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Ray-tracing analysis of intraocularlens power in situ

Thomas Olsen, MD, PhD, Mikkel Funding, MD

PURPOSE: To describe a method for back-solving the power of an intraocular lens (IOL) in situbased on laser biometry and ray-tracing analysis of the pseudophakic eye.

SETTING: University Eye Clinic, Aarhus Hospital, Aarhus, Denmark.

DESIGN: Evaluation of diagnostic test or technology.

METHODS: This study comprised pseudophakic eyes with an IOL power ranging from2.00 toC36.00 diopters (D). Preoperatively, the corneal radius was measured with conventionalautokeratometry and the axial length (AL) with optical biometry. After surgery, the position of theIOL was recorded using laser interferometry. Based on the postoperative refraction and the biomet-

ric measurements, a ray-tracing analysis was performed back-solving for the power of the IOL insitu. The analysis was performed assuming pupil diameters from 0.0 to 8.0 mm with andwithout correction for the Stiles-Crawford effect.

RESULTS: The study evaluated 767 pseudophakic eyes (583 patients). Assuming a 3.0 mm pupil,the mean prediction error between the labeled and the calculated IOL power (G1 standard deviation[SD]) was 0.26 DG 0.65 (SD) (range 2.4 toC1.8 D). The prediction error showed no bias withIOL power or with AL. The calculated IOL power depended on the assumed pupil size and the Stiles-Crawford effect. However, the latter had a modulatory effect on the prediction error for large pupildiameters (>5.0 mm) only.

CONCLUSION: The optics of the pseudophakic eye can be accurately described using exact raytracing and modern biometric techniques.

Financial Disclosure: Dr. Olsen is a shareholder of IOL Innovations Aps, manufacturer of thePhacoOptics IOL calculation software. Dr. Funding has no financial or proprietary interest in anymaterial or method mentioned.

J Cataract Refract Surg 2012; 38:641–647 Q 2012 ASCRS and ESCRS

Numerous methods and formulas have been de-scribed to determine the intraocular lens (IOL) powerto be implanted from preoperative data of the patient,and these methods have improved significantly overthe years.1–4 However, not every patient ends up

exactly as expected and the question then arises as towhat caused this refractive “surprise.” Often, a mea-surement error is the cause. In this case, the error can usually be identified by repeating all biometric mea-surements and comparing them with the preoperativedata. In some cases, the formula or the IOL constant isto blame, especially when the formula or IOL constanthas not been updated recently. In other cases, thepatient had laser in situ keratomileusis (LASIK)many years ago and does not tell the surgeon.

However, in some situations the reason for therefractive surprise is not obvious and the followingquestion then arises: Do you actually know what

power was implanted? Did your staff (and you) takethe wrong box? And consider this: What if the powerof the IOL actually implanted differed from the powerthat was on the label? The aim of the present study wasto describe a method to measure the actual effective

IOL power in the pseudophakic eye based on modern biometry and ray-tracing analysis.

PATIENTS AND METHODS

This retrospective consecutive series comprised patientsreferred for cataract or refractive IOL surgery who hadprimary IOL implantation from January to December 2010.Cases with preoperative or postoperative astigmatismgreater than 4.0 diopters (D), with keratoconus, or with pre-vious trauma or surgery (ie, post-LASIK or corneal trans-plantation) were excluded from the series.

Before surgery, the corneal radius was measured in 2 me-ridians with an automated keratometer (ARK700, Nidek,

Ltd.) and averaged. The axial length (AL) was measured

Q 2012 ASCRS and ESCRS

Published by Elsevier Inc.

0886-3350/$ - see front matter 641doi:10.1016/j.jcrs.2011.10.035

ARTICLE

http://dx.doi.org/10.1016/j.jcrs.2011.10.035

http://dx.doi.org/10.1016/j.jcrs.2011.10.035

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with partial coherence interferometry (PCI) optical biometry(IOLMaster, software version 3.0 or higher, Carl Zeiss Med-itec AG). The instrument was calibrated at weekly intervals.

The surgical technique was small-incision sutureless clearcorneal or scleral tunnel phacoemulsification. The IOLwas implanted in the bag after a continuous curvilinear cap-sulorhexis was created. The IOL types included for the pres-

ent study were from the Acrysof nonaspheric series(SA60AT, MA60AC, MA60MA, Alcon Laboratories Inc.).The IOL powers ranged from 2.00 to C36.00 D.

Postoperatively, the visual acuity and refraction (per-formed at 6 meters) were recorded 1 to 3 weeks after surgeryas part of the routine follow-up. Only cases with good post-operative visual acuity (O20/40) were included in the study.To correct for distance refraction, 0.17 D was subtracted fromthe spherical equivalent measured at 6 m. At the time therefraction was taken, the actual position of the IOL (the post-operative anterior chamber depth [ACD]) was measuredusing the Lenstar LS900 biometer (Haag-Streit AG). Thisoptical biometer uses laser interferometry not only for themeasurement of the AL but also for the corneal thickness,

the ACD, and the lens thickness (Figure 1).

Eye Model

Ray-tracing analysis of any optical system requires thephysical dimensions and refractive indices of all opticalmedia to be clearly defined. For the pseudophakic eye, thecornea, the IOL, the distances between the interfaces, andthe AL have to be defined.

The refractive power of the cornea is often obtained bykeratometry, by which the anterior corneal radius of curva-ture is measured. However, the dioptric reading given bythe keratometer assumes a thin-lens calculation for cornealpower, as follows:

DZ

nc 1

r

where ncZ 1.3375 and r is the corneal radius in meters. Thiskeratometer value is not the true physiologic power, how-ever.5 The correct optical model of the cornea requires it tobe assigned an anterior surface as well as a posterior surface,each with a certain curvature. In most corneal models, theposterior curvature is assumed to be a fixed ratio of the ante-rior curvature assuming a standard corneal shape. For manyyears, the standard shape, and hence the radius of the poste-rior surface, was assumed to be as proposed by Gullstrand.6

It was not until recently that modern studies using Scheimp-flug imaging and other techniques provided detailed infor-mation not only on the curvatures of both surfaces of the

cornea but also on their asphericity.7–9 Therefore, in the pres-ent study, the posterior surface of the cornea was assumed tobe a fixed ratio of the anterior surface according to the modeldescribed by Dubbelman et al.8 so that

R2 Z0:84R1

where R2 is the radius of the posterior surface of the corneaand R1 is the radius of the anterior surface of the cornea.Also from the work of Dubbelman et al., the asphericity of the corneal surfaces was assumed to depend on the age of the patient according to the following equations:

K a Z0:76þ 0:003 age

K p Z0:76þ 0:325K a 0:0072 age

where K a is the asphericity of the anterior surface of thecornea, K p is the asphericity of the posterior surface of thecornea, and age is the age of the patient in years.

The refractive index of the cornea was assumed to bea constant value of 1.376, and the thickness of the corneawas assumed to be a constant value of 0.5 mm. (Initial exper-iments showed negligible effect when using the individuallymeasured corneal thickness.)

For more than a decade, the use of PCI10 has significantlyimproved the precision by which the AL can be measuredcompared with ultrasound. The first commercially availablePCI technique was the IOLMaster instrument. However,precision is not the same as accuracy. For an accurate inter-pretation of the AL, it should be realized that the outputreading of the IOLMaster is not the true optical path length of the eye. The readings given by the commercial version of the IOLMaster device have been calibratedagainst immer-sion ultrasound according to the following formula11,12:

AxZeiss ZðOPL=1:3549 1:3033Þ=0:9571

where AxZeiss is the output reading of the PCI instrumentand OPL is the optical path lengthmeasured by PCI. Accord-ingly, the optical path length can be found as

OPLZðAxZeiss 0:9571þ 1:3033Þ 1:3549

According to Haigis,12 the group refractive index of thephakic eye is 1.3574. Using this value, however, a small in-consistency between the preoperative and postoperativereadings with the IOLMaster device was previously found

Figure 1. Laser biometric scan of a pseudophakic eye showing theposition of the IOL, the corneal thickness, the IOL thickness, andthe AL. The scan includes 3 individual scans indicated by separatecolors (red, yellow, and blue), showing almost identical readings asindicated by the small SD between them (bottom of figure) (AD Z

anterior chamber depth; AL Z axial length; CCT Z central corneal

thickness; LT Z intraocular lens thickness; OS Z left eye; RT Z

retinal thickness).

Submitted: August 5, 2011.Final revision submitted: October 19, 2011.Accepted: October 20, 2011.

From University Eye Clinic, Aarhus Hospital NBG, Aarhus, Denmark.

Corresponding author: Thomas Olsen, MD, PhD, UniversityEye Clinic, Aarhus Hospital NBG, Aarhus C, Denmark. E-mail: [email protected].

642 RAY-TRACING ANALYSIS OF THE IOL POWER IN SITU

J CATARACT REFRACT SURG - VOL 38, APRIL 2012

mailto:[email protected]




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by our group; for this reason, the index of 1.3616 was sug-gested.13 Using this latter value, an estimate of the geometricdistance was obtained according to the formula

AxtrueZðAxZeiss 0:9571þ 1:3033Þ 1:3549=1:3616

This conversion between the IOLMaster reading and the truegeometric AL was used in the present ray-tracing experi-ments. To study the possible bias, however, the ray-tracingcalculations were also performed using the unmodifiedIOLMaster AL readings.

Ray-Tracing Procedure

Ray tracing is a well-known procedure to analyze theoptical properties of any optical system using the Snell lawas follows:

sin ðiÞ

sin ðuÞZ

n2

n1

where i is the incident angle of ray on the surface, u is therefracted angle, n1 is the refractive index of first medium,and n2 is the refractive index of the second medium. Tostudy the optical properties of the pseudophakic eye,a computer program was written in Pascal (Borland Del-phi, Inprise Corp.) to trace the ray intersections of an incoming beam of rays through the optical media andstudy the distribution of the intersections with the retina.To validate the calculations, model eye data were exportedto the Zemax software (Radiant Zemax LLC) for opticaldesign.

When an optical engineer wants to builda complexopticalsystem, he or she may do so by solving for 1 or more vari-ables, giving the best outcome while keeping the other con-ditions equal. The same principle can be applied to thepseudophakic eye: To solve for the optical properties of the

IOL, we can use the actual refraction, the vertex distance,the corneal curvature, the positionof the IOL, and the opticalpath length (rather, the geometric AL) to solve for the curva-tures of the IOL optic that most effectively describes the totaloptical system of the pseudophakic eye.

To solve for the curvatures of the IOL, something aboutthe shape of the optic must be known. Most IOL manufac-turers will provide a cutting chart stating the diopter specificvalues of refractive index, thickness, and curvatures of thefront surface and back surface of the IOL optic. Usuallysuch a chart will show that the IOL has a certain configura-tion; that is, a 1:1, 1:2, or 1:3 biconvex design. The mannerby which the IOL changes its physical curvatures over thepower range (ie, the shape factor) varies between manufac-turers and between IOL models; however, in some cases,

the posterior curvature of the IOL optic is kept constantover a certain power range while the anterior curvaturechanges to produce the labeled power. In other models, theIOL optic may keep its overall configuration constant overthe entire diopter range. The thickness of the IOL often fol-lows a linear relationship with the power (minus-poweredIOLs excluded).

In the present study, the algorithm solving for the frontand back curvatures of the IOL was constructed from themanufacturer’s stated information on the shape factor of the IOL over the diopter range. The IOL curvatures (andpower-dependent thickness) were found by computer itera-tions while observing the width of the point-spread function (PSF) at the retina until a minimum of the root-mean-squaredeviations from the center was found.

The measured power of the IOL was finally calculatedfrom the curvatures according to the paraxial thick-lensformula

D12 ZD1 T

nD1 D2

where D12 is the total dioptric power of the thick lens, D1 isthe dioptric power of front surface, D2 is the dioptric powerof back surface, T is the thickness of the lens in meters, and nis the refractive index of the material. The dioptric power of each IOL surface was calculated according to the thin-lensformula

DZ

n2 n1

r

where D is the dioptric power of the single surface, r is theradius of curvature in meters, n1 is the refractive index of the first medium, and n2 is the refractive index of thesecond medium.

Pupil and the Stiles-Crawford Effect

Although the eye on first sight might look like a physicalsystem, there could be modifications due to biology. Onesuch modification is the directional sensitivity of the retina,which was discovered by Stiles and Crawford in 193314 asa discrepancy between the objective and the functionalarea of the pupil in terms of luminous effectiveness. Theeffect was found to depend on the point of entry of raysthrough the pupil so that peripheral rays had less stimuluseffect on the retina than central rays. For a completedescription of the sensory representation of the physicalimage, the Stiles-Crawford effect therefore has to be taken into account because of the modulatory effect on the spher-ical aberration, as previously described using ray tracing.15

In the present study, the ray-tracing analysis was per-

formed with and without correction for the Stiles-Crawford effect assuming a pupil diameter of 0.1 mm,1.0 mm, 2.0 mm, 3.0 mm, 4.0 mm, 5.0 mm, 6.0 mm,7.0 mm, and 8.0 mm.

Outcome Measures

The error of the prediction was defined as the differencebetween the calculated power and the implanted (labeled)power of the IOL (labeled value minus predicted value). Sta-tistical analysis was performed using distributional methodsand the statistical service of an Excel spreadsheet (MicrosoftCorp.). Distribution methods were used where appropriate.Linear regression analysis was performed using the methodof least squares. A probability level of 0.05 (2 tailed) wasconsidered statistically significant.

RESULTS

The study included 767 eyes of 583 patients. The mean age of the 234 men and 349 women was 69.7 years(range 20 to 95 years). The mean IOL power wasC20.69 D.

The IOL power calculated by ray tracing waslinearly correlated with the implanted (labeled) power(correlation coefficient r O 0.99, P!.00001) (Figure 2).Assuming a 3.0 mm pupil, the mean difference be-

tween the calculated and the observed IOL power

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was 0.26 DG 0.65 (SD). The range of error was from2.4 toC1.8 D. Table 1 shows the results for other pu-pil diameters.

No significant bias between the estimated IOLpower and the labeled IOL power was found, asshown by a difference plot (Figure 3). Also no bias be-tween the IOL power and the AL was found (Figure 4).However, when the unmodified AL readings pro-vided by the IOLMaster were used directly in theray-tracing calculations, a significant bias was ob-served with the IOL power and the AL (Figure 5).

The estimated IOL power was dependent on the as-sumed pupil size. A small overestimation of the la-beled power was seen for pupil sizes close to zero,while an underestimation was found for larger pupils(Figure 6). The effect of the assumed pupil diameterwas found to be dependent on the Stiles-Crawford ef-fect, which tended to correct the underestimation with larger pupils (O5.0 mm). For a 6.0 mm pupil, the dif-ference plus/minus Stiles-Crawford correction wasapproximately 0.25 D of the estimated IOL power.

DISCUSSION

In the present study, we applied exact ray tracing tothe pseudophakic eye to show the accuracy of back-

calculating the IOL power in situ. Using the presentalgorithms, the error between the calculated and theactual (labeled) IOL power showed no bias with theAL or IOL power over the entire range of powersand the mean offset errors were very small. Assuminga 3.0 mm pupil, the overall prediction error was0.26 G 0.65 D in the IOL plane. If one assumesa mean error of zero (which can be obtained retrospec-tively by correcting for the mean offset error), the stan-dard deviation of 0.65 D is the equivalent of determining the IOL power within G1.00 D andG2.00 D in 88% and 99% of cases, respectively.

Considering the accuracy of the current method, weassume that a large part of the spread is the result of measurement errors. However, other sources of error,such as deviations between the labeled and the actualIOL power, should also be considered. The current In-ternational Organization for Standardization (ISO)16

requires the tolerance limits of the labeled IOL powerto be within G0.30 D,G0.40 D,G0.50 D, andG1.00 Dof the true power for the power ranges 0.00 to 15.00 D,15.00 to 25.00 D, 25.00 to 30.00 D, and more than 30.00 D, respectively. If the tolerances of the IOL typesused in the present study were no better than that, it ispossible the “mislabeling” constitutes a considerablesource of error and that the true accuracy of the currentray-tracing method is better than stated in this paper.At present, however, we do not know the exact toler-ance limits of the IOL type used in the present study.

Pupil size is another source of error when calculat-

ing IOL power using ray tracing. We made no attemptin the present study to correct for individual pupildiameter. Due to higher-order aberrations (HOAs)(see below), the actual pupil size is likely to influencethe final refraction, especially with spherical IOLs. Itis therefore possible that the overall prediction accu-racy might have been improved if the pupil diameterhad been measured and used for ray-tracing analysisin the individual case.

Ray-tracing analysis is very sensitive to the assump-tions of the physical model of the pseudophakic eye. In the present study, the cornea was modeled using the

result of recent studies of the radius and asphericityof the anterior and the posterior surfaces of the normalcornea. This model predicts the curvature of the

Figure 2. Correlation between implanted IOL power (labeled value)and the IOL power calculated by ray tracing assuming a 3.0 mmpupil in 767 cases (correlation coefficient r Z 0.99; P!.00001; slope

of y on x linear regression line Z1.0) (IOL Z intraocular lens).

Table 1. The mean error (calculated minus implanted value) predicting the implanted IOL power by ray tracing assuming different pupildiameters in 767 cases.

Pupil (mm)

0.01 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

Mean error

(D) G SD

+0.21 G 0.65 +0.16 G 0.65 +0.00 G 0.65 0.26 G 0.65 0.64 G 0.66 1.16 G 0.67 1.79 G 0.69 2.54 G 0.71 3.45 G 0.75

Range 2.01, +2.34 2.07, +2.28 2.21, +2.08 2.44, +1.78 2.86, +1.38 3.41, +0.91 4.04, +0.47 4.69, +0.11 5.40, 0,27



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posterior surface to be 84% of the front curvature,which is lower than the old, assumed Gullstrand ratioof 88%. The difference between 84% and 88% radiusratio is the equivalent of approximately 0.30 D in cor-neal power, which is the equivalent of 0.50 D in IOLpower. Hence, if the calculations were performed in the paraxial domain (equivalent to ray tracing with an infinitely small pupil size), we would expecta mean difference of approximately 0.50 D in esti-mated IOL power between the 2 ratio models. How-ever, for a complete description of the corneal optics,HOAs should also be taken into account. One HOAis spherical aberration caused by corneal asphericity;this is a significant factor in normal corneas and espe-cially in diseased corneas or corneas with previous la-ser ablation treatment (ie, laser in situ keratomileusis)for refractive errors. In such eyes, paraxial ray tracingdoes not give a complete description of the corneal

optics, and this is why exact ray tracing should beconsidered the method of choice.

For a ray-tracing analysis of the pseudophakic eye tobe realistic, we need a realistic physical model of theIOL optic. Before the modern evolution of asphericIOLs, the shape of most IOLs was sphericalbiconvex. Due to the spherical aberration, we cannotrely on paraxial ray tracing for a realistic representa-tion of the effective power of these spherical IOLs.The aim of modern aspheric IOLs is to counteract theasphericity of the cornea and thereby enhance themodulation transfer function of the eye; that is, nar-rowing the PSF at the retina. Nonetheless, by ISO def-inition, it is the paraxial power that is stated on thelabel of that IOL. However, because the amount of asphericity differs between different aspheric IOLdesigns, the effective power in situ also differs. Thisis the reason for differences in the A-constant that

Figure 4. Prediction error (difference between the labeled and calcu-lated IOL power) versus the AL (767 cases).

Figure 3. Difference plot (Bland-Altman) showing the differencebetween the labeled and the ray tracing–calculated IOL power ver-sus the mean of the IOL power (767 cases) (IOL Z intraocular lens).

Figure 5. Prediction error (difference between the labeled and calcu-lated IOL power) versus the AL when the uncorrected IOLMaster

readings were used in the ray-tracing analysis (767 cases).

Figure 6. Prediction error (difference between the labeled and calcu-lated IOLpower) versus the assumed pupil diameter with andwith-out correction for the Stiles-Crawford effect. The 2 curves differsignificantly from pupil diameters of 5.0 or more due to the modula-tory effect of the Stiles-Crawford effect on spherical aberration. Barsindicate standard deviations (767 cases).



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can be found between IOLs with or without asphericcorrection, other things being equal. In the presentstudy, we only included IOLs of known sphericaldesign. The reason was that the shape factor of theIOL design could be obtained from the manufacturer,and this is essential for the ray-tracing analysis. It isour experience that information on the physical char-acteristics of the aspheric IOL is more difficult toobtain because it is generally considered proprietaryinformation. In the future, however, we expect thisinformation to be more readily obtainable as it willincrease our ability to accurately predict the opticaloutcome of an IOL implant of any type.

The exact optical path length of the eye is anotherimportant variable that is subject to interpretation.As shown in the present study, the uncorrected ALreadings of the IOLMaster device could not be used di-rectly in the ray-tracing analysis to give a satisfactory

unbiased estimation of the IOL power in situ. A signif-icant bias with IOL power as well as with AL wasobserved, indicating that the standard reading of theIOLMaster device is not an ideal representation of the optical path length. The reason for this effectappears to be 2-fold. First, the output readings of theIOLMaster device have been calibrated against(immersion) ultrasound readings. Therefore, to obtain the optical distances, one has to retransform the (ultra-sound-like) readings to their original optical path lengths. Second, to transform the measured opticalpath length into the geometric length, it is necessary

to know the group refractive index valid for the given eye. We previously showed that there is a small incon-sistency between the preoperative and the postopera-tive geometric AL when these distances are based on the group refractive indices originally recommendedby Haigis.12 To correct for this small discrepancy, wesuggested the use of a somewhat higher index forthe crystalline lens, givin g a higher group refractive in-dex for the phakic eye.13 The results in the presentstudy showing a high accuracy in predicting the in situ IOL power with minimal offset error and no biasare supportive of the current eye model. However,

we are aware that other models for the interpretation of the AL readings, such as segmental calculation of the overall optical path based on the individual refrac-tive index of the cornea, the aqueous, the lens, and thevitreous, could also be considered. Future studies areneeded to show whether the current eye model can be improved.

Few studies have dealt with the measurement of IOL power within the eye.17 Several years ago, oneof the authors (T.O.) described a method to measurethe power of an IOL in situ by usin g the Purkinje-Sanson images from the IOL surfaces.18 This method

requires a special setup to detect the size of the images

produced by the convex–concave mirror constitutedby the IOL surface and may not be easy to performin daily clinical practice. The present method usesmodern optical biometry to capture the locations of the IOL position in the eye using laser interferometry,which is a very precise method of performing thismeasurement. This, together with the known accuracyof laser biometry for the AL measurement, makes thebasis for accurate analysis of the optical system consti-tuted by the pseudophakic eye.

In conclusion, in the present study we have shown how the optics of the pseudophakic eye can be accu-rately described by exact ray-tracing analysis of thephysical information provided by modern biometrictechniques. With the present method, it was possibleto achieve a very accurate estimation of the IOL powerover a large power range with no bias with IOL poweror AL. These results are encouraging for exact physical

methods to be applied to the field of IOL powercalculation.

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First author:Thomas Olsen, MD, PhD

University Eye Clinic, Aarhus HospitalNBG, Aarhus, Denmark



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