position.

By curvilinear regression (Pollard 1977) the descriptive parameters Tm, T V and xm as well as their variances were estimated in the particular eye. The standard errors of T m (sE(T~), T V (sE(TV)) and Xm (SE(Xm) are defined as the square root of their respective variances. For each eye the value of Xm was tested to be different from zero by the statistic t = Xm/SE(Xm). Level of significance was chosen as P < 0.05. Based on pooled results from all examined eyes corresponding mean SE of Tm, TV and Xm were calculated. Further, the validity of the descriptive model used (the para- bola) were evaluated by an F-test. Bartlett’s test demonstrated unequal variances in the various measurement positions. Thus, the estimations were performed by a weighted procedure.

Differences between right and left eye in para- meters of Table 1 were evaluated by paired t-tests.

Results

Table 1 lists the pooled results from all indivi- duals. The mean corneal central-peripheral thick- ness profile is expressed by the parameters de- fined in eq (2). Setting TV = 0.0088 rnrn-’, xm = 0, and employing eq (3) will result in an increase in corneal thickness of 0.9% and 22% 1 mm and 5 mm from the visual axis, respectively. Defining

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Mean

the normal interval as k 2 . SD (TV between 0.0 128 mm-2 and 0.0044 mm-2) this correspond to increases in corneal thickness between 0.4- 1.3% and 12-32% 1 mm and 5 mm from apex, respectively.

For the right and the left eye together the normal interval of the position of T m in relation to the visual axis is between 0.39 mm in nasal direc- tion and 0.57 mm in temporal direction (0.09 f 2 . 0.24). Only in 25 eyes (3 1 %) the value of xm was significantly different from zero. Four eyes (5%) showed a median nasal displacement at 0.3 mm (range 0.2-0.5), and 21 eyes (26%) showed a median temporal displacement at 0.4 mm (range 0.2-0.8). In 55 eyes (69%) no significant displace- ment of the T m in relation to the visual axis could be demonstrated.

A comparison between right and left eye de- monstrated no significant differences with respect to the three descriptive parameters. Further T m

and TV were significantly correlated in the two eyes. Concerning Xm no significant correlation were found.

The SE-values of the descriptive parameters in Table 1 expresses the average uncertainty by which the parameters were estimated. The range of SE(Tm), SE(TV), and SE(Xm) were 0.002-0.007 mm, 0.00036-0.00131 mm-2, and 0.06-0.38

SD SE

mm, respectively. The results imply that around the estimates of Tm, T V and Xm the 95% con- fidence intervals (+ 2- SE) are on the average + 0.009 mm, k 0.00136 mm-2 and k 0.26 mm, respectively.

The dispersion for all measurements in a single eye around the descriptive parabola (Fig. 2) de- notes the residual standard deviation SDe which in the actual study is estimated at 0.018 mm as an average value. It may be demonstrated that

SE(Tm) = a . SDe 4) SE(TV) = b . SDe 5) SE(Xm) = C . SDe 6) SD: = d . SD.& + e * SDZl 7)

where SDem is the error of measurement and SDel

is the error introduced by the descriptive model (the lack-of-fit error, Pollard 1977). For given measurement positions and number of measure- ments a, b, c, d, and e are constants.

It appears from eq (4-7) that the SE of the descriptive parameters are proportional with the SDe and consequently with SDem and SDel. Thus the magnitude of SE of the descriptive parameters in the single eye also reflects the validity of the results obtained by the actual examination.

The double determination of the corneal thick- ness in each peripheral measurement position makes it possible to estimate SDem. If SDem is almost equal to SDe, SDel must be small. Based on pooled values of SDem and SDe from all 80 eyes an F-test demonstrated no significant difference be- tween the SDem and SDe. Thus, the hypothetical parabola is accepted as descriptive model.

Discussion

This study presents a new manner to obtain a quantitative continuous description of the central- peripheral thickness profile of the human cornea. The method requires determination of corre- sponding values of corneal thickness and meas- urement position.

Previously, the accurate location of the corneal measuring area has attracted only moderate at- tention. Martola & Baum (1968) simply measured the peripheral corneal thickness at the temporal limbus. Binder et al. (1977) used a commercially available electronic corneal pachometer supplied with 9 fixation lights. They did not describe the

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relation between the fixation lights and the cor- neal measurement point. El Hage & Beaulne ( 1975) measured peripheral corneal thicknesses by moving the slit-lamp with the pachometer in relation to cornea, again without indicating the true corneal location. Mandell & Poke (1969) described how to calculate the corneal measure- ment location when knowing the angle between the fixation light and the central beam of the slit-lamp. Their mode of calculation assumed con- stant radius of the corneal curvature from centre to periphery. As shown by Edmund & SjGntoft (1985), this assumption is not fulfilled even for normal individuals. Thus, the Mandell & Poke ( 1969) calculation method introduces a systematic error in the measurement location resulting in a systematic error in the TV. The magnitude of this error (A TV) is approximately proportional to the size of the coefficient of radius variation (RV) previously defined by Edmund 8c SjGntoft (1985). Empirically it can be shown that

8)

Taken that RV = 0.01 mm-2, the calculation mode of Mandell & Polse (1969) introduces an error in TV at about -0.002 mm-2 or an error of about 20% for normal individuals. In conse- quence of eq (8) it is necessary to use the more exact method of this study when calculating the corneal measurement location. This is particularly true in comparative studies where evidence has proven the coefficient of radius variation not to be a constant from one eye to another.

The central corneal thickness (To) denotes the thickness in the corneal intersection point of the visual axis. The normal upper limits for xm and TV are 0.57 mm and 0.0128 mm-2, respectively. Employing these results and eq (3) one can de- monstrate a maximal difference between To and T m at 0.4% or about 0.002 mm. Thus practically for the normal population To and T m are equal.

Edmund & La Cour (1986) demonstrated the precision (= error as SD) of a single measurement of To to be 0.0152 mm. This precision could be improved by repeated measurements, but the optimum precision is limited by the biological day-to-day variation within the corneal thickness. Expressed as SD this variation was established to be 0.006 mm. The SE of T m is comparable with the precision of To and can like this be improved by repeated measurepents. The SE-values in Table 1 result from an analysis based on double determi-

A T V = -0.2. RV

nation of the corneal thickness in each measure- ment position (usually 28 measurements). Em- ploying only one measurement in each position (i.e. 14 measurements) SE of Tm, T V and xm of 0.0063 mm, 0.00091 mm-2 and 0.20 mm, respec- tively, are obtained. Considering the day-to-day variation mentioned above, the SE obtained with one measurement in each position thus seem to be sufficient.

In the present paper, a new descriptive model is used. Thus, the central-peripheral thickness vari- ation demonstrated in this study may not be directly comparable with others. However, the results are in agreement with several studies, which have produced evidence for the peripheral corneal thickness to surpass the central one (Mar- tola & Baum 1968; Mandell & Poke 1969 Tom- linson 1972; El Hage & Beaulne 1975; Hirji & Larke 1978).

The parameter estimate of Tm is around 0.58 mm. This value exceeds commonly calimed values at about 0.52 mm, but is in agreement with the central corneal thickness established in another study (Edmund & La Cour 1986) in which the possible causes of this discrepancy are discussed.

The angle kappa is the clinical expression for the lacking coincidence between the optic and the visual axis (Duke-Elder 1970). Often the optic axis is a little temporally decentrated in proportion to the visual axis. With respect to the geometrical shape of cornea, Tomlinson & Schwartz (1979) demonstrated a trend of temporal dislocation of the corneal apex and a location within 0.5 mm of the visual axis in 60% of their cases examined. Concerning the position of Tm the same trend is demonstrated in the present study. Thus the optic axis, the geometrical position of the corneal apex, and the minimal corneal thickness seem to be slightly temporally decentrated in relation to the visual axis. Further this implicates that the corneal shape or more precisely the central-peripheral radius variation and the central-peripheral thick- ness variation seem to influence each other. From a mechanical point of view this is interesting because it supports the hypothesis of elastic forces influencing the corneal shape.

The corneal thickness is a function of hydra- tion, amount of tissue and mechanical strech. The corneal hydration reflects the function of the endothelium whereas the corneal tissue mass and its degree of strech reflect quantitative and quali- tative properties of the stromal catabolism. In a

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given point the corneal membrane stress (force per unit cross section) depends on the proportion between radius of curvature and corneal thick- ness. Because water cannot resist mechanical for- ces, a change in corneal hydration may not in- fluence the central-peripheral radius- and thick- ness-variation. On the other hand an alteration in tissue content or tissue strength may be accom- panied by changes in corneal shape and thickness profile. Thus, the in vivo determination of the radius parameters K and RV (Edmund & SjGntoft 1985) and the thickness parameters Tm and T V may be clincially useful in the pathogenetic inter- pretation of observed changes in corneal curva- tu re and thickness.

Appendix

The shape of the cornea in a meridian can be determined from photokeratoscopic examination and described as a conic section (Edmund & Sjmtof t 1985). Therefore, the shape of the cornea in Fig. 1 is given by

Z' = K2/E2- l /E . X' A 1)

where K = radius of the central curvature, E = ( 1 -e2) and e = the eccentricity.

From Fig. 1 it appears that

tan(u) = D/(G+V) = -dz/dx =

x/(E ' z) A2)

where G = 45 . cos(i) and D = 45 . sin(i). The distance MF = 45 m m is defined by the construc- tion of the fixation device.

Further from Fig. 1 it appears that

V = x . sin(u) A3)

Solving eq (A2) with respect to u gives

u = arctan (D/G + V)) A4

and solving eq (A2) with respect to z gives

z = x . (G + V ) / ( C . E) '45)

For a given rotation (i) of the fixation light F and the corneal shape parameters K and E deter- mined by photokeratoscopy, the four equations (Al) , (A3), (A4) and (A5) contain the four un- knowns z, x, u and V. Accordingly, the four mentioned equations can be solved numerically with respect to x which indicates the measurement position.

References

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13: 57-69.

~~~ ~

Received on February 13th, 1986.

Author's address:

C. Edmund, Department of Ophthalmology, Gentofte Hospital, DK-2800 Hellerup, Denmark.

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