ARTICLE

Fluctuations in corneal curvature limit
predictability of intraocular lens
power calculations
Sverker Norrby, PhD, Nino Hirnschall, MD, Yutaro Nishi, MD, Oliver Findl, MD

PURPOSE: To analyze fluctuations in corneal curvature over time.
SETTING: Moorfields Eye Hospital NHS Foundation Trust, London, United Kingdom.
DESIGN: Case series.
METHODS: A 3-piece IOL was implanted in 1 eye and a 1-piece IOL in the other eye through a 3.2 mm
clear corneal temporal incision. Keratometry was performed preoperatively and at several points intime postoperatively. Differences between measurements were analyzed by power vectors.
Statistical significance was assessed by monovariate, bivariate, and trivariate paired t tests. Acuteangle shifts were determined as differences between meridians at 2 points in time.
RESULTS: Fifty patients were enrolled. From preoperatively to 1 year postoperatively, the changes
in vector components (M, J0, J45) were, respectively, 0.02 diopter (D) G 0.23 (SD) (PZ.38),
0.07 G 0.27 D (PZ.02), and C0.04 G 0.25 D (PZ.14). Corresponding changes from 1 year
to 2 years postoperatively were C0.01 G 0.25 D (PZ.73), C0.01 G 0.23 D (PZ.83), andC0.01 G 0.16 D (PZ.40). The meridian shift was 5 G 32 degrees (PZ.13) from preoperatively
to postoperatively and C3 G 22 degrees (PZ.23) from 1 year to 2 years.
CONCLUSIONS: Surgically induced astigmatism was composed of slight flattening in the horizontal
meridian and slight steepening in the oblique meridian but was insignificant in relation to random
fluctuations, which were almost equally large between postoperative measurements 1 year apart.
The fluctuations were not due to imprecision in measurement.
Financial Disclosure: Dr. Norrby is a retiree from Abbott Medical Optics, Inc., and holder of a small
amount of stock. Dr. Findl is scientific advisor to Abbott Medical Optics, Inc. No other author hasa financial or proprietary interest in any material or method mentioned.
J Cataract Refract Surg 2013; 39:174–179 Q 2012 ASCRS and ESCRSIntraocular lens (IOL) power calculation is dependent on the accuracy and precision of the measured
input, usually limited to keratometry for corneal
power and to ultrasound or optical biometry foraxial length. In this paper, we scrutinize the reproducibility of keratometry; that is, how well measurements obtained at widely different points in timeagree. For regular cataract surgery with nontoric
IOLs, it is assumed that the mean corneal power remains the same postoperatively as preoperatively.
This assumption is also made for implantation of toric IOLs, with additional correction for surgically induced astigmatism (SIA). In the simplest case,1
a given amount of flattening is assumed in the meridian of the incision, with corresponding steepening
at 90 degrees to it. This can be expressed as the174

Q 2012 ASCRS and ESCRS
Published by Elsevier Inc.

spherocylinder (C/2 C @ A), where C is the induced positive cylinder of astigmatism, often assumed to be 0.5 diopter (D), and A is its axis,
which coincides with the meridian of incision. The
net spherical equivalent (SE) change is zero. The
more sophisticated approach by Fam and Lim2 takes
into account the complete surgically induced changeby allowing for the possibility that the induced astigmatism is not necessarily in the meridian of the incision and is not necessarily compensated for 90
degrees to it. The resulting surgically inducedchange can thus have a non-zero SE.
PATIENTS AND METHODS
In a study of IOL stability at Moorfields Eye Hospital,
London, United Kingdom, patients had implantation of
0886-3350/$ - see front matter
http://dx.doi.org/10.1016/j.jcrs.2012.09.014

FLUCTUATIONS IN CORNEAL CURVATURE AND IOL POWER CALCULATIONS

a 3-piece Tecnis ZA9003 IOL in 1 eye and a 1-piece Tecnis
ZCB00 IOL (both Abbott Medical Optics, Inc.) in the othereye. The study was approved by the institutional review
board and adhered to the tenets of the Declaration of Helsinki. All patients provided informed consent.
To select patients with essentially healthy eyes except
cataract, the following exclusion criteria were applied: (1)concurrent participation or participation during the past
30 days in any other clinical trial; (2) use of systemic or ocular medications that may affect vision or pupil size; (3)uncontrolled systemic or recurrent ocular disease (eg, recurrent ocular inflammatory condition); (4) requiring an IOL ofless than 15.0 D or more than 26.0 D; (5) history of ocular
trauma or previous ocular surgery; (6) corneal astigmatismof 2.0 D or more; (7) corneal abnormalities (eg, irregularastigmatism, stromal, epithelial or endothelial dystrophies);(8) uncontrolled ocular hypertension or glaucomatouschanges in the retina; (9) pupil abnormalities (nonreactive,
tonic pupils, or abnormally shaped pupils); (10) capsuleor zonule abnormalities that may lead to IOL decentration,
including pseudoexfoliation, trauma, an eccentric anterior
capsulorrhexis, or posterior capsule defects; (11) known pathology that may affect vision or visual field as determinedby the investigator; (12) retinal changes that may affect vision significantly or that may require surgical intervention
during the course of the study (eg, unstable macular
dystrophy).
Surgery was performed using a 3.2 mm temporal incisionin clear cornea close to the limbus. In addition, 1.0 mm paracenteses approximately 2.5 mm in length were made at approximately 60 degrees on either side of the main incision.
The right paracentesis was used for the capsulorrhexis needle and the left for a second instrument during phacoemulsification. Both were used for bimanual irrigation andaspiration.
When scrutinizing the data of the study, there was rather
large variability in keratometry between different points intime. A decision was made to perform a deeper analysis ofit, which is the topic of this paper.
Keratometry was measured with the IOLMaster device(Carl Zeiss Meditec AG) preoperatively and postoperatively.
Changes from 1 timepoint to another were analyzed using
power vectors3 (M, J0, J45) obtained as follows:
M Z S þ C=2

175

J0 Z  C=2  cosð2AÞ
J45 Z  C=2  sinð2AÞ
where S is the power in the flat meridian, C is the positive
power difference between the flat and steep meridians, andA is the axis of the positive cylinder, which coincides withthe flat meridian of the cornea. We prefer positive cylinder
notation for this purpose; however, the method works
equally well for negative notation. M is the same as themean corneal power (SE) and has the dimension of reciprocal
length in meters (m1; equivalent to diopter [D]), which isalso the dimension of J0 and J45. They can be considered as
Jackson crossed cylinders (thus having no spherical power),
1 at 0 degree and the other at 45 degrees. To avoid mirror
symmetry between right eyes and left eyes to cancel out oblique astigmatism, the sign of J45 was reversed for left eyes.
The transformation allows subtraction of 1 vector from another. The difference can be reconverted to spherocylinder inplus cylinder notation by
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
C Z 2  J20 þ J245
S Z M  C=2
A Z 0:5  atanðJ45=J0Þ þ 90
For negative cylinder notation, the negative root is chosen
and the number 90 is omitted.
Power vectors also allow for calculation of means andstandard deviations (SDs) and application of statisticalanalysis. The TTEST function of Excel (Microsoft Corp.)
was used to calculate 2-tailed monovariate, bivariate, andtrivariate paired P values.
For analysis of rotation of the axis from 1 timepoint to another timepoint, the angle shifts Aflat(time 2) – Aflat(time 1)
and Asteep(time 2) – Asteep(time 1) were computed. If thesewere not the same, the 1 with the smallest numerical magnitude was taken, thus obtaining the acute angle shift. These difference angles are bound between 90 degrees and C90
degrees and will have a mean of 0 degree if the direction of rotation is random. Normal statistical analysis could result in
95% limits or even SDs that are outside these bounds andthus not defined. Therefore, the median of the absolute valuesof the difference angles was used to characterize the variability.
This median will always be between 0 degree and 90 degrees.

RESULTS

Submitted: June 9, 2012.
Final revision submitted: September 6, 2012.
Accepted: September 8, 2012.
From Leek, The Netherlands (Norrby); Vienna Institute for Researchin Ocular Surgery (Hirnschall, Findl), Hanusch Krankenhaus,
Vienna, Austria, and Moorfields Eye Hospital, London, United
Kingdom; the Nishi Eye Hospital (Nishi), Osaka, and Departmentof Ophthalmology, Tokyo University, Tokyo, Japan.
Supported by Abbott Medical Optics, Inc., Santa Ana, California,
USA.
Presented in part at the XXIX Congress of the European Society of
Cataract & Refractive Surgeons, Vienna, Austria, September 2011.
Corresponding author: Sverker Norrby, PhD, Landauerlaan 17,
NL-9351 PN Leek, The Netherlands. E-mail: sverker@home.nl.

The study enrolled 50 patients; 91 cases were available
for evaluation at 1 year.
The mean surgically induced changes in vector components (M, J0, J45) were, respectively, 0.02 D G 0.23(SD), 0.07 G 0.27 D, and C0.04 G 0.25 D. This translates to the mean spherocylinder change of 0.10
C0.16 @ 165. The monovariate P values were 0.38,
0.02, and 0.14, respectively. A negative J0 implies flattening in the horizontal meridian. A positive J45 means
steepening in the 45-degree meridian and indicatesthat the SIA is not purely induced in the horizontal meridian but in this case was at 165 degrees. Although themonovariate P value indicates statistical significancefor the change in J0, the bivariate change in J0 and J45

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FLUCTUATIONS IN CORNEAL CURVATURE AND IOL POWER CALCULATIONS

Figure 1. Changes in M (SE) from preoperatively to 1 year postoperatively (n Z 91). Plotted are also the mean (short dashes) and the 95%
probability limits (long dashes).

Figure 2. Changes in J0 and J45 from preoperatively to 1 year postoperatively (n Z 91). Plotted is also the 95% probability ellipse(long dashes). The mean is indicated by the black diamond and its departure from the origin indicates the magnitude of SIA, whichclearly is small compared with the random variation.

had a P value of 0.68. The trivariate P value was 0.28.
Even if there were some surgically induced change, it
was small and insignificant in comparison with therandom variation in the vector components, as seenin Figures 1 and 2. There was no statistically significant
difference between right eyes and left eyes (PZ.57), between men and women (PZ.59), or between age(%72 years and O72 years) (PZ.61).
Table 1 shows the changes in power vector components from preoperatively to several periods postoperatively. To have a coherent data set, only cases thatwere available at all points in time were included,
which reduced the number of cases to 69. There wasa small but statistically significant increase in M (SE)at 1 month and 3 months that disappeared at 1 yearand 2 years. The statistically significant shift in J0 at
1 month disappeared at later timepoints, and the shifts
in J45 did not reach statistical significance at any timepoint. The negative shifts in J0 indicate slight flattening
in the horizontal meridian and the positive shifts in J45
slight steepening in the 45-degree meridian.

Table 1 also shows the changes from 1 year to 2
years postoperatively. As expected, the mean change
in all 3 vector components was practically zero while
the SDs were of the same magnitude as from preoperatively to postoperatively with the exception of J45.
Figures 3 and 4 show the results graphically forcomparison with Figures 1 and 2.
The mean arithmetic change in cylinder was 0.01
G 0.45 D from preoperatively to 1 year postoperatively and 0.05 G 0.34 D from 1 year to 2 years postoperatively. The mean changes were not statisticallysignificant.
The mean shift in the flat meridian from preoperatively to 1 year postoperatively was 5 degrees,
which is not different from 0 degree with statisticalsignificance. The SD was 32 degrees, with a maximumof C82 degrees and a minimum of 89 degrees. The
median absolute shift was 14 degrees. The histogramin Figure 5 shows a more or less random distribution.
As can be seen in Table 2, there was a somewhat erratic tendency toward the median absolute shift

Table 1. Mean changes in vector components and monovariate P values from preoperatively to several postoperative periods and from 1
year to 2 years postoperatively. The slight differences in values between text and table for preoperatively to 1 year postoperatively are explained by the different numbers of cases included.
M (D)PeriodPreop to 1 month
Preop to 3 months
Preop to 1 year
Preop to 2 years
1 year to 2 years

J0 (D)

J45 (D)

Cases (n)Mean G SD

P Value

Mean G SD

P Value

Mean G SD

P Value

69
69
69
69
79

C0.11 G 0.30
C0.11 G 0.28
0.02 G 0.23
C0.01 G 0.25
C0.01 G 0.25

.00
.00
.58
.79
.73

0.14 G 0.31
0.08 G 0.56
0.06 G 0.25
0.05 G 0.30
C0.01 G 0.23

.00
.23
.05
.21
.83

C0.07 G 0.30
C0.02 G 0.41
C0.05 G 0.25
C0.06 G 0.27
C0.01 G 0.16

.05
.67
.10
.06
.40

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FLUCTUATIONS IN CORNEAL CURVATURE AND IOL POWER CALCULATIONS

Figure 3. Changes in M (SE) from 1 year to 2 years postoperatively(n Z 79). Plotted are also the mean (short dashes) and the 95%
probability limits (long dashes).

Figure 4. Changes in J0 and J45 from 1 year to 2 years postoperatively (n Z 79). Plotted is also the 95% probability ellipse (longdashes). The mean is indicated by the black diamond and its departure from the origin is negligible, which could be expected.

being larger for small amounts of corneal cylinderand gradually declining with increasing corneal cylinder; however, the number of observations withineach range was too small for statistical analysis. In
Figure 6, the acute angle shifts are plotted against
preoperative corneal cylinder together with logarithmic fits of median absolute shift against corneal cylinder, displaced by the amount of the (nonsignificant)mean shift.
The mean shift in the flat meridian from 1 year to 2
years postoperatively was C3 degrees (not significant), with a median absolute shift of 8 degrees. The
SD was 22 degrees, with a maximum of C75 degrees
and a minimum of 49 degrees. Detailed data areshown in Table 3, and the results are plotted in
Figure 7. The smaller shifts in comparison with thosein Table 2 and Figure 6 are due to smaller shifts in
the J45 component. Although, as stated earlier, it isquestionable to analyze the distribution in terms of
95% limits, the fit results for comparison with medianare given in Table 4. That the 95% limits fell outside thedefined range of angle shifts (bound between
90 degrees and C90 degrees) for low cylinders demonstrates the inappropriateness of applying this statistical measure to a distribution with upper and lower
bounds.
DISCUSSION
The keratometric measurement itself is rarely questioned, although Shammas and Chan4 recently reported a change of 0.02 G 0.27 D in mean corneal
power between preoperative measurements taken approximately 1 month apart using the IOLMaster device. In a recent study, Shammas and Hoffer5 usedthe Lenstar LS900 device (Haag-Streit AG) and founda change of 0.04 G 0.26 D in mean corneal power between preoperative measurements 1 month apart. Our
results for postoperative measurements 1 year apart
Table 2. Meridian shifts from preoperatively to 1 year postoperatively grouped in 0.25 D intervals of preoperative corneal
cylinder.

Figure 5. Distribution of angle shifts from preoperatively to 1 year
postoperatively (n Z 91).

CylinderInterval (D)Mean Preop
Cylinder (D)Median
Shift ( )Cases (n)0.00 to !0.25
0.25 to !0.50
0.50 to !0.75
0.75 to !1.00
1.00 to !1.25
1.25 to !1.50
1.50 to !1.75
1.75 to !2.00

0.19
0.36
0.64
0.87
1.11
1.30
1.62
1.87

25
43
14
15
5
9
38
12

3
17
31
13
11
5
5
3

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FLUCTUATIONS IN CORNEAL CURVATURE AND IOL POWER CALCULATIONS

Table 3. Meridian shifts from 1 year to 2 years postoperatively
grouped in 0.25 D intervals of corneal cylinder at 1 year.

Figure 6. Acute meridian shifts from preoperatively to 1 year postoperatively plotted against preoperative corneal cylinder. The
mean shift is plotted with short dashes. Logarithmic fits to themedian shifts in 0.25 D intervals shown in Table 2, displaced by
the mean shift, are plotted with long dashes.

are very similar, with a mean change of C0.01 G 0.25
D in M (same as mean corneal power) and a change of
C0.02 G 0.23 D in M from preoperatively to 1 year
postoperatively. All changes in the mean were statistically insignificant. The random variations were small,but not negligible, and far larger than the G0.04 D SD
found for repeat measurements with the IOLMasterdevice by Shirayama et al.6 Shammas and Hoffer5 report a coefficient of variation of 0.003, which translates
to approximately G0.1 D SD in repeat measurementswith the Lenstar LS900 device.
To analyze the variation in our measurements, wetook the pairwise differences in mean keratometry
value of the 3 readings provided by the IOLMasterdevice for the 100 eyes measured preoperatively andfound an SD of G0.10 D. Our conclusion is that thedifferences we and others found in measurement ofcorneal power at different timepoints reflect naturalfluctuations and were not due to measurement error.
Shammas and Chan4 found an arithmetic change incylinder of 0.06 G 0.38 D and Shammas and Hoffer5

CylinderInterval (D)Mean Cylinder
at 1 Year (D)Median
Shift ( )Cases (n)0.00 to !0.25
0.25 to !0.50
0.50 to !0.75
0.75 to !1.00
1.00 to !1.25
1.25 to !1.50
1.50 to !1.75
1.75 to !2.00

0.22
0.38
0.61
0.85
1.13
1.36
1.62
1.86

18
19
11
8
8
5
8
5

3
12
21
13
15
4
7
2

of 0.09 G 0.25 D. This is similar to what we foundfrom preoperatively to 1 year postoperatively (0.01
G 0.45 D) and from 1 year to 2 years postoperatively(0.05 G 0.34 D), although the SDs for cylinder vary
somewhat more than for sphere between the studies.
The mean changes were not statistically significant;thus, we believe the random change was again due
to natural variation in the cornea.
It is slightly misleading to analyze the arithmeticchange in cylinder on its own because there is almost
always a concurrent change in meridian. The correct
analysis is that of the vector components J0 and J45,
for which we found an SD of G0.25 D from preoperatively to 1 year postoperatively. The change from 1
year to 2 years postoperatively had an SD of G0.23 D
in J0 and G0.16 D in J45. We were given access to
the raw data of Shammas and ChanA to perform
power vector analysis, and the results were very similar to ours (C0.02 G 0.24 D in J0 and C0.01 G 0.17 D
Table 4. Logarithmic fit data for median and 95% limits (1.96 
standard deviation) in meridian shifts at different levels of cylinder from preoperatively to 1 year postoperatively and from 1
year to 2 years postoperatively. The nonsignificant changes in
the means are ignored. For low cylinders, the 95% limits are outside the valid range of G90 , which indicates that it is inappropriate to apply this kind of statistics.
Preop to 1 Year
Postop1 Year to 2 Years
Postop95%
95%
Cylinder (D) Median ( ) Limits ( ) Median ( ) Limits ( )Figure 7. Acute meridian shifts from 1 year to 2 years postoperatively plotted against corneal cylinder at 1 year. The mean shift isplotted with short dashes. Logarithmic fits to the median shifts in
0.25 D intervals shown in Table 3, displaced by the mean shift, areplotted with long dashes.

0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00

G39
G30
G23
G18
G14
G10
G8
G6
G5

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G174
G145
G122
G102
G85
G71
G59
G50
G41

G20
G16
G13
G11
G9
G7
G6
G5
G4

G252
G165
G108
G71
G46
G30
G20
G13
G9

FLUCTUATIONS IN CORNEAL CURVATURE AND IOL POWER CALCULATIONS

in J45). The random variation in J0 was thus approximately G0.25 D for measurements at different timepoints both preoperatively and postoperatively,
while the variation in J45 was slightly less. The higher
SD in J45 from preoperatively to 1 year postoperatively is thus the only indication of surgically inducedchange and also explains the higher variation in axis
from preoperatively to 1 year postoperatively thanfrom 1 year to 2 years postoperatively.
A reflection that can be drawn from Figure 2 is thatthe application of correction for SIA in toric IOL
calculation is insignificant in relation to the randomvariation, even for the relatively large incision(3.2 mm) in our study. There was flattening (negativeJ0) in the axis of incision in more than half the cases
but significant steepening (positive J0) in many cases.
Of course, there is no harm in applying correction for
SIA.
We found a mean change in meridian of 5 G 32
degrees and a median absolute change of 14 degrees
from preoperatively to 1 year postoperatively andof C3 G 22 degrees and 8 degrees, respectively,from 1 year to 2 years postoperatively. This compares well with the C4 G 24 degrees and median
12 degrees found by Shammas and Chan4 and the3 G 12 degrees and median 9 degrees found by
Shammas and Hoffer.5 The mean changes were notstatistically significant in any of the studies. Achange of approximately 10 degrees in median appears normal for measurements both preoperatively
and postoperatively due to natural fluctuations in
the cornea. Logically, determination of meridians isless accurate for corneas with low astigmatism(Table 4). From preoperatively to 1 year postoperatively, it was 14 degrees at 1.0 D of astigmatism and5 degrees at 2.0 D. From 1 year to 2 years postoperatively, it was 9 degrees and 4 degrees, respectively.
By definition, half the cases are below the medians
and the other half above. This has repercussions fortoric IOL power calculations. For a 15-degree error
in meridian, half the effect of the cylinder correction
is lost.
Because the time of day of the measurements was
not controlled for in any of the studies, diurnal
variations can be a reason for the fluctuations. Other
environmental factors, such as temperature and humidity, could also play a role. The fact remains thatthe measured corneal power could be different byapproximately 0.5 D from 1 occasion to another in approximately 5% of cases. In IOL power calculation, thatdifference will affect refractive outcome with about thesame amount. The same applies to toric IOL power calculation, with the added uncertainty due to naturalfluctuations in magnitude and axis of cylinder.

179

WHAT WAS KNOWN
 In IOL power calculation, the cornea is treated as a static
component of the optical system of the eye.
 Reasons for postoperative refractive error after cataract
surgery are attributed to errors in measurement of corneal
power and axial length and in the estimation of the effective IOL position.
WHAT THIS PAPER ADDS Our study suggests that temporal (varying in time) fluctuation in corneal curvature is an additional source of error
in IOL power calculation.
 It appears that the cornea is a dynamic optical component
preoperatively as well as postoperatively.
 The fluctuations were much larger than the surgicallyinduced change in cataract surgery.
REFERENCES
1. Hill W. Expected effects of surgically induced astigmatism on
AcrySof toric intraocular lens results. J Cataract Refract Surg
2008; 34:364–367
2. Fam HB, Lim KL. Meridional analysis for calculating the expected
spherocylindrical refraction in eyes with toric intraocular lenses.
J Cataract Refract Surg 2007; 33:2072–2076
3. Thibos LN, Wheeler W, Horner D. Power vectors: an application of
Fourier analysis to the description and statistical analysis of refractive error. Optom Vis Sci 1997; 74:367–375. Available at: http://
journals.lww.com/optvissci/Abstract/1997/06000/Power_Vectors__
An_Application_of_Fourier_Analysis.19.aspx. Accessed September
20, 2012
4. Shammas HJ, Chan S. Precision of biometry, keratometry, andrefractive measurements with a partial coherence interferometry–keratometry device. J Cataract Refract Surg 2010;36:1474–1478
5. Shammas HJ, Hoffer KJ. Repeatability and reproducibility ofbiometry and keratometry measurements using a noncontact
optical low-coherence reflectometer and keratometer. Am J Ophthalmol 2012; 153:55–61
6. Shirayama M, Wang L, Weikert MP, Koch DD. Comparison of corneal powers obtained from 4 different devices. Am J Ophthalmol
2009; 148:528–535

OTHER CITED MATERIAL
A. Personal communication from John Shammas to Sverker Norrby,
September 24, 2010

J CATARACT REFRACT SURG - VOL 39, FEBRUARY 2013

First author:
Sverker Norrby, PhD
Leek, The Netherlands