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Effect of keratometer and axial length measurement errors on primary implant power calculations
Effect of keratometer and axial length measurement errors on primary implant power calculations Jack R. McEwan, M.D., R.K. Massengill, M.D., Samuel D. Friedel, M.D.
ABSTRACT Analytical predictions of primary implant power using presumptive errors in keratometer and axial length measurements were performed using the modified Binkhorst, modified Colenbrander, Holladay, Hoffer, and SRK liT\' equations. These predictions demonstrate that the contributions to primary implant power error resulting from inaccurate axial length and keratometer measurements are algebraically additive. In eyes with a normal axial length, the resulting implant power determination ermr can be larger· than differences in implant power prediction among these five IOL equations. Calculations using measurement et-rors of 0.2 mm in axial length and 0.50 dioptet· (D) in corneal curvature predicted a worst case primary implant power error of ± 1.17 D. These calculations were performed using an axial length and corneal curvature near the population mean. In contrast, implant equation variability was determined to be ± 0.19 D by calculating the standard deviation of the five implant powet· formulas with the measurement et·rors set to zero. Implant power prediction errors were inct·eased when a Rat cornea was paired with an axial hyperopic or an axial myopic eye. These combinations maximize the implant power error resulting from both implant formula variation and inaccurate measurements. Primary implant power error prediction tables are presented for emmetropic, axial hyperopic, and axial myopic eyes, as a function of presumed errors in axial length and corneal curvature. These error predictions cleady show that inaccuracy in axial length mea urements and keratometer readings can be first order determinants of postoperative spherical refractive error. Key Words: axial length measurement, cataract surgery, implant power formula , intraocular lens, intraocular len equation , keratometer measurem nt, posterior chamber lens
A fundamental knowledge of the factors determining the accuracy of primary implant power calculations is essential for the cataract surgeon. Unfortunately the literature does not contain a plethora of quantitative data characterizing this vital subject. This situation led to the development of an analytical protocol to analyze the first order determinants of primary implant power prediction accuracy. These determinants included axial length and keratometer measurement errors and implant formula prediction variability.
Prediction variability originates from differences between implant formulas and is a function of axial length and corneal curvature. A goal of the analysis was to identify axial lengths and corneal curvatures that maximized implant power prediction variability. The standard deviation of primary implant power predicted by five popular implant formulas was selected as a measure of prediction variability. Measurement errors tend to be either anomalous or routine. Routine measurement errors are specified by
Reprint requests to jack R. McEwan, M.D., 401 E Lake Vista Circle, Cockeysville, Maryland 21030. J CATARACT REFRACT SURG-VOL 16, JANUARY 1990
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the manufacturer and represent the anticipated accuracy of a properly calibrated instrument. In contrast, anomalous measurement errors result from instrument malfunction or operator error, and characteristically exceed the magnitude of routine measurement errors. Errors in primary implant power calculations result from anomalous or routfne axial length and keratometer measurement errors. Primary implant power calculation errors were predicted by introducing presumptive measurement errors into implant formulas. Since different algebraic combinations of axial length and keratometer measurement errors are possible, it was necessary to establish those combinations that maximized the implant prediction error as a starting point for the analysis.
Binkhorst equation
K -
Kerror
am =
a -
aerror
K,n
K
am
K,n
=
+ Kerror
= a = K -
24
22
22
20
20
0
18
-[ '0 .!:
16
.... ~ c. .!:'
28
~
26
26
24
24
E
~
30~~------------------------------------,30 28
22
20 18
(2) (3)
aerror
Kerror
(4)
am = a + where am is the measured axial length in millimeters; a is the true axial length in millimeters; aerror is axial length measurement error in millimeters; K.n is the measured keratometer reading in diopters; K is the true keratometer reading in diopters; Kerror is the keratometer measurement error in diopters. Two of these four sets of equations will establish maximum and minimum error bounds about a primary implant power curve in which the error contributions are set to zero. Computer simulations using the modified Binkhorst, modified Colen brander, Holladay, Hoffer, and SRK II equations 1 -5 consistently identified equation sets (1) and (2) with the minimum and maximum error bounds, respectively. These equations are referenced in the appendix with their respective anterior chamber depth scaling algorithms. An anterior chamber depth scaling algorithm was added to the Colen brander equation making it consistent with other second generation formulas. The error simulations were performed with an extensively modified version of the IOLSTAT program on an IBM PS/2 Model 50 computer. 6 The maximum and minimum error bounds for each of the five implant formulas are presented in Figure l. 0 errvr
62
26
24
~
MATERIALS AND METHODS Clinical measurements of axial length and keratometer readings may be considered to be the sum of the actual parameter ± measurement error term. The sign of the error term may differ between the axial length and the keratometer reading. The following combinations of axial length and keratometer measurement errors are possible. (1) K,n = K + Kerror am = a + 0 error K,n =
26
~ 0
~
30~--------------------------------------~30 '',~ . .- , Hoffer equation 28 28
·~
26
'0
~
;
%
-~
26
''-...,,::'--,,
24
...............
24
. . . . . .......................... ............ ,,,~"--
22
20
--
18
22 - ................ _
....................................... _
20 ........................ .......
18
~
16
- . . . . ___ ------~
E ~
', 14+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-~~~+14
16
30~--------------------------------------,30
:: ~------------
SRK II equation
26
24
24
-- "'----:---------
22
-- -- -- -- --
~---_-- -- --
~
28
18
--
------
-- ......... ....._
16
22
20 \.... ..........
18
\. ...................................................... ........
'-......
'-
16
14+-+-+-+-+-+-+-+-+-+-+-+-+-------+-~~~+14
21.00
22.00
23.00
24.00
25.00
26.00
Axial length in millimeters
Fig. l.
(McEwan) Primary implant power as a function of axial length. Dashed lines represent upper and lower error bounds using aerror = 0.2 mm and Kerror = 0.5 D. Plot parameters K = 43.0 D, v = 12.0 mm, A = 116.7 D, SF = 0.5 mm, and Rs = - 1.0 D.
J CATARACT REFRACT SURG-VOL 16, JANUARY 1990
The upper and lower error bounds were calculated using presumed measurement errors of0.2 mm in axial length and 0.50 D in corneal curvature. Note the relative symmetry about the zero error implant power curve. In each of these cases, the axial length and keratometer error terms were of similar sign, which maximized the magnitude of the difference between either error bound and the zero error power curve. When error terms are of similar sign, an additive implant power error results from the contribution of individual measurement errors. However, when the axial length and keratometer errors are of opposite sign the difference between the upper and lower error bounds is significantly reduced. This implies that the effect of a keratometer error may be partially compensated for by an axial length error of opposite sign. The observation that keratometer and axial length measurement errors of the same sign reinforce one another can be explained by simple optics. A positive keratometer error will act to increase the refracting power of the cornea in the intraocular lens (IOL) equation. Such an error will move the posterior focal point anteriorly. A similar effect will be noted by a positive error in axial length measurement, inducing a myopic shift and thereby displacing the posterior focal point anteriorly. A determination of the variability in IOL power prediction among the five IOL equations was performed for eyes of normal axial length, axial hyperopia, and axial myopia using corneal curvatures of39, 43, and 47 D. These equations included the modified Binkhorst, modified Colenbrander, Holladay, Hoffer, and SRK II equations. An average primary implant power and standard deviation was calculated for each axial length and corneal curvature. Axial length and keratometer measurement errors were set to zero for these calculations. The average IOL power predictions and
standard deviations are tabulated as a function of corneal curvature and anticipated postoperative refraction in Table 1. These calculations were compared to similar predictions using axial length and keratometer measurement errors ofidentical sign. For each of the five equations, a zero error implant power calculation was performed. Five sets of upper and lower error bound predictions were then generated with axial length errors that ranged between 0 and 0.5 mm and keratometer errors that ranged between 0 and 1.0 D. The magnitude of the difference between upper or lower error bound and the zero error prediction was calculated, generating five sets of upper and lower error bound differences. The average upper and lower error bound difference and standard deviation was then calculated for an anticipated postoperative refraction of -1.0 D. A sample calculation is shown in Table 2, and the resulting error table is presented in Table 3. The average upper and lower error bound entries occupy respective positions in this table. These calculations were performed using an axial length of 24 mm and a corneal curvature of 43 D, which are near the population mean in Sorsby et al.'s series of 408 eyes. 7 To maximize the error bound differences, this procedure was repeated associating a flat cornea with either an axial hyperopic or an axial myopic eye. The results appear in Tables 4 and 5, respectively. Parameters other than axial length and corneal curvature were held constant for these calculations so that differences between the tables would reflect single parameter variation. DISCUSSION The variability in IOL power prediction among the modified Binkhorst, modified Colen brander, Holladay, Hoffer, and SRK II equations is best represented by the
Table 1. Average primary implant power ± standard deviation as a function of axial length and corneal curvature. The average primary implant powers were computed using the modified Binkhorst, modified Colenbrander, Holladay, Hoffer, and SRK II equations. The analysis parameters were A = 116.7 D, v = 12.0 mm, and SF = 0.5 mm. Corneal Curvature, K = 39 D
+3.0
21 mm
24 mm
28.18± 1.18 18.56±0.46
+2.0
29.47 ± 1.21 19.91 ±0.50
+ 1.0
30.75± 1.24 21.22 ± 0.53
0
32.01 ± 1.26 22.52±0.56
29 mm
21 mm
24 mm
=
47 D
Axial Length
Axial Length
Axial Length Refraction
Corneal Curvature, K
Corneal Curvature, K = 43 D
21 mm
29 mm
5.79±0.25 23.58±0.56 13.76±0.30 0.60±0.79 7.11±0.37 24.90±0.60 15.13±0.25 1.97 ± 0.58 8.42±0.53 26.19±0.64 16.47±0.22 3.32±0.38 9.71±0.69 27.48±0.67 17.80±0.20 4.65±0.22
24 mm
29 mm
8.77± 1.09
<0
20.20±0.34 10.17± 1.03
<0
21.53±0.35 11.56±0.98
<0
22.83±0.37 12.93 ± 0.95
<0
18.86±0.32
24.12±0.39 14.27±0.93 0.69±0.98
-2.0
33.25± 1.29 23.80±0.58 10.97±0.85 28.74±0.70 19.11±0.19 5.95±0.15 34.48 ± 1.31 25.06±0.60 12.22± 1.00 29.99±0.72 20.40±0.18 7.24±0.24
-3.0
35.70± 1.33 26.30±0.63 13.45± 1.15 31.22±0.75 21.67±0.18 8.51±0.38
26.65±0.42 16.91 ±0.89 3.34±0.66
-1.0
25.39±0.40 15.60±0.91 2.02±0.81
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63
Table 2. Sample calculation demonstrating the method used to calculate the error tables. Equation parameters K = 43 D, a = 24 mm, Rs = -1.0 D, SF = 0.50 mm, A = ll6. 7 D, aerror = 0.2 mm, Kerror = 0.5 D. Error Bound
Implant Power
I Upper or Lower Bound- Zero Error I
Lower Zero Upper
17.69 18.94 20.19
1.25
Colen brander Lower Zero Upper
17.72 19.01 20.30
Holladay
Lower Zero Upper
18.26 19.36 20.47
Lower Zero Upper
17.73 18.97 20.21
Lower Zero Upper
18.30 19.25 20.20
Lower Zero Upper
17.94 ± 0.31 19.11 ± 0.19 20.27 ± 0.12
Equation Binkhorst
Hoffer
SRK II
Mean± standard deviation
1.26 1.29 1.29 1.10 1.10 1.24 1.24 0.95 0.95 1.17 ± 0.14 1.17 ± 0.14
standard deviations listed in Table 1. These calculations clearly indicate that the largest standard deviations occurred for the axial hyperope with a flat cornea. Although rare, this association was observed in Sorsby et al. 's series of 408 eyes. 7 Table 1 lists the average primary implant power of the five implant formulas ± standard deviation as a function of anticipated postoperative refraction. The largest standard deviations in Table 1 occurred when a 39 D corneal curvature was paired with a 21 mm axial length. In this case, the standard deviation of the implant formulas varied from ± 1.18 to ± 1.33 D, over a range of+ 3.00 to -3.00 D in desired postoperative refraction. Three graphs demonstrating the individual variability of primary implant power prediction as a function of axial length in the hyperopic eye are presented in Figure 2. Corneal curvatures of 39, 43, and 47 D were used for this analysis. To permit an accurate visual comparison each graph was assigned a vertical axis increment of 0.5 D. A comparison of the graphs presented in Figure 2 demonstrates that among these equations the variability in primary implant power prediction increased with decreasing corneal curvature. In contrast to the hyperopic eye, the standard deviations in zero error implant power prediction were somewhat less in the myopic eye. However, the ratio of the standard deviation to the zero error implant power
Table 3. Average primary implant power errors as predicted by the modified Binkhorst, modified Colenhrander, Holladay, Hoffer, and SRK II equations. Table entries represent the Iupper error hound- zero error implant power I ± standard deviation, and the Ilower error hound - zero error implant power I ± standard deviation. The upper and lower error hound differences occupy respective positions in this table. The analysis was performed with a= 24 mm, K = 43.0 D, A= ll6.7 D, v = 12.0 mm, SF = 0.5 mm, and Rs = - 1.0 D. Axial Length Error (mm) 0
0.10
0.20
0.30
0.40
0.50
64
Keratometer Error in Diopters 0
0.25
0.50
0.75
1.00
0
0.30 ± 0.05
0.59 ± 0.09
0.89 ± 0.14
1.19 ± 0.19
0
0.30 ± 0.05
0.60 ± 0.09
0.89 ± 0.14
1.19 ± 0.19
0.29 ± 0.02
0.58 ± 0.07
0.88 ± 0.12
1.18 ± 0.16
1.47 ± 0.21
0.29 ± 0.02
0.58 ± 0.07
0.88 ± 0.12
1.18 ± 0.17
1.48 ± 0.21
0.58 ± 0.05
0.87 ± 0.10
1.17 ± 0.14
1.46 ± 0.19
1.76 ± 0.23
0.57 ± 0.05
0.87 ± 0.09
1.17 ± 0.14
1.47 ± 0.19
1.77 ± 0.24
0.87 ± 0.08
1.16 ± 0.12
1.46 ± 0.17
1.75 ± 0.21
2.05 ± 0.26
0.85 ± O.o7
1.15 ± 0.12
1.45 ± 0.16
1.75 ± 0.21
2.05 ± 0.26
1.16 ± 0.11
1.46 ± 0.15
1.75 ± 0.19
2.04 ± 0.24
2.34 ± 0.29
1.13 ± 0.09
1.43 ± 0.14
1.73 ± 0.18
2.03 ± 0.23
2.34 ± 0.28
1.45 ± 0.13
1.75 ± 0.18
2.04 ± 0.22
2.34 ± 0.27
2.63 ± 0.31
1.51 ± 0.15
1.81 ± 0.12
2.11 ± 0.12
2.41 ± 0.13
2.72 ± 0.15
J CATARACT REFRACT SURG-VOL 16, JANUARY 1990
Table 4. Average hyperopic primary implant power errors as predicted by the modified Binkhorst, modified Colenbrander, Holladay, Hoffer, and SRK II equations. Table entries represent the Iupper error bound-zero error implant power I ± standard deviation and the Ilower error bound - zero error implant power I ± standard deviation. The upper and lower error bound differences occupy respective positions in this table. The analysis was performed with a = 21 mm, K = 39.0 D, A = 116.7 D, v = 12.0 mm, SF = 0.5 mm, and Rs = -1.0 D. Axial Length Error (mm)
0
0.25
0
0
0.28 ± 0.04
0.56 ± 0.08
0.84 ± 0.12
1.11 ± 0.16
0
0.28 ± 0.04
0.56 ± 0.08
0.84 ± 0.12
1.12 ± 0.16
0.54 ± 0.40
0.10
0.20
0.30
0.40
0.50
Keratometer Error in Diopters
0.50
0.75
1.00
0.82 ± 0.37
1.10 ± 0.34
1.38 ± 0.32
1.65 ± 0.30
0.34 ± 0.05
0.62 ± 0.09
0.90 ± 0.13
1.18 ± 0.17
1.46 ± 0.21
0.89 ± 0.35
1.17 ± 0.32
1.44 ± 0.30
1.72 ± 0.28
2.00 ± 0.26
0.68 ± 0.11
0.96 ± 0.14
1.24 ± 0.18
1.52 ± 0.22
1.80 ± 0.26
1.23 ± 0.30
1.51 ± 0.27
1.79 ± 0.25
2.07 ± 0.24
2.34 ± 0.23
1.01 ± 0.16
1.29 ± 0.20
1.57 ± 0.23
1.85 ± 0.27
2.13 ± 0.31
1.58 ± 0.25
1.86 ± 0.23
2.14 ± 0.22
2.42 ± 0.21
2.69 ± 0.21
1.34 ± 0.21
1.63 ± 0.25
1.91 ± 0.28
2.19 ± 0.32
2.47 ± 0.36
1.94 ± 0.21
2.21 ± 0.20
2.49 ± 0.19
2.77 ± 0.20
3.04 ± 0.20
1.67 ± 0.26
1.96 ± 0.30
2.24 ± 0.33
2.52 ± 0.37
2.80 ± 0.41
Table 5. Average myopic primary implant power errors as predicted by the modified Binkhorst, modified Colen brander, Holladay, Hoffer, and SRK II equations. Table entries represent the Iupper error bound- zero error implant power I ± standard deviation and the Ilower error bound-zero error implant power! ± standard deviation. The upper and lower error bound differences occupy respective positions in this table. The analysis was performed with a = 29 mm, K = 39.0 D, A = 116.7 D, v = 12.0 mm, SF = 0.5 mm, and Rs -1.0 D.
=
Axial Length Error (mm)
0
0.25
0.50
0.75
1.00
0
0
0.31 ± 0.05
0.61 ± 0.10
0.92 ± 0.15
1.22 ± 0.20
0
0.31 ± 0.05
0.62 ± 0.10
0.93 ± 0.16
1.24 ± 0.21
0.10
0.20
0.30
0.40
0.50
Keratometer Error in Diopters
0.22 ± 0.01
0.53 ± 0.04
0.84 ± 0.09
1.14 ± 0.14
1.45 ± 0.19
0.22 ± 0.01
0.53 ± 0.04
0.84 ± 0.09
1.15 ± 0.14
1.46 ± 0.20
0.45 ± 0.03
0.76 ± 0.03
1.06 ± 0.08
1.37 ± 0.13
1.67 ± 0.18
0.45 ± 0.03
0.75 ± 0.03
1.06 ± 0.08
1.37 ± 0.13
1.68 ± 0.18
0.68 ± 0.04
0.98 ± 0.03
1.29 ± 0.07
1.59 ± 0.12
1.90 ± 0.17
0.67 ± 0.05
0.98 ± 0.03
1.28 ± 0.07
1.59 ± 0.12
1.91 ± 0.17
0.90 ± 0.05
1.21 ± 0.03
1.52 ± 0.06
1.82 ± 0.11
2.12 ± 0.16
0.89 ± 0.06
1.20 ± 0.03
1.51 ± 0.06
1.82 ± 0.11
2.13 ± 0.16
1.13 ± 0.07
1.44 ± 0.03
1.74 ± 0.05
2.05 ± 0.10
2.35 ± 0.15
1.11 ± 0.08
1.41 ± 0.05
1.72 ± 0.06
2.04 ± 0.10
2.35 ± 0.15
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65
39
15.00 . - - - - - - - - - - - - - - - - - - - - - - , 1 5 . 0 0
39 Corneal Curvature, K
= 39
D
37
37
35
35
33
33
31
31
29 35
29 35
13.00
13.00
11.00
11.00
9.00
9.00
...."' ..... tj)
a_ 0
Corneal Curvature, K
'0 c .....
33
;!: 0 a_
31
= 43
D
.2 a_
-~
~
E
a.
7.00 -1----+----1---+----t---o----1----+-~7.00 9.00 +---+----1---+----t---o----1----+---+9.00
.Q
s.....
Corneal Curvature, K
31
;!: 0 a_
....c
29
0..
27
27
~
E
25
30
= 47
D
28
3.00
SRK I I -
E
a.
1.00 -1----+----1---+----t---o----1----+---+ 1.00 8.00 +---+----1---+----t---o----1----+---+8.00 Corneal Curvature, K
26
--...... -........ ........... ....
24
24
---.............:_ .... ~,. ·-............... ..
,
------.::..::.::.::_~ ~ ,.._
22 20 20.00
..
22
----::: ':::::--::-.....
20.50
21.00
21.50
=
47 D
6.00
6.00
4.00
4.00
28
--------------------::..._
5.00
Hoffer---
3.00
30 Corneal Curvature, K
26
:::~:;:.~,-''"""">~oo-,,,,>
5.00
c
·c 25
D
7.00
c
29
= 43
7.00
tj)
c
·;::
tj)
"0
33
tj)
....c
..... ...."'a_
2.00
---
:::::·:::-:::-:::.::::.::::.::;.::;:::;_:;~_~-- ·-·-
2.00
0.00 +---+----1---+-----="--=t-___;"----t----1-----+-""""---+0.00 28.00 28.50 29.00 29.50 30.00
----::: 20 22.00
Axial length in millimeters
Axial length in millimeters Fig. 2.
(McEwan) Primary implant power as a function of axial length in the hyperopic eye. Plot parameters v = 12.0 mm, A = 116.7 D, SF = 0.5 mm, and Rs = -1.0 D. Legend: Binkhorst-short dashes, Colenbrander-long dashes, Holladay-dots, Hoffer-dash double dot, and SRK II- solid line.
was much larger. Consequently, the percentage error introduced by variation in implant power prediction was higher in the myopic eye. This observation was based upon an anticipated postoperative refraction of -1.00 D. If the patient were made more myopic, this ratio would decrease as the power of the implant increased to satisfy the negative shift in desired postoperative refraction. Graphs of primary implant power in the myopic eye are presented in Figure 3 as a function of axial length and corneal curvature. Marked variability in implant power is demonstrated at the extremes of39 and 47 D. Most of the variation was contributed by the SRK II formula, which is a piecewise linear regression formula. A new SRK formula for myopic eyes is currently being developed. The remainder of the equations were 66
Fig. 3.
(McEwan) Primary implant power as a function of axial length in the myopic eye. Plot parameters v = 12.0 mm, A = 116.7 D, SF = 0.5 mm, and Rs = -1.0 D. Legend: Binkhorst-short dashes, Colenbrander-long dashes, Holladay-dots, Hoffer-dash double dot, and SRK 11solid line.
theoretically derived and exhibited a variation of less than 1.0 D. Note that all five equations demonstrated minimal variation in both hyperopic and myopic eyes with a corneal curvature of 43 D. The implant power prediction variability was further reduced when a corneal curvature of 43 D was paired with an axial length of24 mm. These values are close to the population means of 43.1 D and 24.2 mm in Sorsby et al. 's study of 408 eyes. 7 Using these values the standard deviations of the zero error implant power predictions varied between 0.18 and 0.30 D over a -3.0 to + 3.0 D range in postoperative refraction. These values are presented in Table l. However, large errors in primary implant power specification can result when relatively small axial length and keratometer measurement errors are intro-
J CATARACT REFRACT SURG-VOL 16, JANUARY 1990
duced in an eye representing the population mean. Referring to Table 3, an error of ± 1.17 D can result from measurement errors of0.2 mm and 0.5 Din axial length and corneal curvature. This error represents the average magnitude of the difference between the upper or lower error bound curve, and the zero error curve for the five equations considered in this study. Note that for eyes with axial lengths and corneal curvatures near the population mean an implant power error of 1.17 D exceeds the maximum standard deviation in zero error implant power by a factor of nearly four. This demonstrates that measurement errors as opposed to implant equation variability can play a dominant role in determining errors in postoperative spherical refraction in most eyes. Using the procedure described in Table 2, measurement error tables were calculated for axial hyperopic and axial myopic eyes with flat corneas. These calculations were performed to assess the effect of equation variability upon the measurement error calculations. The hyperopic implant power errors appear in Table 4. An axial length of 21 mm and a corneal curvature of 39 D were used in these calculations. Measurement errors of0.2 mm and 0.5 D produced an implant power error ranging between -1.44 and+ 1.24 D. This result was slightly larger than the ± 1.17 D that was calculated using an axial length of 24 mm and a corneal curvature of 43 D. The myopic implant power errors were evaluated using an axial length of 29 mm paired with a corneal curvature of 39 D. These predictions are presented in Table 5. By comparison, measurement errors of0.2 mm and 0.5 D produced an implant power error of ± 1.06 D. When considering the clinical relevance of these calculations, it is important to emphasize that these predictions represent worst case implant power errors based upon errors in axial length and keratometer readings. These errors result from a multitude of factors, and consist of a constant plus a variable component. The constant component may reflect a calibration error in the instrumentation or a consistent operator error or bias. The variable component may be associated with inconsistent technique, measurements performed by different technicians, poor patient cooperation, difficulty in obtaining a precise reading on an older A-scan, corneal surface irregularities in the central3.0 mm, or an improperly adjusted keratometer eyepiece. The net result will most likely skew the implant error in either a positive or negative direction. However, the magnitude of error should not exceed the calculations presented in Tables 3 through 5. Considerations for improving the predictability of primary implant power calculations were identified by Holladay et al. 8 Holladay found that in 92% of 26 patients with postoperative errors of greater than 2.0 D
Table 6. Average deviation ± standard deviation from the anticipated postoperative refraction experienced by surgeon A in a study by Richards et al. 9 Average Deviation ± Standard Deviation (D) Equation
lntermedics 019E
AMO PC llF
Modified Binkhorst
-0.17 ± 1.12
+0.45 ± 1.31
Colen brander
-0.74 ± 1.35
-0.11 ± 1.26
Hoffer
-0.90 ± 1.37
-0.35 ± 1.12
the theoretical and linear regression formulas differed by more than 1.0 D . He suggested that an independent observer repeat the axial length and keratometer measurements when a 1.0 D difference or more was observed between the two formulas. Other factors such as errors in predicted postoperative anterior chamber depth, surgically induced astigmatism, and IOL style contribute to errors in postoperative refraction. Three surgeons and 304 patients participated in a study by Richards et al.9 that measured the deviation from the anticipated postoperative refraction. The results were tabulated as the average deviation ± standard deviation with four different IOL styles. The most negative deviation was associated with surgeon A and the Intermedics 019E IOL; the same surgeon experienced the most positive deviation with the American Medical Optics PC-llF. These data are summarized in Table 6. Note that the standard deviation shows less variation than the average deviation and that the magnitude of the standard deviation is comparable to the deviations resulting from measurement error predicted in Tables 3 through 5. However, these predictions represent primary implant power errors, not errors in postoperative refraction. These errors are similar but they are not identical in magnitude. The disparity results from the factors previously described in addition to a round-off error. This error results from the commercial availability of IOLs in 0.5 D steps, requiring that the predicted IOL power be rounded off to the nearest 0.5 D. The magnitude of this error was evaluated using the graphical analysis section in IOLSTAT and is approximately ± 0.2 D. CONCLUSIONS Errors in primary implant power prediction, attributable to routine axial length and keratometer measurement inaccuracy, can be significantly greater than errors resulting from variability in the five IOL power prediction formulas. Measurement inaccuracies of 0.2 mm in axial length and 0.50 Din corneal curvature can produce an error of ± 1.17 D in primary implant power. These calculations were performed with an axial length and corneal curvature near the population
J CATARACT REFRACT SURG-VOL 16, JANUARY 1990
67
mean. Using an identical axial length and corneal curvature, the standard deviation in primary implant power was ± 0.19 D for the five implant formulas. The magnitude of the error in implant power is dependent upon the sign of the axial length and keratometer error terms. Measurement errors of opposite sign compensate for one another; errors of similar sign reinforce one another. Maximum implant power prediction errors resulting from equation variability occur when a hyperopic eye is paired with a flat cornea. These results are equation specific and apply to the five implant equations considered in this study. In the case of the myopic eye, the implant power errors are slightly less than those predicted using a mean axial length and corneal curvature. However, the ratio of implant power error to the specified IOL power is much greater in the myopic eye. Equation variability in the myopic eye was largely attributable to the SRK II formula. These predictions suggest that axial length and keratometer measurement errors can play a dominant role in determining the postoperative spherical result. Improvements in predicting the postoperative spherical result will occur as equipment manufacturers develop more sophisticated instrumentation capable of reducing measurement errors.
REFERENCES l. Binkhorst RD: Intraocular Lens Calculation Manual. A Guide to the Author's TI 58!59 IOL Power Module, ed 2, New York, RD Binkhorst, 1981 2. Colen brander MC: Calculation of the power of an iris clip lens for distant vision. Br J Ophthalmol 57:735-740, 1973 3. Hoffer KJ: Intraocular lens calculation: The problem of the short eye. Ophthalmic Surg 12:269-272, 1981 4. Holladay JT, Prager TC, Chandler TY, Musgrove KH, eta!: A three-part system for refining intraocular lens power calculations . .T Cataract Refract Surg 14:17-24, 1988 5. Sanders DR, Retzlaff}, KraffMC: Comparison of the SRK IP" formula and other second generation formulas. ] Cataract
Refract Surg 14:136-141, 1988 6. McEwan J: IOLSTAT- A program for comprehensive intraocular lens equation comparison with statistical analysis. .T Cataract Refract Surg 12:543-549, 1986 7. Sorsby A, Leary GA, Richards MJ: Correlation ametropia and component ametropia. Vision Res 2:309-313, 1962 8. Holladay JT, Prager TC, Ruiz RS, Lewis JW, eta!: Improving the predictability of intraocular lens power calculations. Arch Ophthalmol 104:539-541, 1986 9. Richards SC, Olson RJ, Richards WL, Brodstein RS, et a!: Clinical evaluation of six intraocular lens calculation formulas. Am Intra-Ocular Implant Soc] 11:153-158, 1985 10. Binkhorst RD: The optical design of intraocular lens implants. Ophthalmic Surg 6(3):17-31, 1975 11. Binkhorst RD: Pitfalls in the determination of intraocular lens power without ultrasound. Ophthalmic Surg 7(3):69-82, 1976 12. Hoffer KJ: The effect of axial length on posterior chamber lens and posterior capsule position. Curr Concepts Ophthalmic Surg 1:20-22, 1984
APPENDIX Intraocular Lens Equations Used to Perform the Measurement Error Simulations Modified Binkhorst Equation
1336 (acor - d){i3r - d - O.OOlR, [v(i3r- d)+udr]} where a=
1.0
13 = __n..c.v-
with nv = 1.336 and nc =
337.5 r= _ __
t = 1.3333
as recommended by Binkhorst to compensate for a postoperative reduction in the refracting power of the cornea.l.l 0 • 11 An axial length correction factor was included to compensate for the difference between the true axial length and the axial length as measured by ultrasound.! This correction is required because the ultrasound wave is reflected at the vitreoretinal interface and does not measure the distance between the interface and the sensory layer of the retina. The measured axial length is corrected for by retinal thickness using the equation,
'""'
~ '"' +
T, - t, (
-
n~..)
where nv = 1.336 nP = 1.49166
68
Tr = 0.250 mm t; = 0.500 mm J CATARACT REFRACT SURG-VOL 16, JANUARY 1990
The postoperative anterior chamber depth is estimated by
d d
for d :S 4.43 mm for d > 4.43 mm
0.17am + 0.017 4.43 mm
Holladay Equation - __0_.0_0_1__R_s_[v_(_~_r___a_c~·o_r)__+__a__ac~·o~rr_J_______________ 1336 ______~_r__- __a~c<_><__ (acor- d- SF){~r- d- SF- 0.001RJv(~r- d- SF)+ a(d + SF)r]} Ra~ = Ra~ =
r 7 mm
for r ~ 7 mm for r < 7 mm
AG AG
0.533am 13.5 mm
for AG :S 13.5 mm for AG > 13.5 mm
with
d=0.56+
Rag-~
Holladay compensates for retinal thickness with the following axial length correction factor, 4 where Tr = 0.200 mm The polynomials and constants a, ~, nc, and nv are identical to those used in the Binkhorst equation.
Modified Colenbrander Equation Colenbrander's equation was originally developed for an iris clip lens. 2 The original formula has been modified by the addition of a postoperative anterior chamber depth scaling algorithm and a term to produce desired ammetropia. These changes make this equation consistent with the other second generation formulas used in this analysis. nv --------- -
~- 0.001d
[0.001(am - d) - 0.00005] [
Km
l.336Rs
0.00005]
The postoperative anterior chamber depth in millimeters is estimated by
d
d
O.l7am + 0.017 6.0mm
for d :S 6.0 mm ford> 6.0 mm
Hoffer Equation 1336 (am :- d - 0.05)
[
1.336 1.336
(Km
+
(d + 0.05) RJ
1000
]
The Hoffer equation was derived without the use of axial length correction factors. Hoffer derived the following postoperative anterior chamber depth scaling algorithm,
d d
0.292am - 2.93 6.0 mm
d :S 6.0 mm d > 6.0 mm
which was used with his equation in the measurement error simulations.l2
SRK II Equation
J CATARACT REFRACT SURG-VOL 16, JANUARY 1990
69
where
A+ 3 A+2 A+ 1 A A 0.5
A1 A1 A1 A1 A1
for for for for for
am < 20.0 20.0 S am 21.0 S am 22.0 S am am ~ 24.5
mm < 21.0 mm < 22.0 mm < 24.5 mm mm
and 1.00 1.25
'Y 'Y
for A1 - 0 9 K - 2.5a S 14.00 mm for A1 - o:g K:: - 2.5a:: > 14.00 mm
No axial length corrections are used with the SRK II formula. ·5
Statistical Equations Average primary implant power:
Standard Deviation:
5
Davg
I
i= 1
s
Di
~ i=l
5
4
Symbols Used in Error Analysis Equations A
the SRK constant in diopters
am
the axial length as measured on ultrasound, in millimeters
acor
the corrected axial length in millimeters
D
the dioptric power in aqueous of the intraocular lens
D1
the primary implant power predicted by the modified Binkhorst equation
D2
the primary implant power predicted by the Holladay equation
D.3
the primary implant power predicted by the modified Colenbrander equation
D4
the primary implant power predicted by the Hoffer equation
D5
the primary implant power predicted by the SRK liT" formula
d
the calculated postoperative anterior chamber depth in millimeters
Km
the measured average radius of curvature of the anterior surface of the cornea in diopters
nc
the index of refraction of the cornea
nP
the index of refraction of poly( methyl methacrylate)
nv
the index of refraction of the aqueous or vitreous
Rs
the desired postoperative refraction in diopters
SF
Holladay's surgeon factor in millimeters
ti
the thickness of the implant in millimeters
Tr
the distance between the vitreoretinal interface and the sensory layer of the retina in millimeters
v
the vertex distance in millimeters
70
J CATARACT REFRACT SURG-VOL 16, JANUARY 1990
ABSTRACT Analytical predictions of primary implant power using presumptive errors in keratometer and axial length measurements were performed using the modified Binkhorst, modified Colenbrander, Holladay, Hoffer, and SRK liT\' equations. These predictions demonstrate that the contributions to primary implant power error resulting from inaccurate axial length and keratometer measurements are algebraically additive. In eyes with a normal axial length, the resulting implant power determination ermr can be larger· than differences in implant power prediction among these five IOL equations. Calculations using measurement et-rors of 0.2 mm in axial length and 0.50 dioptet· (D) in corneal curvature predicted a worst case primary implant power error of ± 1.17 D. These calculations were performed using an axial length and corneal curvature near the population mean. In contrast, implant equation variability was determined to be ± 0.19 D by calculating the standard deviation of the five implant powet· formulas with the measurement et·rors set to zero. Implant power prediction errors were inct·eased when a Rat cornea was paired with an axial hyperopic or an axial myopic eye. These combinations maximize the implant power error resulting from both implant formula variation and inaccurate measurements. Primary implant power error prediction tables are presented for emmetropic, axial hyperopic, and axial myopic eyes, as a function of presumed errors in axial length and corneal curvature. These error predictions cleady show that inaccuracy in axial length mea urements and keratometer readings can be first order determinants of postoperative spherical refractive error. Key Words: axial length measurement, cataract surgery, implant power formula , intraocular lens, intraocular len equation , keratometer measurem nt, posterior chamber lens
A fundamental knowledge of the factors determining the accuracy of primary implant power calculations is essential for the cataract surgeon. Unfortunately the literature does not contain a plethora of quantitative data characterizing this vital subject. This situation led to the development of an analytical protocol to analyze the first order determinants of primary implant power prediction accuracy. These determinants included axial length and keratometer measurement errors and implant formula prediction variability.
Prediction variability originates from differences between implant formulas and is a function of axial length and corneal curvature. A goal of the analysis was to identify axial lengths and corneal curvatures that maximized implant power prediction variability. The standard deviation of primary implant power predicted by five popular implant formulas was selected as a measure of prediction variability. Measurement errors tend to be either anomalous or routine. Routine measurement errors are specified by
Reprint requests to jack R. McEwan, M.D., 401 E Lake Vista Circle, Cockeysville, Maryland 21030. J CATARACT REFRACT SURG-VOL 16, JANUARY 1990
61
the manufacturer and represent the anticipated accuracy of a properly calibrated instrument. In contrast, anomalous measurement errors result from instrument malfunction or operator error, and characteristically exceed the magnitude of routine measurement errors. Errors in primary implant power calculations result from anomalous or routfne axial length and keratometer measurement errors. Primary implant power calculation errors were predicted by introducing presumptive measurement errors into implant formulas. Since different algebraic combinations of axial length and keratometer measurement errors are possible, it was necessary to establish those combinations that maximized the implant prediction error as a starting point for the analysis.
Binkhorst equation
K -
Kerror
am =
a -
aerror
K,n
K
am
K,n
=
+ Kerror
= a = K -
24
22
22
20
20
0
18
-[ '0 .!:
16
.... ~ c. .!:'
28
~
26
26
24
24
E
~
30~~------------------------------------,30 28
22
20 18
(2) (3)
aerror
Kerror
(4)
am = a + where am is the measured axial length in millimeters; a is the true axial length in millimeters; aerror is axial length measurement error in millimeters; K.n is the measured keratometer reading in diopters; K is the true keratometer reading in diopters; Kerror is the keratometer measurement error in diopters. Two of these four sets of equations will establish maximum and minimum error bounds about a primary implant power curve in which the error contributions are set to zero. Computer simulations using the modified Binkhorst, modified Colen brander, Holladay, Hoffer, and SRK II equations 1 -5 consistently identified equation sets (1) and (2) with the minimum and maximum error bounds, respectively. These equations are referenced in the appendix with their respective anterior chamber depth scaling algorithms. An anterior chamber depth scaling algorithm was added to the Colen brander equation making it consistent with other second generation formulas. The error simulations were performed with an extensively modified version of the IOLSTAT program on an IBM PS/2 Model 50 computer. 6 The maximum and minimum error bounds for each of the five implant formulas are presented in Figure l. 0 errvr
62
26
24
~
MATERIALS AND METHODS Clinical measurements of axial length and keratometer readings may be considered to be the sum of the actual parameter ± measurement error term. The sign of the error term may differ between the axial length and the keratometer reading. The following combinations of axial length and keratometer measurement errors are possible. (1) K,n = K + Kerror am = a + 0 error K,n =
26
~ 0
~
30~--------------------------------------~30 '',~ . .- , Hoffer equation 28 28
·~
26
'0
~
;
%
-~
26
''-...,,::'--,,
24
...............
24
. . . . . .......................... ............ ,,,~"--
22
20
--
18
22 - ................ _
....................................... _
20 ........................ .......
18
~
16
- . . . . ___ ------~
E ~
', 14+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-~~~+14
16
30~--------------------------------------,30
:: ~------------
SRK II equation
26
24
24
-- "'----:---------
22
-- -- -- -- --
~---_-- -- --
~
28
18
--
------
-- ......... ....._
16
22
20 \.... ..........
18
\. ...................................................... ........
'-......
'-
16
14+-+-+-+-+-+-+-+-+-+-+-+-+-------+-~~~+14
21.00
22.00
23.00
24.00
25.00
26.00
Axial length in millimeters
Fig. l.
(McEwan) Primary implant power as a function of axial length. Dashed lines represent upper and lower error bounds using aerror = 0.2 mm and Kerror = 0.5 D. Plot parameters K = 43.0 D, v = 12.0 mm, A = 116.7 D, SF = 0.5 mm, and Rs = - 1.0 D.
J CATARACT REFRACT SURG-VOL 16, JANUARY 1990
The upper and lower error bounds were calculated using presumed measurement errors of0.2 mm in axial length and 0.50 D in corneal curvature. Note the relative symmetry about the zero error implant power curve. In each of these cases, the axial length and keratometer error terms were of similar sign, which maximized the magnitude of the difference between either error bound and the zero error power curve. When error terms are of similar sign, an additive implant power error results from the contribution of individual measurement errors. However, when the axial length and keratometer errors are of opposite sign the difference between the upper and lower error bounds is significantly reduced. This implies that the effect of a keratometer error may be partially compensated for by an axial length error of opposite sign. The observation that keratometer and axial length measurement errors of the same sign reinforce one another can be explained by simple optics. A positive keratometer error will act to increase the refracting power of the cornea in the intraocular lens (IOL) equation. Such an error will move the posterior focal point anteriorly. A similar effect will be noted by a positive error in axial length measurement, inducing a myopic shift and thereby displacing the posterior focal point anteriorly. A determination of the variability in IOL power prediction among the five IOL equations was performed for eyes of normal axial length, axial hyperopia, and axial myopia using corneal curvatures of39, 43, and 47 D. These equations included the modified Binkhorst, modified Colenbrander, Holladay, Hoffer, and SRK II equations. An average primary implant power and standard deviation was calculated for each axial length and corneal curvature. Axial length and keratometer measurement errors were set to zero for these calculations. The average IOL power predictions and
standard deviations are tabulated as a function of corneal curvature and anticipated postoperative refraction in Table 1. These calculations were compared to similar predictions using axial length and keratometer measurement errors ofidentical sign. For each of the five equations, a zero error implant power calculation was performed. Five sets of upper and lower error bound predictions were then generated with axial length errors that ranged between 0 and 0.5 mm and keratometer errors that ranged between 0 and 1.0 D. The magnitude of the difference between upper or lower error bound and the zero error prediction was calculated, generating five sets of upper and lower error bound differences. The average upper and lower error bound difference and standard deviation was then calculated for an anticipated postoperative refraction of -1.0 D. A sample calculation is shown in Table 2, and the resulting error table is presented in Table 3. The average upper and lower error bound entries occupy respective positions in this table. These calculations were performed using an axial length of 24 mm and a corneal curvature of 43 D, which are near the population mean in Sorsby et al.'s series of 408 eyes. 7 To maximize the error bound differences, this procedure was repeated associating a flat cornea with either an axial hyperopic or an axial myopic eye. The results appear in Tables 4 and 5, respectively. Parameters other than axial length and corneal curvature were held constant for these calculations so that differences between the tables would reflect single parameter variation. DISCUSSION The variability in IOL power prediction among the modified Binkhorst, modified Colen brander, Holladay, Hoffer, and SRK II equations is best represented by the
Table 1. Average primary implant power ± standard deviation as a function of axial length and corneal curvature. The average primary implant powers were computed using the modified Binkhorst, modified Colenbrander, Holladay, Hoffer, and SRK II equations. The analysis parameters were A = 116.7 D, v = 12.0 mm, and SF = 0.5 mm. Corneal Curvature, K = 39 D
+3.0
21 mm
24 mm
28.18± 1.18 18.56±0.46
+2.0
29.47 ± 1.21 19.91 ±0.50
+ 1.0
30.75± 1.24 21.22 ± 0.53
0
32.01 ± 1.26 22.52±0.56
29 mm
21 mm
24 mm
=
47 D
Axial Length
Axial Length
Axial Length Refraction
Corneal Curvature, K
Corneal Curvature, K = 43 D
21 mm
29 mm
5.79±0.25 23.58±0.56 13.76±0.30 0.60±0.79 7.11±0.37 24.90±0.60 15.13±0.25 1.97 ± 0.58 8.42±0.53 26.19±0.64 16.47±0.22 3.32±0.38 9.71±0.69 27.48±0.67 17.80±0.20 4.65±0.22
24 mm
29 mm
8.77± 1.09
<0
20.20±0.34 10.17± 1.03
<0
21.53±0.35 11.56±0.98
<0
22.83±0.37 12.93 ± 0.95
<0
18.86±0.32
24.12±0.39 14.27±0.93 0.69±0.98
-2.0
33.25± 1.29 23.80±0.58 10.97±0.85 28.74±0.70 19.11±0.19 5.95±0.15 34.48 ± 1.31 25.06±0.60 12.22± 1.00 29.99±0.72 20.40±0.18 7.24±0.24
-3.0
35.70± 1.33 26.30±0.63 13.45± 1.15 31.22±0.75 21.67±0.18 8.51±0.38
26.65±0.42 16.91 ±0.89 3.34±0.66
-1.0
25.39±0.40 15.60±0.91 2.02±0.81
J CATARACT REFRACT SURG-VOL 16, JANUARY 1990
63
Table 2. Sample calculation demonstrating the method used to calculate the error tables. Equation parameters K = 43 D, a = 24 mm, Rs = -1.0 D, SF = 0.50 mm, A = ll6. 7 D, aerror = 0.2 mm, Kerror = 0.5 D. Error Bound
Implant Power
I Upper or Lower Bound- Zero Error I
Lower Zero Upper
17.69 18.94 20.19
1.25
Colen brander Lower Zero Upper
17.72 19.01 20.30
Holladay
Lower Zero Upper
18.26 19.36 20.47
Lower Zero Upper
17.73 18.97 20.21
Lower Zero Upper
18.30 19.25 20.20
Lower Zero Upper
17.94 ± 0.31 19.11 ± 0.19 20.27 ± 0.12
Equation Binkhorst
Hoffer
SRK II
Mean± standard deviation
1.26 1.29 1.29 1.10 1.10 1.24 1.24 0.95 0.95 1.17 ± 0.14 1.17 ± 0.14
standard deviations listed in Table 1. These calculations clearly indicate that the largest standard deviations occurred for the axial hyperope with a flat cornea. Although rare, this association was observed in Sorsby et al. 's series of 408 eyes. 7 Table 1 lists the average primary implant power of the five implant formulas ± standard deviation as a function of anticipated postoperative refraction. The largest standard deviations in Table 1 occurred when a 39 D corneal curvature was paired with a 21 mm axial length. In this case, the standard deviation of the implant formulas varied from ± 1.18 to ± 1.33 D, over a range of+ 3.00 to -3.00 D in desired postoperative refraction. Three graphs demonstrating the individual variability of primary implant power prediction as a function of axial length in the hyperopic eye are presented in Figure 2. Corneal curvatures of 39, 43, and 47 D were used for this analysis. To permit an accurate visual comparison each graph was assigned a vertical axis increment of 0.5 D. A comparison of the graphs presented in Figure 2 demonstrates that among these equations the variability in primary implant power prediction increased with decreasing corneal curvature. In contrast to the hyperopic eye, the standard deviations in zero error implant power prediction were somewhat less in the myopic eye. However, the ratio of the standard deviation to the zero error implant power
Table 3. Average primary implant power errors as predicted by the modified Binkhorst, modified Colenhrander, Holladay, Hoffer, and SRK II equations. Table entries represent the Iupper error hound- zero error implant power I ± standard deviation, and the Ilower error hound - zero error implant power I ± standard deviation. The upper and lower error hound differences occupy respective positions in this table. The analysis was performed with a= 24 mm, K = 43.0 D, A= ll6.7 D, v = 12.0 mm, SF = 0.5 mm, and Rs = - 1.0 D. Axial Length Error (mm) 0
0.10
0.20
0.30
0.40
0.50
64
Keratometer Error in Diopters 0
0.25
0.50
0.75
1.00
0
0.30 ± 0.05
0.59 ± 0.09
0.89 ± 0.14
1.19 ± 0.19
0
0.30 ± 0.05
0.60 ± 0.09
0.89 ± 0.14
1.19 ± 0.19
0.29 ± 0.02
0.58 ± 0.07
0.88 ± 0.12
1.18 ± 0.16
1.47 ± 0.21
0.29 ± 0.02
0.58 ± 0.07
0.88 ± 0.12
1.18 ± 0.17
1.48 ± 0.21
0.58 ± 0.05
0.87 ± 0.10
1.17 ± 0.14
1.46 ± 0.19
1.76 ± 0.23
0.57 ± 0.05
0.87 ± 0.09
1.17 ± 0.14
1.47 ± 0.19
1.77 ± 0.24
0.87 ± 0.08
1.16 ± 0.12
1.46 ± 0.17
1.75 ± 0.21
2.05 ± 0.26
0.85 ± O.o7
1.15 ± 0.12
1.45 ± 0.16
1.75 ± 0.21
2.05 ± 0.26
1.16 ± 0.11
1.46 ± 0.15
1.75 ± 0.19
2.04 ± 0.24
2.34 ± 0.29
1.13 ± 0.09
1.43 ± 0.14
1.73 ± 0.18
2.03 ± 0.23
2.34 ± 0.28
1.45 ± 0.13
1.75 ± 0.18
2.04 ± 0.22
2.34 ± 0.27
2.63 ± 0.31
1.51 ± 0.15
1.81 ± 0.12
2.11 ± 0.12
2.41 ± 0.13
2.72 ± 0.15
J CATARACT REFRACT SURG-VOL 16, JANUARY 1990
Table 4. Average hyperopic primary implant power errors as predicted by the modified Binkhorst, modified Colenbrander, Holladay, Hoffer, and SRK II equations. Table entries represent the Iupper error bound-zero error implant power I ± standard deviation and the Ilower error bound - zero error implant power I ± standard deviation. The upper and lower error bound differences occupy respective positions in this table. The analysis was performed with a = 21 mm, K = 39.0 D, A = 116.7 D, v = 12.0 mm, SF = 0.5 mm, and Rs = -1.0 D. Axial Length Error (mm)
0
0.25
0
0
0.28 ± 0.04
0.56 ± 0.08
0.84 ± 0.12
1.11 ± 0.16
0
0.28 ± 0.04
0.56 ± 0.08
0.84 ± 0.12
1.12 ± 0.16
0.54 ± 0.40
0.10
0.20
0.30
0.40
0.50
Keratometer Error in Diopters
0.50
0.75
1.00
0.82 ± 0.37
1.10 ± 0.34
1.38 ± 0.32
1.65 ± 0.30
0.34 ± 0.05
0.62 ± 0.09
0.90 ± 0.13
1.18 ± 0.17
1.46 ± 0.21
0.89 ± 0.35
1.17 ± 0.32
1.44 ± 0.30
1.72 ± 0.28
2.00 ± 0.26
0.68 ± 0.11
0.96 ± 0.14
1.24 ± 0.18
1.52 ± 0.22
1.80 ± 0.26
1.23 ± 0.30
1.51 ± 0.27
1.79 ± 0.25
2.07 ± 0.24
2.34 ± 0.23
1.01 ± 0.16
1.29 ± 0.20
1.57 ± 0.23
1.85 ± 0.27
2.13 ± 0.31
1.58 ± 0.25
1.86 ± 0.23
2.14 ± 0.22
2.42 ± 0.21
2.69 ± 0.21
1.34 ± 0.21
1.63 ± 0.25
1.91 ± 0.28
2.19 ± 0.32
2.47 ± 0.36
1.94 ± 0.21
2.21 ± 0.20
2.49 ± 0.19
2.77 ± 0.20
3.04 ± 0.20
1.67 ± 0.26
1.96 ± 0.30
2.24 ± 0.33
2.52 ± 0.37
2.80 ± 0.41
Table 5. Average myopic primary implant power errors as predicted by the modified Binkhorst, modified Colen brander, Holladay, Hoffer, and SRK II equations. Table entries represent the Iupper error bound- zero error implant power I ± standard deviation and the Ilower error bound-zero error implant power! ± standard deviation. The upper and lower error bound differences occupy respective positions in this table. The analysis was performed with a = 29 mm, K = 39.0 D, A = 116.7 D, v = 12.0 mm, SF = 0.5 mm, and Rs -1.0 D.
=
Axial Length Error (mm)
0
0.25
0.50
0.75
1.00
0
0
0.31 ± 0.05
0.61 ± 0.10
0.92 ± 0.15
1.22 ± 0.20
0
0.31 ± 0.05
0.62 ± 0.10
0.93 ± 0.16
1.24 ± 0.21
0.10
0.20
0.30
0.40
0.50
Keratometer Error in Diopters
0.22 ± 0.01
0.53 ± 0.04
0.84 ± 0.09
1.14 ± 0.14
1.45 ± 0.19
0.22 ± 0.01
0.53 ± 0.04
0.84 ± 0.09
1.15 ± 0.14
1.46 ± 0.20
0.45 ± 0.03
0.76 ± 0.03
1.06 ± 0.08
1.37 ± 0.13
1.67 ± 0.18
0.45 ± 0.03
0.75 ± 0.03
1.06 ± 0.08
1.37 ± 0.13
1.68 ± 0.18
0.68 ± 0.04
0.98 ± 0.03
1.29 ± 0.07
1.59 ± 0.12
1.90 ± 0.17
0.67 ± 0.05
0.98 ± 0.03
1.28 ± 0.07
1.59 ± 0.12
1.91 ± 0.17
0.90 ± 0.05
1.21 ± 0.03
1.52 ± 0.06
1.82 ± 0.11
2.12 ± 0.16
0.89 ± 0.06
1.20 ± 0.03
1.51 ± 0.06
1.82 ± 0.11
2.13 ± 0.16
1.13 ± 0.07
1.44 ± 0.03
1.74 ± 0.05
2.05 ± 0.10
2.35 ± 0.15
1.11 ± 0.08
1.41 ± 0.05
1.72 ± 0.06
2.04 ± 0.10
2.35 ± 0.15
J CATARACT REFRACT SURG-VOL 16, JANUARY 1990
65
39
15.00 . - - - - - - - - - - - - - - - - - - - - - - , 1 5 . 0 0
39 Corneal Curvature, K
= 39
D
37
37
35
35
33
33
31
31
29 35
29 35
13.00
13.00
11.00
11.00
9.00
9.00
...."' ..... tj)
a_ 0
Corneal Curvature, K
'0 c .....
33
;!: 0 a_
31
= 43
D
.2 a_
-~
~
E
a.
7.00 -1----+----1---+----t---o----1----+-~7.00 9.00 +---+----1---+----t---o----1----+---+9.00
.Q
s.....
Corneal Curvature, K
31
;!: 0 a_
....c
29
0..
27
27
~
E
25
30
= 47
D
28
3.00
SRK I I -
E
a.
1.00 -1----+----1---+----t---o----1----+---+ 1.00 8.00 +---+----1---+----t---o----1----+---+8.00 Corneal Curvature, K
26
--...... -........ ........... ....
24
24
---.............:_ .... ~,. ·-............... ..
,
------.::..::.::.::_~ ~ ,.._
22 20 20.00
..
22
----::: ':::::--::-.....
20.50
21.00
21.50
=
47 D
6.00
6.00
4.00
4.00
28
--------------------::..._
5.00
Hoffer---
3.00
30 Corneal Curvature, K
26
:::~:;:.~,-''"""">~oo-,,,,>
5.00
c
·c 25
D
7.00
c
29
= 43
7.00
tj)
c
·;::
tj)
"0
33
tj)
....c
..... ...."'a_
2.00
---
:::::·:::-:::-:::.::::.::::.::;.::;:::;_:;~_~-- ·-·-
2.00
0.00 +---+----1---+-----="--=t-___;"----t----1-----+-""""---+0.00 28.00 28.50 29.00 29.50 30.00
----::: 20 22.00
Axial length in millimeters
Axial length in millimeters Fig. 2.
(McEwan) Primary implant power as a function of axial length in the hyperopic eye. Plot parameters v = 12.0 mm, A = 116.7 D, SF = 0.5 mm, and Rs = -1.0 D. Legend: Binkhorst-short dashes, Colenbrander-long dashes, Holladay-dots, Hoffer-dash double dot, and SRK II- solid line.
was much larger. Consequently, the percentage error introduced by variation in implant power prediction was higher in the myopic eye. This observation was based upon an anticipated postoperative refraction of -1.00 D. If the patient were made more myopic, this ratio would decrease as the power of the implant increased to satisfy the negative shift in desired postoperative refraction. Graphs of primary implant power in the myopic eye are presented in Figure 3 as a function of axial length and corneal curvature. Marked variability in implant power is demonstrated at the extremes of39 and 47 D. Most of the variation was contributed by the SRK II formula, which is a piecewise linear regression formula. A new SRK formula for myopic eyes is currently being developed. The remainder of the equations were 66
Fig. 3.
(McEwan) Primary implant power as a function of axial length in the myopic eye. Plot parameters v = 12.0 mm, A = 116.7 D, SF = 0.5 mm, and Rs = -1.0 D. Legend: Binkhorst-short dashes, Colenbrander-long dashes, Holladay-dots, Hoffer-dash double dot, and SRK 11solid line.
theoretically derived and exhibited a variation of less than 1.0 D. Note that all five equations demonstrated minimal variation in both hyperopic and myopic eyes with a corneal curvature of 43 D. The implant power prediction variability was further reduced when a corneal curvature of 43 D was paired with an axial length of24 mm. These values are close to the population means of 43.1 D and 24.2 mm in Sorsby et al. 's study of 408 eyes. 7 Using these values the standard deviations of the zero error implant power predictions varied between 0.18 and 0.30 D over a -3.0 to + 3.0 D range in postoperative refraction. These values are presented in Table l. However, large errors in primary implant power specification can result when relatively small axial length and keratometer measurement errors are intro-
J CATARACT REFRACT SURG-VOL 16, JANUARY 1990
duced in an eye representing the population mean. Referring to Table 3, an error of ± 1.17 D can result from measurement errors of0.2 mm and 0.5 Din axial length and corneal curvature. This error represents the average magnitude of the difference between the upper or lower error bound curve, and the zero error curve for the five equations considered in this study. Note that for eyes with axial lengths and corneal curvatures near the population mean an implant power error of 1.17 D exceeds the maximum standard deviation in zero error implant power by a factor of nearly four. This demonstrates that measurement errors as opposed to implant equation variability can play a dominant role in determining errors in postoperative spherical refraction in most eyes. Using the procedure described in Table 2, measurement error tables were calculated for axial hyperopic and axial myopic eyes with flat corneas. These calculations were performed to assess the effect of equation variability upon the measurement error calculations. The hyperopic implant power errors appear in Table 4. An axial length of 21 mm and a corneal curvature of 39 D were used in these calculations. Measurement errors of0.2 mm and 0.5 D produced an implant power error ranging between -1.44 and+ 1.24 D. This result was slightly larger than the ± 1.17 D that was calculated using an axial length of 24 mm and a corneal curvature of 43 D. The myopic implant power errors were evaluated using an axial length of 29 mm paired with a corneal curvature of 39 D. These predictions are presented in Table 5. By comparison, measurement errors of0.2 mm and 0.5 D produced an implant power error of ± 1.06 D. When considering the clinical relevance of these calculations, it is important to emphasize that these predictions represent worst case implant power errors based upon errors in axial length and keratometer readings. These errors result from a multitude of factors, and consist of a constant plus a variable component. The constant component may reflect a calibration error in the instrumentation or a consistent operator error or bias. The variable component may be associated with inconsistent technique, measurements performed by different technicians, poor patient cooperation, difficulty in obtaining a precise reading on an older A-scan, corneal surface irregularities in the central3.0 mm, or an improperly adjusted keratometer eyepiece. The net result will most likely skew the implant error in either a positive or negative direction. However, the magnitude of error should not exceed the calculations presented in Tables 3 through 5. Considerations for improving the predictability of primary implant power calculations were identified by Holladay et al. 8 Holladay found that in 92% of 26 patients with postoperative errors of greater than 2.0 D
Table 6. Average deviation ± standard deviation from the anticipated postoperative refraction experienced by surgeon A in a study by Richards et al. 9 Average Deviation ± Standard Deviation (D) Equation
lntermedics 019E
AMO PC llF
Modified Binkhorst
-0.17 ± 1.12
+0.45 ± 1.31
Colen brander
-0.74 ± 1.35
-0.11 ± 1.26
Hoffer
-0.90 ± 1.37
-0.35 ± 1.12
the theoretical and linear regression formulas differed by more than 1.0 D . He suggested that an independent observer repeat the axial length and keratometer measurements when a 1.0 D difference or more was observed between the two formulas. Other factors such as errors in predicted postoperative anterior chamber depth, surgically induced astigmatism, and IOL style contribute to errors in postoperative refraction. Three surgeons and 304 patients participated in a study by Richards et al.9 that measured the deviation from the anticipated postoperative refraction. The results were tabulated as the average deviation ± standard deviation with four different IOL styles. The most negative deviation was associated with surgeon A and the Intermedics 019E IOL; the same surgeon experienced the most positive deviation with the American Medical Optics PC-llF. These data are summarized in Table 6. Note that the standard deviation shows less variation than the average deviation and that the magnitude of the standard deviation is comparable to the deviations resulting from measurement error predicted in Tables 3 through 5. However, these predictions represent primary implant power errors, not errors in postoperative refraction. These errors are similar but they are not identical in magnitude. The disparity results from the factors previously described in addition to a round-off error. This error results from the commercial availability of IOLs in 0.5 D steps, requiring that the predicted IOL power be rounded off to the nearest 0.5 D. The magnitude of this error was evaluated using the graphical analysis section in IOLSTAT and is approximately ± 0.2 D. CONCLUSIONS Errors in primary implant power prediction, attributable to routine axial length and keratometer measurement inaccuracy, can be significantly greater than errors resulting from variability in the five IOL power prediction formulas. Measurement inaccuracies of 0.2 mm in axial length and 0.50 Din corneal curvature can produce an error of ± 1.17 D in primary implant power. These calculations were performed with an axial length and corneal curvature near the population
J CATARACT REFRACT SURG-VOL 16, JANUARY 1990
67
mean. Using an identical axial length and corneal curvature, the standard deviation in primary implant power was ± 0.19 D for the five implant formulas. The magnitude of the error in implant power is dependent upon the sign of the axial length and keratometer error terms. Measurement errors of opposite sign compensate for one another; errors of similar sign reinforce one another. Maximum implant power prediction errors resulting from equation variability occur when a hyperopic eye is paired with a flat cornea. These results are equation specific and apply to the five implant equations considered in this study. In the case of the myopic eye, the implant power errors are slightly less than those predicted using a mean axial length and corneal curvature. However, the ratio of implant power error to the specified IOL power is much greater in the myopic eye. Equation variability in the myopic eye was largely attributable to the SRK II formula. These predictions suggest that axial length and keratometer measurement errors can play a dominant role in determining the postoperative spherical result. Improvements in predicting the postoperative spherical result will occur as equipment manufacturers develop more sophisticated instrumentation capable of reducing measurement errors.
REFERENCES l. Binkhorst RD: Intraocular Lens Calculation Manual. A Guide to the Author's TI 58!59 IOL Power Module, ed 2, New York, RD Binkhorst, 1981 2. Colen brander MC: Calculation of the power of an iris clip lens for distant vision. Br J Ophthalmol 57:735-740, 1973 3. Hoffer KJ: Intraocular lens calculation: The problem of the short eye. Ophthalmic Surg 12:269-272, 1981 4. Holladay JT, Prager TC, Chandler TY, Musgrove KH, eta!: A three-part system for refining intraocular lens power calculations . .T Cataract Refract Surg 14:17-24, 1988 5. Sanders DR, Retzlaff}, KraffMC: Comparison of the SRK IP" formula and other second generation formulas. ] Cataract
Refract Surg 14:136-141, 1988 6. McEwan J: IOLSTAT- A program for comprehensive intraocular lens equation comparison with statistical analysis. .T Cataract Refract Surg 12:543-549, 1986 7. Sorsby A, Leary GA, Richards MJ: Correlation ametropia and component ametropia. Vision Res 2:309-313, 1962 8. Holladay JT, Prager TC, Ruiz RS, Lewis JW, eta!: Improving the predictability of intraocular lens power calculations. Arch Ophthalmol 104:539-541, 1986 9. Richards SC, Olson RJ, Richards WL, Brodstein RS, et a!: Clinical evaluation of six intraocular lens calculation formulas. Am Intra-Ocular Implant Soc] 11:153-158, 1985 10. Binkhorst RD: The optical design of intraocular lens implants. Ophthalmic Surg 6(3):17-31, 1975 11. Binkhorst RD: Pitfalls in the determination of intraocular lens power without ultrasound. Ophthalmic Surg 7(3):69-82, 1976 12. Hoffer KJ: The effect of axial length on posterior chamber lens and posterior capsule position. Curr Concepts Ophthalmic Surg 1:20-22, 1984
APPENDIX Intraocular Lens Equations Used to Perform the Measurement Error Simulations Modified Binkhorst Equation
1336 (acor - d){i3r - d - O.OOlR, [v(i3r- d)+udr]} where a=
1.0
13 = __n..c.v-
with nv = 1.336 and nc =
337.5 r= _ __
t = 1.3333
as recommended by Binkhorst to compensate for a postoperative reduction in the refracting power of the cornea.l.l 0 • 11 An axial length correction factor was included to compensate for the difference between the true axial length and the axial length as measured by ultrasound.! This correction is required because the ultrasound wave is reflected at the vitreoretinal interface and does not measure the distance between the interface and the sensory layer of the retina. The measured axial length is corrected for by retinal thickness using the equation,
'""'
~ '"' +
T, - t, (
-
n~..)
where nv = 1.336 nP = 1.49166
68
Tr = 0.250 mm t; = 0.500 mm J CATARACT REFRACT SURG-VOL 16, JANUARY 1990
The postoperative anterior chamber depth is estimated by
d d
for d :S 4.43 mm for d > 4.43 mm
0.17am + 0.017 4.43 mm
Holladay Equation - __0_.0_0_1__R_s_[v_(_~_r___a_c~·o_r)__+__a__ac~·o~rr_J_______________ 1336 ______~_r__- __a~c<_><__ (acor- d- SF){~r- d- SF- 0.001RJv(~r- d- SF)+ a(d + SF)r]} Ra~ = Ra~ =
r 7 mm
for r ~ 7 mm for r < 7 mm
AG AG
0.533am 13.5 mm
for AG :S 13.5 mm for AG > 13.5 mm
with
d=0.56+
Rag-~
Holladay compensates for retinal thickness with the following axial length correction factor, 4 where Tr = 0.200 mm The polynomials and constants a, ~, nc, and nv are identical to those used in the Binkhorst equation.
Modified Colenbrander Equation Colenbrander's equation was originally developed for an iris clip lens. 2 The original formula has been modified by the addition of a postoperative anterior chamber depth scaling algorithm and a term to produce desired ammetropia. These changes make this equation consistent with the other second generation formulas used in this analysis. nv --------- -
~- 0.001d
[0.001(am - d) - 0.00005] [
Km
l.336Rs
0.00005]
The postoperative anterior chamber depth in millimeters is estimated by
d
d
O.l7am + 0.017 6.0mm
for d :S 6.0 mm ford> 6.0 mm
Hoffer Equation 1336 (am :- d - 0.05)
[
1.336 1.336
(Km
+
(d + 0.05) RJ
1000
]
The Hoffer equation was derived without the use of axial length correction factors. Hoffer derived the following postoperative anterior chamber depth scaling algorithm,
d d
0.292am - 2.93 6.0 mm
d :S 6.0 mm d > 6.0 mm
which was used with his equation in the measurement error simulations.l2
SRK II Equation
J CATARACT REFRACT SURG-VOL 16, JANUARY 1990
69
where
A+ 3 A+2 A+ 1 A A 0.5
A1 A1 A1 A1 A1
for for for for for
am < 20.0 20.0 S am 21.0 S am 22.0 S am am ~ 24.5
mm < 21.0 mm < 22.0 mm < 24.5 mm mm
and 1.00 1.25
'Y 'Y
for A1 - 0 9 K - 2.5a S 14.00 mm for A1 - o:g K:: - 2.5a:: > 14.00 mm
No axial length corrections are used with the SRK II formula. ·5
Statistical Equations Average primary implant power:
Standard Deviation:
5
Davg
I
i= 1
s
Di
~ i=l
5
4
Symbols Used in Error Analysis Equations A
the SRK constant in diopters
am
the axial length as measured on ultrasound, in millimeters
acor
the corrected axial length in millimeters
D
the dioptric power in aqueous of the intraocular lens
D1
the primary implant power predicted by the modified Binkhorst equation
D2
the primary implant power predicted by the Holladay equation
D.3
the primary implant power predicted by the modified Colenbrander equation
D4
the primary implant power predicted by the Hoffer equation
D5
the primary implant power predicted by the SRK liT" formula
d
the calculated postoperative anterior chamber depth in millimeters
Km
the measured average radius of curvature of the anterior surface of the cornea in diopters
nc
the index of refraction of the cornea
nP
the index of refraction of poly( methyl methacrylate)
nv
the index of refraction of the aqueous or vitreous
Rs
the desired postoperative refraction in diopters
SF
Holladay's surgeon factor in millimeters
ti
the thickness of the implant in millimeters
Tr
the distance between the vitreoretinal interface and the sensory layer of the retina in millimeters
v
the vertex distance in millimeters
70
J CATARACT REFRACT SURG-VOL 16, JANUARY 1990