The Olsen Formula

50
Thomas Olsen

The Olsen Formula The paraxial approach allows for thick-lens

calculations whereby the cornea and the IOL can
The Olsen formula was developed at the time be represented as the two-surface optical lenses
when the Sanders-Retzlaff-Kraff (SRK) method they are. The advantage is that different optic
was popular. Although the SRK formula was configurations can be dealt with, and the refrac-
working all right in the normal range, errors were tive effect of a, say 1:1 biconvex, 1:2 biconvex, or
frequent in the extreme range and the lack of a a meniscus concave-convex IOL, can be calcu-
flexible, optical model was frustrating. So, the lated independently from the IOL position. All it
ambition was to develop a thick-lens formula requires is a knowledge of the shape of the IOL,
based on paraxial ray tracing as assumption-free which must be provided by the IOL
as possible allowing for the use of real physical manufacturer.
dimensions—including the physical position of One disadvantage of the paraxial approach is
the IOL— to be used in the formula. that higher-order aberrations are not taken into
The first step for the author was to realize that account. The most significant aberration is spher-
the K-reading of the keratometer using the stan- ical aberration, which plays a role in normal eyes,
dard index of 1.3375 was wrong (see the but can be excessive in abnormal corneas like
“Keratometry” chapter). To avoid confusion, the post-LASIK cases and keratoconus. Hence, from
author has always preferred to input the radius of 2012 the Olsen formula was modified to allow
the K-reading rather than the diopter value. The exact ray tracing on aspheric surfaces in order to
conversion to corneal power is then done inter- include the effect of spherical aberration in the
nally by the formula. From the beginning, a ficti- calculated effective refraction. This meant a
tious index of 1.3315 based on the Gullstrand change in Gullstrand ratio to 0.83 (which is the
ratio of 0.883 was found to give a more realistic value also demonstrated in many Scheimpflug
value for effective corneal power. This value has reports) but now in addition using the Q-value of
later been used by other authors, i.e., Haigis and the front and back surface of the cornea for a
Barrett, and there seems to be growing consensus more detailed calculation of the corneal power. If
among newer formulas that the lower value is a no Q-values are stated, the program will assume
better choice for IOL power calculation. the default normal values. In this way, it was pos-
sible to include the effect of the wavefront-­
corrected spherical aberration of an aspheric
IOL.
T. Olsen (*)
Aros Private Hospital, Aarhus, Denmark

© The Author(s) 2024 731

J. Aramberri et al. (eds.), Intraocular Lens Calculations, Essentials in Ophthalmology,
https://doi.org/10.1007/978-3-031-50666-6_50
732 T. Olsen

A realistic corneal power is required to predict constant calculated from the SRK/T A-constant,
the refractive effect of the IOL using the physical as a first go. However, it is recommended to keep
position of the IOL. Once it was found that the track of the outcome and adjust the ACD constant
position of the IOL could be predicted, the next as more data become available.
step was to improve the ELP prediction. Over the
years, a number of ELP predictors have been Data Entry
studied by the author: 1) K-reading, ACD and lens Data entry can be made manually or by importing
thickness (Olsen 1986) [1], 2) K-reading, ACD from biometers via a data bridge (xml files or
and axial length, K-reading, ACD, lens thickness, similar). The following biometers are supported
axial length, corneal diameter distance, and for data bridge import:
refraction [2], and finally 3) ACD and lens thick-
ness measured by laser biometry to arrive at the 1. Haag-Streit Lenstar LS900
novel concept called the C-constant approach 2. Oculus Pentacam (full cornea analysis)
(Olsen & Hoffmann 2014) [3]. The latter method 3. Zeiss IOLMaster 700
represented a “heureka” moment in its simple 4. Topcon Aladdin
form that proved to be effective and robust with- 5. Tomey OA-2000
out the indirect predictors such as the K-reading, 6. Ziemer Galilei G6 (full cornea analysis)
axial length, corneal diameter, refraction, and age
with previous methods. The advantage of this The K-readings can be expanded (double-­
approach is that it should work equally effectively click on the field) to allow entry of posterior cur-
in abnormal corneas such as post-LASIK cases, vatures and Q-values if these are available. If no
keratoconus, megalocornea, scleral buckling pro- data are input for the posterior surface, the pro-
cedure, and horses, if you may. gram will assume a default value. In this way,
corneal astigmatism can be calculated based on
the default posterior cylinder or based on exact
PhacoOptics® Software measurements. This allows for a full-thickness
analysis of the corneal power from tomography
A stand-alone PC software for Microsoft data, i.e., captured with the Oculus Pentacam or
Windows (www.phacooptics.com) was released the Ziemer Galilei G6. This is particularly useful
by the author in 2009. Using paraxial and exact when dealing with post-LASIK cases or other
ray tracing, the software package offers a com- abnormal corneas.
prehensive system for IOL power calculation and The Olsen formula has also been implemented
data management. as a dynamic library into the software of the
Because of the ray tracing, the physical data of Haag-Streit Lenstar, the Topcon Aladdin, the
the IOL need to be stated in more detail than in Tomey OA-2000, and the Oculus Pentacam.
most formulas. The IOL constants are: The IOL power calculation algorithm follows
the principles described in this chapter. The pre-
1. Refractive index diction of the ELP (rather: the physical IOL posi-
2. Anterior and posterior radius of curvature of tion) has been given the flexibility of a 2-­factor
an average-powered IOL version and a 4-factor version (selectable by the
3. Thickness of an average-powered IOL user). Both versions use the C-constant, which is
4. Wavefront Z(4,0) correction for spherical based on the ACD and the lens thickness, but the
aberration 4-factor version uses an additional corrective
5. ACD constant (average value in representa- term based on the K-reading and the axial length.
tive population) The 4-factor version may have a little more accu-
racy than the 2-factor version as shown by Cooke
When the physical parameters 1–4 have been and Cooke [4, 5], but is only applicable to nor-
entered, it is possible to have item 5, the ACD mal, virgin eyes. The 2-factor version is indepen-
50 The Olsen Formula 733

dent of the K-reading and the axial length and is son. A full-thickness analysis of the right cornea
therefore more robust in post-­LASIK cases and was done by importing the values from the
other abnormal cases. Oculus Pentacam (highlighted fields). The
detailed information can be viewed (and edited)
 ata Quality Is the Key
D by right- or double-clicking the K-reading fields
All calculations depend on the quality of the (insert lower right). In this case, the Gullstrand
input data. Garbage in means garbage out, as ratio was 0.779 on the post-LASIK right eye and
everybody knows. To help filter out typing errors 0.883 (default) on the virgin left eye. An abnor-
or other mistakes, the program will evaluate the mal Q-value for the front surface of the right eye
plausibility of all data input when in manual due to the LASIK procedure is noted.
entering mode. This plausibility check is per- Figure 50.2 shows the IOL power calculation
formed at three different levels: screen of the same post-LASIK case. The IOL
type has been selected from a drop-down menu.
1. The out-of-range plausibility of the individual Both the power, the cylinder, and the axis can be
variable changed by scrolling up and down, and the result-
2. The intra-eye plausibility of the input com- ing sphere cylinder and axis are displayed below.
pared to other variables of the same eye (e.g., By default, the optimum placement axis of the
a flat cornea in a short eye) toric has been calculated based on the complete
3. The inter-eye plausibility of the input com- corneal data. The axis can be confirmed by press-
pared to existing data of the contralateral eye ing the small button marked? “Cyl axis.” Here, a
small cylinder was chosen to minimize the astig-
The threshold of the plausibility levels can be matism of the postoperative refraction. The surgi-
set in the program settings. cally induced astigmatism (SIA) can also be
As is the case with any IOL formula, it is added in a detail window (not shown).
important that the K-readings and the axial length For the post-LASIK case, the ELP prediction
are accurate. In addition, the Olsen formula is was done using a 2-factor algorithm (identical to
particularly sensitive to measurement errors of the C-constant) because the post-LASIK
the anterior chamber depth and the lens thick- K-reading is unsuited for this purpose. The selec-
ness. This is because the C-constant is entirely tion was done after double-clicking the ACD
dependent on these two variables. It is good clini- field. Note the nearly identical values for the right
cal practice to check the consistency of the read- and left eye despite the post-LASIK state of the
ings, especially for the lens thickness, which may right eye.
be hard for the biometer software to pick up with
good spikes of the anterior and the posterior
surface. Formula Validation
Finally, the pupil size should be mentioned.
Unlike most other formulas, PhacoOptics does The aim of the Olsen formula was to “divide and
take the pupil size into account as it will play a conquer” the unknowns of IOL power calcula-
role when the spherical aberration is high. Care tion. On the one side, we have the measurements
should be taken, however, to check the pupil size of corneal power, axial length, and optical prop-
if you are importing data from an external biom- erties of the IOL. All measurements must be rep-
eter, and the patient was dilated at the examina- resentative of the physical reality. Also, the
tion. A safe procedure is to leave the pupil blank, physical properties of the IOL must be known so
which is the equivalent of a standard pupil size of that we can calculate the refractive effect for a
3 mm assumed by the program. given IOL location. On the other side, we have an
Figure 50.1 shows a PhacoOptics screenshot issue with the prediction of the IOL position for
of the preoperative data of a post-LASIK case of which empirical studies are needed.
the right eye and untouched left eye for compari-
734 T. Olsen

Fig. 50.1 Preoperative data screen of a post-LASIK case wanted. The two inserts at the bottom show the detailed
on the right eye with untouched left eye. You may note the information of the K1-reading (double-click in the K1
right-left difference in K-readings. The K-readings of the field) with complete data on the right eye and default data
right eye are highlighted in yellow after Pentacam import, on the left eye
because a full-thickness analysis of the corneal power is

A critical question is as follows: What if the years ago. The database contained 1622 cases of
exact IOL position was known, and would the 1269 university clinic patients with an implanted
formula be able to predict the refractive outcome power ranging from −3.0 to +39.0 D. Ninety
accurately? The question can be answered by percent of the IOLs were of the Alcon Acrysof
recording the actual IOL position after surgery family (SA60AT, SN60AT, and torics and
and using this value in the “predictions.” This MA60MA for the low IOL power), and 10%
was done by Olsen and Hoffmann [3] in a subset were of the Abbott Tecnis types. The pseudopha-
of cases, demonstrating a drop in MAE from 0.39 kic ACD was recorded after surgery with Lenstar
to 0.36 for a public university series and from laser biometry.
0.30 D to 0.26 D in a private series, respectively, The refractive prediction mean error was
when the actual, measured postoperative IOL found to be −0.13 D ± 0.469 D (SD) with the
position was substituted for the predicted value in standard Olsen procedure and −0.019 D ± 0.436
retrospect. D (SD) when the postoperative, actual ACD was
For this book chapter, the study concept was used in the “predictions.” The mean error with
repeated with a larger database collected some the postoperative ACD was not significantly
50 The Olsen Formula 735

Fig. 50.2 IOL power calculation screen of the right eye field. An aspheric IOL with a small cylinder has been
post-LASIK case. The ELP prediction was done using a selected. The IOL details (insert) were called by double-­
2-factor algorithm (identical to the C-constant) because clicking the IOL power field. The program calculates the
the post-LASIK K-reading is unsuited for this purpose. exact curvatures of the front and back surfaces of the IOL
The selection was done after double-clicking the ACD to be used for ray tracing

d­ ifferent from zero. The standard deviation of actual position of the IOL after surgery was
±0.436 D corresponded to a mean absolute error recorded using Haag-Streit Lenstar laser inter-
(MAE) of 0.35 D, which was significantly lower ferometry. Based on the postoperative refraction
than that of the normal predictions (p < 0.01) and the biometric measurements, a ray tracing
(Fig. 50.3). In conclusion, when the IOL position analysis was performed back-solving for the
was known, the formula was able to predict the power of the IOL in situ. The results showed the
refraction with no bias or offset error (!) and a calculated IOL power to be in good agreement
corresponding improvement in accuracy. This with the labeled power over the entire power
finding means that if the ELP prediction would range with no offset or bias. This finding was
improve as a result of newer biometry techniques, another “heureka” moment for the author show-
the Olsen formula can utilize this information ing that the optics of the pseudophakic eye can
and improve the accuracy accordingly. be described by ray tracing and modern biome-
Another method of verification is to reverse try techniques.
the calculations: From the known postoperative For the present book chapter, the study was
refraction and the IOL position, it is possible to repeated on the same database as mentioned
back-solve for the IOL power using ray tracing. above. Figure 50.4 shows the correlation between
This was originally done by Olsen and Funding the calculated IOL power in situ and the labeled
(2012) [6] who studied 767 eyes with an power for the 1622 cases. The correlation coeffi-
implanted IOL power of the old Alcon Acrysof cient was 0.99, and the slope of the linear regres-
type ranging from −2.00 D to +36.0 D. The sion equation was not significant from unity. This
736 T. Olsen

Fig. 50.3 Prediction accuracy of the Olsen formula with and without the usage of the postop ACD in the
“predictions”

Fig. 50.4 IOL power in

situ calculated by exact
ray tracing compared to
the labeled value

finding can be regarded as a verification of the a highly standardized and controlled environment
optical algorithms used in the Olsen formula. for IOL power calculation.
In Fig. 50.5, the accuracy observed by the
author has been tabulated for a period of 30+
Own History of Calculation Accuracy years, covering both ultrasound and later optical
biometry. The number of cases within 0.5 D
The author has over 30 years of experience with accuracy has been computed from the standard
IOL power calculation. Looking back, it is amaz- deviation of the prediction error observed in each
ing how the accuracy has been ever-increasing series. Except for the last column (year 2020), all
over time. One reason for the improvement in columns have been constructed from the papers
accuracy has been the unsurpassed accuracy of published by the author and associates [3, 7–17].
optical biometry, but other factors such as stan- The last column showing 90% of cases within
dardization of surgery and improvement in for- ±0.5 D was the result of an independent study of
mula (ELP prediction) have combined to produce 469 refractive lens exchange cases using
50 The Olsen Formula 737

Fig. 50.5 History of IOL calculation accuracy (author’s own series)

IOLMaster 700 and the Olsen formula One of the largest comparative studies ever
(unpublished). was the study by Melles et al. (2018) [15] who
investigated the accuracy of seven different for-
mulas in a total of 18,501 cases of AcrySof
Recent Clinical Studies SN60WF (13,301 cases) and SA60AT (5200
cases) implants using Haag-Streit Lenstar biom-
There is a plethora of publications dealing with etry. The lowest prediction error was found with
IOL power calculation, and many new IOL for- the Barrett Universal II, followed by Olsen,
mulas have evolved. The interest comes from the Haigis, Holladay 2, Holladay 1, SRK/T, and
fact that modern lens surgery with a perfect IOL Hoffer Q.
power calculation holds the promise to free the The Melles 2018 study was later repeated with
spectacle dependence of the patient. As discussed updated versions of the Olsen formula (4-factor
in the section “The History of IOL Power version rather than the 2-factor version studied in
Calculation Accuracy,” the accuracy is approach- the first paper), the Hill RBF formula (newest
ing 90% of cases within 0.5 D of the target. version 2), the Holladay 2 (newest version, axial
As the Olsen formula requires good measure- length adjusted for the hyperopic error in long
ments of the anterior chamber depth and of the eyes), and 2 newer formulas: the Kane formula
lens thickness for the prediction of the IOL posi- and the EVO formula. The most accurate formu-
tion, it is not possible to evaluate the performance las were the Kane, the Olsen, and the Barret for-
of the Olsen formula using the traditional PCI mula all achieving more than 80% of the
optical biometry (IOLMaster 500) that does not predictions within ±0.50 D of the target, followed
measure the lens thickness. However, more and by the EVO, the Hill RBF, the Holladay 2, the
more studies have emerged using OLCR or Haigis, the Holladay 1, the SRK/T, and the Hoffer
swept-source OCT (SS-OCT) that does offer Q formulas in that order, respectively.
measurements of all intraocular distances by the The 2-factor version of the Olsen formula was
laser. the version that was originally implemented on
738 T. Olsen

the Lenstar biometer. The 2-factor version only mean absolute error (MAE) was 0.30 D with
takes the anterior chamber depth and lens thick- 81.8% of the cases within ±0.5 D. The material
ness as parameters and uses the unmodified was analyzed for possible bias with the axial
C-constant concept for the prediction of the IOL length. As shown in Fig. 50.6, no correlation was
position. However, as found by Cooke and Cooke found between the numerical error and the axial
29, 30
there seems to be a marginal higher accuracy length. This finding is noteworthy as a hyperopic
using the 4-factor version that also takes the axial error has been reported for some formulas in the
length and the corneal curvature as additional long eyes, giving rise to the Wang-Koch adjust-
parameters in the prediction of effective lens ment of the Holladay 1 and the SRK/T formula.
position. The 4-factor version is the default ver- The absolute error showed a trend toward
sion of the stand-alone PC software available on higher error in the short eyes and lower error in
the website www.phacooptics.com. the long eyes (Fig. 50.7). The short eyes remain
The author has had the opportunity to review the group of eyes with the highest error, first of
the large database of the Melles study and check all because all measurement errors have a rela-
the prediction accuracy. The database consists of tively higher impact on a short eye and also
outcome data for many surgeons from many clin- because the error of the ELP estimation has a
ics, and therefore, some variation can be found in much higher impact on the short eyes (see
data quality. Some cases were noted to have Fig. 50.8).
recorded highly unlikely values for the lens thick- When analyzing for bias with the K-reading,
ness: for example, a lens thickness of 2.5 in a no correlation was found between the prediction
76 years old, which is virtually impossible and error and the K-reading (Fig. 50.8). Hence,
must be due to a measurement mistake of the whether the eye is long, short, or has a steep or
Lenstar biometer. Therefore, all cases with lens flat cornea did not appear to have a significant
thickness <3 mm were excluded from the present bias on the formula performance.
review. None were excluded because of a high Finally, a note on the gender bias would be
prediction error per se. appropriate since some formulas use gender as a
Thus, after the exclusion of 92 cases with co-predictor. For example, gender was taken as a
unlikely lens thickness, the Melles database con- parameter by the Hoffer H formula [18] and is
sisted of 13,209 cases of SA60WF implants suit- also included as a parameter in the newer Kane
able for analysis. The standard deviation of the formula [19]. The rationale behind this is that
prediction error was found to be ±0.38 D, and the female eyes tend to be a little shorter, have a

Fig. 50.6 Numerical

error vs axial length in
13,209 cases
50 The Olsen Formula 739

Fig. 50.7 Absolute error vs axial length in 13,209 cases

Fig. 50.8 Numerical error vs keratometry reading in 13,209 cases

steeper K-reading, and have a shallower anterior females. The mean difference was 0.06 D
chamber than men. Therefore, one might suspect between males and females. Although statisti-
different behavior with respect to IOL constants cally significant (p < 0.01), the difference is not
and possibly introducing a bias in the IOL power clinically relevant. The lack of systematic bias
prediction. may be due to the use of the C-constant, which is
Table 50.1 shows the accuracy of the Olsen based on the position and thickness of the crystal-
formula according to gender. The mean numeri- line lens and works independently of the
cal error (± SD) was found to be +0.034 D (± K-reading, the axial length, and anterior chamber
0.387) in males and − 0.029 D (± 0.392) in depth.
740 T. Olsen

Table 50.1 Influence of gender on the prediction accuracy of the Olsen formula
Gender Error (± SD) MAE Range
Males (n = 5409) +0.034 (± 0.387) 0.307 −1.66 to +1.82
Females (n = 7800) −0.029 (± 0.392) 0.311 −1.93 to +1.80

10. Olsen T, Gimbel H. Phacoemulsification, capsu-

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