Iol Calculation

Acta Ophthalmologica Scandinavica 2007

Review Article

Calculation of intraocular lens power: a review Thomas Olsen University Eye Clinic, Aarhus Hospital, Aarhus, Denmark

ABSTRACT. This review describes the principles and practices involved in the calculation of intraocular lens (IOL) power. The theories behind formulas for calculating IOL power are described, using regression and optical methods employing ‘thin lens’ and ‘thick lens’ models, as well as exact ray-tracing methods. Numerical examples are included to illustrate the points made. The paper emphasizes the importance of establishing an accurate estimation of corneal power as well as an accurate technique for the measurement of axial length and accurate methods of predicting postoperative anterior chamber depth (ACD). It is concluded that current improvements in diagnostic and surgical technology, combined with the latest generation IOL power formulas, make the calculation and selection of appropriate IOL power among the most effective tools in refractive surgery today.

comparable results (Haigis 2001; Vogel et al. 2001; Kiss et al. 2002; Packer et al. 2002; Findl et al. 2003). Finally, the development of aspheric (Holladay et al. 2002; Packer et al. 2004) and multifocal IOLs of various designs (Olsen & Corydon 1990; Javitt et al. 2000; Bellucci 2005; Dick 2005; Chiam et al. 2006) brings within reach the ultimate goal of spectacle freedom after lens surgery, provided we have accurate methods of controlling the optics of the pseudophakic eye.

Key words: accuracy – biometry – cataract – IOL power calculation – lens surgery – optics – prediction error – ray tracing – refraction

History

Acta Ophthalmol. Scand. 2007: 85: 472–485 ª 2007 The Author Journal compilation ª 2007 Acta Ophthalmol Scand

doi: 10.1111/j.1600-0420.2007.00879.x

Introduction It is often said that cataract surgery is refractive surgery, even when no intraocular lens (IOL) is implanted. However, whereas in the old days the cataract was removed first and the spectacle prescription given last, the situation today is reversed: we prescribe an IOL to obtain a certain refractive effect and this may represent the indication for lens surgery. The difference between the past and the present lies in the development of modern diagnostic and surgical techniques that control refractive outcome with ever-increasing accuracy. There are several reasons why methods of calculating IOL power have come into focus in recent years.

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Firstly, the use of small, sutureless incisions has greatly reduced surgically induced astigmatism (Kershner 1991; Olsen et al. 1996; Olson & Crandall 1998), making the spherical component of the refraction critical to spectacle dependency after surgery. Secondly, the introduction of optical biometry by partial coherence interferometry (PCI) (Drexler et al. 1998) for the measurement of axial length as performed with the Zeiss IOLMasterª (Carl Zeiss Meditec, Jena, Germany) has introduced new standards for the measurement of axial length. Not only is this optical biometry highly reproducible and therefore potentially more accurate, but, as it is observer-independent, it allows surgeons in different parts of the world to obtain

The optics of the eye represents one of the oldest fields in ophthalmology; readers of this journal will be well aware of important contributions made a century ago by Scandinavians such as Alvar Gullstrand (Nobel Laureate, Sweden) (Gullstrand 1909) and Marius Tscherning (Denmark) (Norn & Jensen 2004). The original scientific works by these giants are still highly recommended reading for anyone who wants to understand the principles of ocular optics. The history of IOL power calculation began in 1949 when Harold Ridley implanted the first IOL in a blind eye. The surgery was reported to be successful (the patient still could not see) but the refractive error was found to be ) 20 D! The error was soon identified by Ridley as involving the optical design of the lens. Ridley had tried to copy the curvatures of the natural lens as described by Gullstrand, but failed to recognize the effect of the higher index of refraction of the IOL material (Perspex) (Apple 2006).

Acta Ophthalmologica Scandinavica 2007

My own experience with IOL optics began as a resident in the early 1980s when implantation of standard-power IOLs was still popular. When our senior professor was asked why no effort was made to select an individual power for the patient, he would answer, ‘In my department we restore the patient’s basic refraction,’ claiming that Gullstrand had found the natural lens to have a constant power of 19 D! Although it is true that ametropia is strongly correlated with the length of the eye, it has been known for some time that biological lens power has a significant statistical distribution of its own (Sorsby 1956; Stenstro¨m 1946; Olsen et al. 2007). Furthermore, clinical studies have shown that a fixed IOL power would leave 5% of patients with refractive errors that differed from their basic refraction by > 5 D (Olsen 1988a). Not only would such patients be highly dependent on spectacles after surgery, but, clearly, anisometropia of this magnitude might cause significant aniseikonia and have profound influence on binocular vision. Although much of the following will deal with methods of controlling the dioptric outcome of lens surgery, we must not forget that a complete optical description involves not only refraction in terms of the sphere and cylinder of the spectacles, but also the magnification of the eye)spectacle system, possible aberrations, depth of focus, the question of accommodation, contrast sensitivity, pupil dependency, colour perception and other optical properties of the pseudophakic eye.

Some basic optical formulas Assuming paraxial imagery, the refractive effect of any spherical surface can be calculated as described in the following formula (Bennett & Rabbetts 2006): n2  n1 F¼ ð1Þ r where F ¼ refractive power of surface in dioptres (D), n1 ¼ index of refraction of the material before the surface, n2 ¼ index of refraction of the material after the surface, and r ¼ radius of

curvature in metres. There is a signage convention dictating that anterior convex surfaces are given a plus sign and posterior convex surfaces a minus sign. Vergence is another important concept. It is described as the reciprocal of the ‘reduced’ distance to the focal point, defined as: n V¼ ð2Þ d where V ¼ vergence of paraxial rays in dioptres, d ¼ distance in metres from vergence plane to focal point and n ¼ refractive index of the material. When a refractive system (i.e. a lens) of power F is added to a bundle of rays of vergence V1, the vergence V2 of the rays leaving the lens can be calculated by addition: V2 ¼ V1 þ F

ð3Þ

‘Thin lens’ IOL power calculation formula

Assuming the corneal power (K), the axial length (Ax), the effective lens plane (d) and the refractive index (n) of the eye are known, what power of IOL is needed for emmetropia? Answer

When incoming (parallel) rays leave the cornea, the focal distance is given by 1 ⁄ K. When the ray bundle enters the effective lens plane, the focal distance is reduced by the distance d ⁄ n1 where d is the effective anterior chamber depth (ACD) in metres and n1 is the refractive index in the anterior segment. The vergence V1 at the front surface of the lens plane can therefore be calculated as the reciprocal of the new focal distance according to equation 2: V1 ¼

1 ðK1  nd1 Þ

ð4Þ

In order to be focused on the retina, rays leaving the lens plane must have a vergence V2 defined by the distance from the lens plane to the retina, that is: n2 V2 ¼ ð5Þ ðAx  dÞ where Ax is the axial length of the eye in metres, d is the effective ACD in metres and n2 is the refractive index in the posterior segment. According to equation 3, the power P0 of the IOL can now be found by

subtracting the two vergences V2 and V1 to give: P0 ¼

n2 1  1 d ðAx  dÞ ðK  n Þ

ð6Þ

1

Equation 6 is a thin lens formula, identical in format to the early so-called ‘theoretical’ IOL power calculation formulas (Colenbrander 1973; Fyodorov et al. 1975; Binkhorst 1975, 1979). Although the format looks rather simple, it involves several unknowns that should be dealt with if the formula is to be applied in clinical practice. Some of these unknowns include the refractive index, how to accurately calculate the corneal power, how to predict the effective lens plane, the correction of principal planes of a ‘thick lens’ model, the accuracy of axial length measurements, and the significance of higher-order aberrations, etc. In general, optical formulas used for power calculation can be ranked into orders of increasing complexity: (1) thins lens formulas using simplified thin lens models for the cornea and the lens; (2) thick lens formulas that regard the cornea and lens as having finite thicknesses with separate curvatures on their surfaces (paraxial ray tracing), and (3) exact ray tracing (wavefront techniques), including higher orders of aberrations of the cornea and lens.

The statistical (regression) approach In the first years of IOL power calculation, the accuracy of early theoretical formulas was unconvincing; better results were reported with a statistical regression approach, first represented by the Sanders)Retzlaff)Kraff (SRK I, SRK II) formulas (Sanders & Kraff 1980; Sanders et al. 1988). (Note that the latest version, the SRK ⁄ T formula [Sanders et al. 1990] is not a regression formula but, rather, a modified Binkhorst formula with modified ACD-prediction algorithms.) The advantage of any empirical approach is that the formula is based on actual measurements, which, to some extent, eliminates the need to make assumptions on, for example,

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Acta Ophthalmologica Scandinavica 2007

how to calculate corneal power, how to adjust for principal planes, how to correct axial length for retinal thickness, and how to make any clinical measurements work in the physical sense. The working principle of a regression formula is that it generates a mean value and incorporates a correction (through regression coefficients) to deviations from mean values. Properly derived, the arithmetical mean errors of a regression equation should sum to zero in a representative patient sample. The original SRK I formula consisted of a simple linear regression equation (Sanders & Kraff 1980): P0 ¼ A  0:9  K  2:5  Ax

ð7Þ

where P0 ¼ power of implant for emmetropia, K ¼ dioptric keratometry reading (using index 1.3375), Ax ¼ axial length of the eye as measured by ultrasound and A ¼ the A-constant according to the type of IOL and the mean values of the K-readings and axial length readings. The disadvantage of any empirical approach is that the formula in principle only works for the dataset from which it is derived. For example, if the axial length is measured by a different technique in another clinical setting, the A-constant (and maybe the regression coefficients) will change accordingly. This would be true when changing biometric technique from ultrasound to optical coherence interferometry (PCI) (Zeiss IOLMasterª), which tends to produce longer readings than ultrasound. However, the formula might also be sensitive to differences in surgical technique, such as whether the IOL is placed inside or outside the capsular bag, a difference that alters the average position and refractive effect of the IOL. Thus, in order to overcome problems with differences in measuring or surgical technique, it is recommended that the formula is personalized and that the A-constant in a representative number of cases is backsolved in order to make it accurate in the average case (see below).

A numerical example

To investigate the accuracy of a statistical regression approach on a modern dataset, the records of 1000 recent

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cases were extracted from electronic case records at the University Eye Clinic, Aarhus, Denmark. These referred to patients who fulfilled the following criteria:

Table 1 shows clinical data for these subjects. The IOL power that would have produced emmetropia was calculated from:

(1) they were consecutive patients aged 40–100 years, who had been admitted for senile cataract; (2) they had not previously undergone anterior or posterior segment surgery; (3) they had undergone preoperative keratometry performed with the same autokeratometer (Nidek ARK 700; Nidek Ltd, Gamagori, Japan), the results of which showed no astigmatism > 4 D; (4) their axial lengths had been measured with the Zeiss IOLMasterª; (5) the same type of IOL implant (Alcon Acrysof SA60AT; Alcon Laboratories, Fort Worth, TX, USA) had been used in all of them; (6) the IOL had been placed inthe-bag, and (7) final manifest refraction was recorded at least 2 weeks after surgery with a visual acuity ‡ 20 ⁄ 40.

P0 ¼ Pi þ 1:5  Rx

ð8Þ

where Pi ¼ actual power of implant and Rx ¼ actual postoperative refraction. A multiple regression analysis using the method of least square gave the following regression equation (r ¼ 0.96, p < 0.0001): P0 ¼ 151:3  1:2  K  3:3  Ax ð9Þ Again, P0 ¼ power for emmetropia, K ¼ K-reading in dioptres (using common keratometer index 1.3375) and Ax ¼ axial length using optical biometry (Zeiss IOLMasterª). Note that the present regression equation (equation 9) is quite different from the old SRK formula (equation 7) derived over 20 years earlier. When this newly derived regression equation (equation 9) was used in retrospect to ‘predict’ the observed actual refraction, the mean numerical error was observed to be

Table 1. Clinical data for 1000 consecutive cataract surgeries with recorded final refraction. Axial length was measured with the Zeiss IOLMasterª. The K-reading was calculated from the corneal radius using an assumed index of 1.3375. Axial length (mm)

Corneal radius (mm)

K-reading (D)

IOL power (D)

Postop Rx (D)

Mean (± SD) 23.30 (± 1.14) 7.74 (± 0.27) 43.66 (± 1.54) 22.33 (± 3.45) ) 0.56 (± 0.73) Range 20.56–30.41 6.88–8.73 38.66–49.06 7.00–33.00 ) 4.00 to + 1.75 IOL ¼ intraocular lens; D ¼ dioptre; Postop Rx ¼ recorded final refraction; SD ¼ standard deviation.

Fig. 1. The correlation between observed and predicted refraction in 1000 consecutive cases using a regression formula derived from the same dataset (P0 ¼ 151.3–1.2*K ) 3.3*Ax, where P0 ¼ intraocular lens power for emmetropia (D), K ¼ the keratometry reading of corneal power (D) and Ax ¼ axial length as measured by optical biometry (in mm). Obs Rx ¼ observed refraction error; Predicted Rx ¼ predicted refraction error.

Acta Ophthalmologica Scandinavica 2007

0.00 ± 0.64 D (± standard deviation [SD]), as plotted in Fig. 1. The corresponding mean absolute error was 0.49 ± 0.42 D (± SD). For comparison, the mean numerical prediction error using the latest generation IOL power calculation formula (Olsen 2007) on the same dataset was found to be 0.00 ± 0.58 D (± SD) with a mean absolute error of 0.47 D. Using the original SRK I approach (equation 7) and an optimized A-constant of 119.05 for the same dataset, the mean numerical error was found to be 0.00 ± 0.87 D (± SD) with a mean absolute error of 0.66 D (Table 2, Fig. 2). (Note that the numerical mean error was zero in all instances due to the optimization of the predictions.)

Measurement of corneal power Corneal power accounts for about two-thirds of the total dioptric power of the eye and is an important component of the ocular refractive system. If the calculation of corneal power is inaccurate, it will induce error propagation and have profound consequences on the remaining steps in the calculation of IOL power. Unfortunately, calculating corneal power is not a straightforward process. No keratometer measures corneal power directly. What is actually measured is the size of the image reflected from the convex mirror constituted by the tear film of the corneal surface.

Table 2. The accuracy of intraocular lens power calculation in 1000 consecutive cases using three different IOL power calculation formulas: a modern thick lens optical formula according to Olsen; a newly derived regression formula, and the old SRK I formula using optimized IOL constants. Values are expressed as observed refraction minus expected refraction in dioptres.

Numerical error (mean ± SD) Absolute error (mean ± SD)  Range (minimum–maximum)

Olsen optical

Olsen regression

SRK I regression

0.00 ± 0.60* 0.47à ± 0.39 ) 2.29 to + 2.30

0.00 ± 0.64 0.49 ± 0.42 ) 2.29 to + 2.30

0.00 ± 0.87 0.66 ± 0.56 ) 2.29 to + 2.30

* Significantly different from column 2 (p < 0.05) and column 3 (p < 0.001) by F-test.   The SD is not statistically meaningful here because the absolute error is not normally distributed. à Significantly different from column 2 (p < 0.01) and column 3 (p < 0.001) by Wilcoxon non-parametric paired comparison. IOL ¼ intraocular lens; SD ¼ standard deviation.

Fig. 2. The distribution of the prediction error in 1000 consecutive cases using three different intraocular lens (IOL) power calculation formulas: a modern thick lens optical formula according to Olsen; a newly derived regression formula, and the old SRK I formula using optimized IOL constants.

A magnification is calculated from this image size, which is directly related to the radius of curvature of the reflecting corneal surface. To do this, the cornea is normally assumed to be represented by a spherocylinder. Also to be considered is the fact that the reading is not taken from the most important central area, but usually from a 3-mm diameter midperipheral zone area, depending on the instrument. For the sake of simplicity, most keratometers regard the cornea as a ‘thin lens’ with a single refractive surface, the dioptric power (D) of which can be calculated according to equation 10: n1 D¼ ð10Þ r where r ¼ radius of the front surface in metres. Assuming n ¼ 1.3375, this equation becomes: 337:5 D¼ ð11Þ r where r ¼ front radius, now in mm. This index has been used as the common calibration setting of most keratometers (mostly in the USA and not always in Europe). Thus for a cornea with a 7.5-mm radius, the K-reading would be 45.00 D. (To check the calibration, observe the keratometer reading corresponding to a 10.0-mm radius of curvature. If it reads 33.75 D, the calibration index is 1.3375.) However, we have known for some time, that the thin lens model of the cornea does not give a physiological estimate of power, which is desirable in IOL power calculation (Olsen 1986a). The cornea has two refracting surfaces and, in order to calculate the total corneal power, it is necessary to know the curvature of not only the front but also the back of the cornea. Because the latter is not easily measured by current clinical methods (see below), most methods have assumed posterior curvature to represent a fixed ratio of the front surface. If, for example, we assume this ratio to be the same as for the Gullstrand exact schematic eye (i.e. 6.8 : 7.7), it is possible to calculate the power at each surface and calculate the total power according to the conventional thick lens formula: T D12 ¼ D1 þ D2   D1  D2 ð12Þ n

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Acta Ophthalmologica Scandinavica 2007

where D12 ¼ total dioptric power of the thick lens, D1 ¼ dioptric power of the front surface, D2 ¼ dioptric power of the back surface, T ¼ thickness of the lens (in metres) and n ¼ refractive index. Hence, for a ‘standard’ cornea with a 7.7-mm front surface and 0.5-mm thickness, the calculation is straightforward (assuming the refractive indices of air, cornea and aqueous to be 1.0, 1.376 and 1.336, respectively): D1 ¼

D2 ¼

ð1:376  1Þ  1000 ¼ 48:83 D 7:7 ð13Þ

ð1:336  1:376Þ  1000 ¼ 5:88 D 7:7  ð6:8=7:7Þ ð14Þ

and hence 0:5 1:376  1000  48:83  ð5:88Þ ¼ 43:05 D ð15Þ

D12 ¼ 48:83  5:88 

Note that the back surface of the cornea has a negative power of about ) 6 D. Also note that the total power is about 0.8 D lower than the value obtained with the common keratometer index calibration of 1.3375. If we apply a refractive index to the front surface that would produce the same result as the thick lens calculation, the index can be calculated by reversing equation 10 as: n¼

43:05  7:7 þ 1 ¼ 1:3315 1000

ð16Þ

This value was used by Olsen (1987a, 1987b, 1988b) and later by Haigis (2004). For several years it has been the lowest value used for the fictitious refractive index of the cornea among current IOL power calculation formulas. There is recent evidence, however, that the old Gullstrand ratio of 6.8 : 7.7 (¼ 0.8831) for the ratio between the back and front curvatures of the central cornea may be too high. As Dunne et al. (1992) show, using keratometry readings derived from Purkinje I + II images, a better value may be 0.823. Recent work by Dubbelman et al. (2002, 2006) using Scheimpflug photography show the Gullstrand ratio to range from 0.82 to

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0.84. These new results may call for a modification of the corneal power model. However, before we simply replace the old Gullstrand ratio with the newer values, it may be necessary to consider the aberrations of the cornea, firstly, the spherical aberration.

(1) corneal anterior asphericity (ka) as a function of age: ka ¼ 0:76 þ 0:003  age and (2) corneal posterior (kp) as a function of age:

asphericity

kp ¼ 0:76 þ 0:325  ka  0:0072  age Spherical aberration of the cornea

The effect of a (positive) spherical aberration is to increase the effective power of the cornea. For example, if we assume the old Gullstrand ratio of 6.8 : 7.7 represents the back surface : front surface of the cornea, and subject this spherical model to an exact ray tracing technique (wavefront analysis), the Gullstrand cornea will show a spherical aberration of almost 0.5 D for a 4-mm pupil (Fig. 3). The biological cornea also shows spherical aberration. However, as the cornea flattens somewhat towards the periphery the shape is more like a prolate and the amount of asphericity therefore has to be quantified. The advent of various topography methods has resulted in a considerable amount of data in the literature on the front corneal surface but this is insufficient because the posterior surface may also contribute significantly to total optical power. Recent studies, particularly by Dubbelman et al. (2002, 2006), using Scheimpflug photography, have provided data for normal values of the front and back surfaces of the cornea and their dependency on age. Dubbelman et al. (2002, 2006) used regression analysis to derive the following formulas to express:

Using this model, the conic coefficients of the front and back surfaces of the cornea can be estimated to be ) 0.06 and ) 0.37 on average in a 60year-old subject. Using exact ray tracing, the total effective power of the cornea can thus be calculated as a function of pupil size, as shown in Fig. 4. Note that effective corneal power increases with pupil size as a result of the spherical aberration. To illustrate and compare different cornea models, corneal power was calculated using: (1) the keratometry reading (refractive index of 1.3375); (2) paraxial ray tracing according to the Gullstrand schematic eye, and (3) exact ray tracing on the Dubbelman aspheric cornea, assuming a 4-mm pupil and a 60-year-old subject. Figure 5 shows the results. As expected, the results show that the standard keratometer reads the corneal power about 0.75 D higher than the Gullstrand value. The surprising result is, however, that the effective power of the Dubbelman aspheric cornea is very close to that of the paraxial Gullstrand spheric model. For a 7.8-mm cornea, the Dubbelman value is 0.13 D higher than the Gullstrand value! This result can be attributed to

Fig. 3. Spherical aberration of the Gullstrand cornea (front radius 7.7 mm) expressed as the difference between the effective and paraxial power as a function of pupil size.

Acta Ophthalmologica Scandinavica 2007

Fig. 4. Dubbelman corneal power as a function of front radius and pupil size.

Fig. 5. Corneal power as a function of front central radius for three different corneal models: common keratometry calibration using index 1.3375; Gullstrand paraxial model assuming the ratio between the back and front curvature is 6.8 : 7.7, and exact ray tracing on the Dubbelman aspheric corneal model, assuming the ratio between the back and front curvature is 6.53 : 7.79 and a 4-mm pupil.

the spherical aberration of the cornea, which increases the effective power of the cornea over the paraxial value. Corneal power after refractive surgery

A growing population of patients who have undergone corneal refractive surgery presents a considerable problem for the calculation of IOL power. This is due to changes in normal corneal anatomy, which makes the calculation of corneal power very difficult. For example, if a cornea has undergone radial keratotomy (RK), its topographic profile will have been altered by the procedure, with flattening in the central region and steepening in the periphery. These topographic changes will also be seen in a postphotorefractive keratotomy (PRK) or post-laser in situ keratomileusis (LASIK) cornea, but, by contrast with the post-RK case, a post-PRK or post-LASIK case will also show

changes in the corneal optic configuration relating to the anterior and posterior curvatures of the cornea. Difficulties in the calculation of corneal power can therefore be divided into: (1) topographical problems common to all post-refractive cases, and (2) problems relating to the corneal model, which is abnormal in the case of a post-PRK or post-LASIK cornea. Although the conventional keratometer measures the central 3-mm zone of the cornea, assuming the cornea to be a spherocylinder, this assumption is certainly not appropriate when the corneal surface has been changed as a result of refractive surgery. After myopic keratorefractive surgery, the flattest area of the cornea is the central area, which is not covered by the keratometer. A better reading may be obtained by corneal topography, which usually gives more central

readings. However, even the best topographer may not give reliable readings in the very central area; most topographers (and all keratometers) tend to give readings that are too steep for the effective central zone of the cornea. This may result in a hyperopic error after IOL implantation. Several methods have been proposed to overcome this problem. These comprise the clinical history method, refraction-derived correction according to Shammas et al. (2003), the correcting factor according to Rosa et al. (2002), the variable refractive index according to Ferrara et al. (2004), a formula and nomograms according to Feiz et al. (2001, 2005), the corneal thick-lens formula according to Speicher (2001) and the regression formulas developed by Latkany et al. (2005). The different formulas have been reviewed by Savini et al. (2006). The clinical history method, generally considered to represent the gold standard, is based on the logical assumption that a surgically induced change in refraction should be explained by the change in effective corneal power (assuming that no lenticular myopia has developed). To calculate post-surgical corneal power, the observed increase in refraction (vertexed to the corneal plane) is subtracted from the preoperative corneal power. This method therefore requires preoperative K-readings. Another method is the hard contact lens technique (Holladay 1989; cf. Haigis 2003), which determines the difference between refraction with and without a plano hard contact lens of zero power and subtracts this difference from the base curve of the contact lens. However, although it is theoretically sound, the variability of this method is difficult to control in clinical practice (Joslin et al. 2005; Savini et al. 2006). According to the refraction-derived method of Shammas et al. (2003), the keratometric value (using the 1.3375 index) can be calculated as Kc ¼ Kpost () 0.23*CR), where Kc ¼ corrected keratometry, Kpost ¼ postoperative keratometry and CR ¼ amount of myopia corrected at the corneal plane. A variant of this formula is the clinically derived correction Kc ¼ 1.14*Kpost ) 6.8, which does not take preoperative refraction into account.

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The method described by Rosa et al. (2002) utilizes axial length in a regression formula to account for the induced refractive change. This formula depends on the association of axial length with myopia, which is known to represent the strongest correlation in the normal population. According to Rosa et al. (2002), the corneal radius as measured by topography should be corrected by a factor varying between 1.01 and 1.22 according to the axial length of the eye. The corneal power is then obtained using the formula (1.3375–1) ⁄ rc, where rc ¼ the corrected corneal radius. Other formulas use a variable refractive index (Ferrara et al. 2004), according to which the corrected refractive index of the cornea can be calculated as n ¼ ) 0.0006* (Ax*Ax) + 0.0213*Ax + 1.1573, where Ax ¼ axial length in mm and n ¼ corrected refractive index of the cornea, that assuming corneal power ¼ (n ) 1) ⁄ r, where r ¼ the measured central corneal curvature in metres. This formula also assumes that emmetropia is the result of the refractive procedure and is based on axial length being a strong predictor of the preoperative ametropia. Many of these methods use the preoperative status of the patient to calculate the changes induced in corneal anatomy. To help in evaluating the post-LASIK patient for lens surgery, it would be desirable if all refractive surgeons kept records of preoperative keratometry and refraction values and gave this information to the patient, as is the case with implant surgery. In addition, it would be helpful if axial length was measured at the time of refractive surgery in order to compensate for the possible development of lenticular myopia, which might hamper the calculation of induced corneal change using the history method.

Measurement of axial length Measurement of axial length remains one of the most crucial steps in IOL power calculation. As a 0.1-mm error in axial length is equivalent to an error of about 0.27 D in the spectacle plane (assuming normal eye dimensions), accuracy within 0.1 mm is necessary. For comparison, other

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Table 3. Deviation from the mean values of different variables and corresponding refraction errors. Variable

Error

Rx error

Corneal radius Axial length Postoperative ACD IOL power

1.0 1.0 1.0 1.0

5.7 D 2.7 D 1.5 D 0.67 D

mm mm mm D

Rx error ¼ refraction error; ACD ¼ anterior chamber depth; IOL ¼ intraocular lens.

measurement errors and their influence on refractive error are shown in Table 3. The conversion from IOL power error to error in the spectacle plane is about 1.5 (cf. equation 8). For many years ultrasound was the only technique by which the length of the eye could be measured in clinical practice. What is really measured by ultrasound is the transit time taken by the ultrasonic beam to travel through the ocular media while it is deflected from the internal structures of the eye. The best signal is obtained when the ultrasonic beam strikes a surface at normal incidence that gives rise to a steep spike on the echogram. With good alignment along the ocular axis, it is possible to detect a corneal signal (sometimes a double-spike), the front and back surfaces of the lens and the retina at the same time. The ‘retinal’ spike is generally assumed to arise at the internal limiting membrane of the retina. This may call for correction to account for retinal thickness when the readings are to be used in an IOL power formula. It is important to know the velocity of ultrasound in order to calculate the distances in question. For the normal phakic eye, velocity is generally assumed to be 1532 m ⁄ second for the anterior chamber and the vitreous and 1641 m ⁄ second for the lens (Jansson & Kock 1962). In an average eye, this is equivalent to 1550 m ⁄ second for the whole eye. However, if we assume a constant lens thickness, this average velocity is lower in a long eye and higher in a short eye, and should be corrected to obtain an unbiased prediction in these unusual eyes (Olsen et al. 1991). The pitfalls of ultrasound measurements are numerous: readings should be coaxial with the ocular axis. This requires a steep spike from the retina as well as good spikes from the anterior and posterior surfaces of the lens.

Some eyes do not have perfectly parallel structures, however, and readings can be difficult to obtain in eyes with dense cataracts and eyes with posterior staphyloma. Care should be taken not to indent the cornea if contact measurements are used. For this reason immersion readings are generally considered more accurate than contact measurements. The introduction of optical biometry using partial coherence interferometry (Drexler et al. 1998) (commercially available as the Zeiss IOLMasterª) has significantly improved the accuracy with which axial length can be measured. The fact that the retinal pigment epithelium is the end-point of an optical measurement, whereas the interface between the vitreous and the neuroretina is the end-point of an ultrasonic measurement, makes measurements by PCI longer than those taken with ultrasound. However, just as distance measurements taken with ultrasound are dependent on the assumed ultrasound velocity, optical biometry is dependent on the assumed group refractive indices of the phakic eye. The indices used by the Zeiss IOLMasterª were estimated by Haigis (2001) and were partly based on extrapolated data. There is some evidence, however, that index calibration of the phakic eye may need some modification in order to ensure consistency between preoperative and postoperative readings (Olsen & Thorwest 2005) and more studies may be needed to investigate the index calibration of the PCI technique. It should be acknowledged that readings taken with the commercial version of the Zeiss IOLMasterª do not provide a direct measure of the true optical path length of the eye. In order not to change the system of A-constants and other formula constants that have been used for years with ultrasound, readings taken with the commercial version of the Zeiss IOLMasterª were calibrated (Haigis et al. 2000; Haigis 2001) against immersion ultrasound according to the formula: ALðZeissÞ ¼ ðOPL=1:3549  1:3033Þ =0:9571 ð17Þ where AL(Zeiss) ¼ output reading of the Zeiss instrument and OPL ¼ the

Acta Ophthalmologica Scandinavica 2007

optical path length measured by PCI. This formula makes the reading achieved with the commercial version of the Zeiss IOLMasterª equal to that obtained by immersion ultrasound in the average case. This need not be the case with contact ultrasound, however, as ultrasound measurements may be confounded by indentation of the cornea. Equation 17 can be rearranged to give the optical path: OPL ¼ ðALðZeissÞ  0:9571 þ 1:3033Þ  1:3549 ð18Þ Assuming a refractive index of 1.3574 for the phakic eye (Haigis 2001), we can obtain the true axial length according to: ALðTrueÞ ¼ ðALðZeissÞ  0:9571 þ 1:3033Þ  1:3549=1:3574 ð19Þ The above considerations are valid for phakic eyes with normal vitreous compartments. In pseudophakic eyes and ⁄ or silicone-filled eyes the axial length should be deduced with regard to the altered ultrasonic velocity or refractive index of the eye in question. In the case of silicone oil in the posterior segment, the situation is further complicated by the fact that the optical path length of the posterior segment is altered and the refractive effect of the posterior surface of the (posterior convex) IOL may be reduced (see next section). In such cases it might be necessary to use a thick lens formula with a corrected refractive index for the vitreous cavity and corrected calculation of the dioptric power of the posterior surface of the lens. As a rule of thumb, the refractive power of the IOL needs to be increased by an amount that compares with the clinical observation of an average + 6-D shift in the refraction of the silicone-filled phakic eye. Another option would be to use a planoconvex IOL with all the refraction on the front surface.

The refractive effect of the IOL The refractive effect of the IOL depends on the optic configuration

(shape factor) of the IOL, its position within the eye, the power of the implant and the amount of spherical aberration. The optic configuration of the IOL determines the effective lens plane, which represents the principal plane when dealing with paraxial ray tracing. All the dioptric power of a planoconvex lens is on one surface and thus that surface represents the effective lens plane. With a biconvex lens, the effective lens plane is ‘inside’ the lens. For example, an IOL with a 2 : 1 biconvex optic configuration has a radius of curvature on the front surface which is twice the radius of curvature on the back surface. In other words, the power of the back surface is twice the power of the front surface. The optic design and hence the position of the principal plane have significant influence on the refractive effect of the IOL. Table 4 shows the effect of varying the design on the refractive effect of the IOL in an average eye, assuming a constant position of the anterior surface of the IOL. The changes in refraction will translate into an equivalent change in the A-constant of a given lens. For example, if the design of one lens changes from a 1 : 2 configuration to a 2 : 1 configuration, the total refractive effect in the spectacle plane is about 0.26 + 0.18 D, thus ¼ + 0.44 D, which is equivalent to a 0.44*1.5 D ¼ 0.66 D change in IOL power. This value would be the corresponding correction needed for the A-constant. In addition to the shift in principal planes, the amount of spherical aberration of the IOL may also have significant influence on the refractive effect of the IOL. As IOL power according to the American National Standards Institute (ANSI) definition refers to paraxial power, the more aspheric the IOL, the higher the labelled IOL

power to produce the same refractive effect in the spectacle plane.

Prediction of postoperative anterior chamber depth At the time when early theoretical formulas were being developed, very little was known about the actual position of the implant after surgery. For example, the Binkhorst I formula (Binkhorst 1979) used a fixed ACD value to predict the position of the implant in each case. It soon became obvious, however, that the fixed ACD model was inappropriate because it resulted in predictions that were actually worse than empirically derived formulas. Modern progress in IOL power calculation formulas largely reflects advances in methods of predicting the position of the implant after surgery based on preoperative measures. Today, there is strong evidence that postoperative ACD is positively correlated with axial length. The fixedACD model therefore predicted ACDs that were too short in long eyes and too deep in short eyes. As a consequence, a myopic error would be produced in a short eye and a hyperopic error in a long eye. To avoid this effect, the prediction of postoperative ACD should in some way be corrected for axial length. The following simple ACD formula was implemented in the Binkhorst II formula: ACDpost ¼ ACDmean  Ax=23:45 ð20Þ where ACDmean ¼ average ACD (socalled ACD constant) of a given IOL type, and Ax ¼ axial length in mm (Ax < 26 mm). The Binkhorst II ACD method was an example of a method to avoid large bias with the axial length. However, before we go further and use more sophisticated methods, it might

Table 4. Refractive effect of a shift in optic configuration. The refractive effect is expressed as relative to a standard 1 : 1 biconvex 22 D intraocular lens (assuming acrylic material and average eye dimensions and constant anterior chamber depth of the IOL). 22 D IOL

2 : 1 configuration

1 : 1 configuration

1 : 2 configuration

Refractive effect (spectacle plane)

+ 0.26 D

0.00 D

) 0.18 D

IOL ¼ intraocular lens.

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be worthwhile considering the following question: if postoperative ACD were to be predicted by an accurate method, would refractive predictions show no bias according to axial length? This question was studied by the author by measuring the actual postoperative ACD in a large series (about 1000 cases) and substituting the predicted postoperative ACD with the true, postoperative ACD in each case (Olsen & Corydon 1993). The results failed to demonstrate any significant bias according to axial length! A very important conclusion to be drawn from this study was that, apart from a constant used to correct axial length (called ‘retinal thickness’, which may in fact correct for other errors), no fudge factors were necessary to adjust IOL power calculations. All that was needed was an accurate prediction of the physical position of the IOL in each case. This study was encouraging of further research into better means of predicting ACD. Today, estimation of an effective postoperative ACD is based on observations of the statistical association between various preoperative measurements of the eye and effective ACD (Table 5, Fig. 6). Thus, estimation of ACD remains the true empirical content in every IOL calculation formula and different models for doing this cause optical IOL power formulas to differ in accuracy.

Fig. 6. Distances used in the prediction of postoperative anterior chamber depth (ACDpost). Ax ¼ axial length; ACDpre ¼ preoperative ACD; LT ¼ lens thickness; R ¼ front radius of cornea; H ¼ corneal height.

Models for anterior chamber depth

Definitions of ACD vary according to context and this should be acknowledged in any discussion of ACD. The clinical definition of ACD in the normal phakic eye is straightforward, and refers to the distance from the cornea to the anterior surface of the lens. Anatomically, ACD is often reckoned from the posterior surface of the cornea, but in an optical context, such as

Table 5. Variables used by various formulas in the prediction of postoperative anterior chamber depth or effective lens plane. ACD predictor

Formula ⁄ author

Axial length

Binkhorst Hoffer SRK T Holladay Haigis Olsen

Corneal height

Fyodorov SRK T Holladay I + II Hoffer Olsen Haigis Holladay II, Olsen Holladay II, Olsen

Binkhorst 1979 Hoffer 1993 Sanders et al. 1990 Holladay et al. 1988 Haigis 2004 Olsen 1987a, 2006; Olsen & Gimbel 1993; Olsen et al. 1995 Fyodorov et al. 1975 Sanders et al. 1990 Holladay et al. 1988 Hoffer 1993 Olsen 1986b, 2006 Haigis 2004 Olsen 1986b, 2006 Olsen et al. 1995, Olsen 2006

Holladay II Olsen Holladay II, Olsen

Unpublished data Olsen 2006 Olsen 2006

Preoperative ACD Lens thickness Others Age Refraction

ACD ¼ anterior chamber depth.

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when discussing ACD in an IOL power formula, the distance is normally measured from the anterior surface of the cornea and includes the corneal thickness. This is partly justified by the position of the second principal plane of the cornea, which is close to the anterior surface (actually about 0.05 mm in front of the cornea). However, the end-point of the ACD distance is much more complex. Many formulas do not use the anterior surface of the IOL as the reference point, but rather the ‘effective lens plane’ (ELP), defined as the effective distance from the anterior surface of the cornea to the lens plane as if the lens was of infinite thickness (thin lens). The ELP may be back-calculated as the effective ACD ‘predicting’ the actual postoperative refraction on a given dataset. Hence, the ELP is formula-dependent and need not reflect the true ACD in the anatomical sense. This is the case for an ACD defined by the manufacturer on an IOL container along with the A-constant. The ACD in this context is most often based on the Binkhorst formula and cannot be taken to reflect the true physical value of the ACD of the pseudophakic eye. Corneal curvature and corneal height

One of the earliest models for the prediction of postoperative ACD used the

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base of the anterior spherical segment as the reference plane (Fyodorov et al. 1975). This plane can be calculated from the corneal curvature and corneal diameter, the latter by taking an average value or by using the white-towhite distance of the cornea. The Fyodorov formula was intended for irisclip lenses and was adopted by the author for anterior chamber lenses (Olsen 1986b) and later for posterior chamber lenses (Olsen et al. 1990, 1992). The Fyodorov corneal height formula was reintroduced for the calculation of the so-called ‘surgeon’s factor’, defined as the difference between the corneal height and the effective optical plane of the IOL (Holladay et al. 1988) and was adopted at around the same time as the SRK ⁄ T approach (Sanders et al. 1990). However, recent work by the author seems to indicate that there is no significant information in corneal height based on corneal diameter, compared with corneal curvature itself. Other predictors, such as axial length, preoperative ACD and lens thickness, have been found to be significantly more important (Olsen 2006). Preoperative ACD

Today, most newer generation IOL power calculation formulas recognize the importance of factors other than axial length in predicting ACD. Preoperative ACD is one such predictor; it has been used in formulas such as the Haigis formula (Haigis 2004) and

the Olsen formula (Olsen et al. 1992, 1995; Olsen 2006). The importance of preoperative ACD is ranked second to axial length in statistical significance as shown by multiple regression analysis (Olsen 2006).

Lens thickness

If we accept the importance of preoperative ACD to postoperative ACD, it seems logical to assume that preoperative lens thickness also has some influence. This is due to the thickening of the lens with age and the statistical negative correlation between ACD and lens thickness in the normal eye. Despite this logical assumption and the fact that most ultrasound equipment is capable of measuring lens thickness, it is surprising how little lens thickness has been used in ACD prediction algorithms. One exception to this rule is the Olsen formula, which has used this predictor since 1995 (Olsen et al. 1995). More recently, it has also been considered by other authors (Norrby 2004; Norrby et al. 2005). Recent studies on large series have confirmed that lens thickness is important to accurate ACD prediction, especially in combination with preoperative ACD (Olsen 2006). Whichever method is used in the prediction of postoperative ACD, it should be realized that the error in refraction produced by an error in postoperative ACD is strongly

Fig. 7. Intraocular lens prediction error (spectacle plane) as a result of a 0.25-mm error in anterior chamber depth (ACD) prediction versus axial length calculated on an actual large dataset (n ¼ 7418). Note that the error increases five-fold from a long eye of 30 mm (about 0.1 D refractive error) to a short eye of 20 mm (about 0.5 D refractive error). Bars indicate ± 1 standard deviation.

dependent on axial length. This can be clearly seen in Fig. 7, which was constructed from an actual dataset (number of observations: n ¼ 7418) examined to establish the corresponding IOL prediction error (spectacle plane) resulting from a 0.25-mm error in postoperative ACD versus axial length. As Fig. 7 shows, the corresponding prediction error increases five-fold from a 0.1-D error in a 30-mm long eye to a 0.5-D error in 20-mm short eye. Accurate prediction of ACD therefore remains much more important in short eyes compared with long eyes. It should be noted that the means to predict postoperative ACD or ELP to a large extent is based on the statistical relationship between several preoperatively defined measures and the actual position of the implant. This requires the eye to exhibit a normal anatomy. If normality is compromised, as a result of keratorefractive surgery (or the axial length has been altered as a result of a scleral buckling procedure), the statistical model behind the prediction of ACD may no longer be valid and it may be necessary to ‘normalize’ the anatomy. This is the rationale behind the ‘double K-method’ reported by Aramberri (2003). Assuming the total prediction error in IOL power calculation to be the sum of the error associated with the main variables, namely: (1) measurement of axial length; (2) measurement of corneal power, and (3) estimation of postoperative ACD, it is possible to calculate the relative magnitude of each of these errors using ultrasound biometry (Olsen 1992). Assuming current axial length measurement error using optical biometry and the latest generation of ACD prediction algorithm (Olsen 2006), the relative contribution of these three sources of error was estimated (Fig. 8). Although ACD prediction is probably one of the most accurate algorithms currently available, the error contribution from axial length measurements (36%) was found to be less than the error from ACD predictions (42%). This result contrasts with previous error estimation, showing axial length measurements

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Fig. 8. Breakdown of total intraocular lens power prediction error into the three main sources of error: axial length measurement by optical biometry (AX); measurement of corneal power (K), and estimation of postoperative anterior chamber depth (IOL ACD). The graph was constructed assuming average eye data and the current statistical measurement error of each variable. RX ¼ refractive error.

with ultrasound to be responsible for > 50% of total error. The reason for the relatively lower contribution from the axial length source and the higher contribution from the ACD source is the higher accuracy of the optical biometry using Zeiss IOLMasterª. Therefore, the degree of accuracy with which postoperative ACD can be predicted is the major limiting factor in the accuracy of modern IOL power calculation.

Optimization and accuracy Most of what has been said in the foregoing sections has been under the assumption that predictions have been

‘optimized’. Optimization means that the numerical error of predictions counting both negative and positive deviations will average to zero for the entire dataset. The IOL power calculation formula works at its best when this is so, leaving only statistical (measurement) errors to influence the result. If a systematic error arises anywhere in the system, the Gaussian distribution of the predictions would gather around a certain offset value. Optimization aims to correct for this error (Fig. 9). There are different sources of systematic error. Systematic deviations in measurement error translate directly into equivalent errors in IOL power prediction (cf. Table 3) and should be dealt with accordingly. If, for instance, ultrasound measurements are compared with those of PCI, it is likely that the readings will be somewhat longer with optical biometry. Thus, if one surgeon achieves perfect results using a particular SRK A-constant with ultrasound, it is likely that he ⁄ she will need to increase the A-constant by the equivalent value of the shift in axial length measurement from ultrasound to optical biometry. To some surgeons this may seem surprising, as the A-constant is often regarded as an IOL-related constant. The truth is, however, that the A-constant is a ‘black box’ that contains any offset errors that arise in the entire clinical environment, as demonstrated in the numerical example given above. The introduction of optical biometry by coherence interferometry has

Fig. 9. The distribution of prediction error with and without correction for an offset error of 0.5 D by optimization.

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vastly improved conditions for greater standardization in calculating IOL power. For the first time, we now have a highly accurate, observer-independent technique by which axial length can be measured. This allows surgeons around the world to compare results and exchange information regarding the most appropriate A-constant or other IOL constants. It should be remembered, however, that the A-constant by nature depends on the population average. If, for example, the population in one country has on average longer eyes than that of another region, this would call for an adjustment of the A-constant, although the axial length measurements were made with the same equipment. Every IOL has its own refractive effect depending on the optic configuration, the amount of spherical aberration and the effective ACD (see section above). The existence of a ‘personalized’ ACD may have been viable in the old days, when surgeons had different ways of opening the capsule and therefore different ways of anchoring the IOL in the eye. If, for example, one surgeon implanted every IOL within the capsular bag, while another placed the IOL in the sulcus, they would achieve different refractive results. As anterior displacement of the IOL enhances its refractive effect, the sulcus surgeon would need to use IOLs with a power 0.5–1.0 D lower than those used by the in-the-bag surgeon in the average case. However, since the introduction of the continuous curvilinear capsulorhexis technique (Gimbel & Neuhann 1991), most surgeons have developed a very standardized way of implanting the IOL, thus decreasing the variability of postoperative ACD. Theoretically, however, it is still possible that a capsulorhexis with a large diameter may cause the IOL to be more anteriorly located than a small capsulorhexis due to postoperative shrinking of the capsule, which leaves some room for personalized ACDs. The amount of spherical aberration has significant influence on the effective power of the IOL. Due to the positive spherical aberration of a conventional, spherical IOL, the effective power of that IOL is actually higher than its paraxial (labelled) power and higher than that of an aspherical IOL

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with negative spherical aberration. To obtain the same refractive result with an aspherical IOL as with a spherical IOL, the surgeon would need to use a higher labelled power. The amount of ‘extra’ power needed equals the amount of corrected spherical aberration, which may be estimated from the physical constants of the IOL (i.e. the conic coefficients of the surfaces). However, due to the Stiles)Crawford effect, which tends to correct for the spherical aberration of the ocular system (Olsen 1993), the effective power of an IOL may not be 100% deducible from its optical bench power. Therefore, only by clinical studies is it possible to evaluate the true effective IOL power and obtain the necessary IOL constants to optimize and finetune IOL power calculations. In the event of unexpected refraction after surgery, every effort should be made to identify the error. A recommended procedure would be to verify all measurements by remeasuring the corneal curvature and axial length and comparing the results with preoperative measurements. In most cases this will reveal the error to be a measurement error. However, other errors, including labelling errors, should also be considered. The author has obtained good results by measuring actual ACD in the pseudophakic eye and comparing this with the predicted value. The measurement of actual postoperative ACD together with a repeat K-reading and (optical) biometry makes it possible based on the observed refraction to back-calculate IOL power in situ within an accuracy of ± 1 D. This allows for the diagnosis of a case of IOL mislabelling with reasonable accuracy. The mislabelling of IOLs has been a problem in some cases in the past (Olsen & Olesen 1993), but it seems to have been seldom reported in recent literature. The higher accuracy of optical biometry has greatly improved measuring conditions for diagnosing mislabelled IOLs.

Conclusions Although IOL power calculation began as an optical approach using theoretical formulas, the majority of methods used in clinical practice over the past 25 years have been based on

empirical methods that have used ‘fudged’ formulas to compensate for the unknowns in the system. However, the advent of better diagnostic equipment and ever-improving surgical techniques has decreased the number of unknowns, and optical methods now hold sway in IOL power calculation. To conclude the discussion of regression versus a theoretical approach, it would be unfair to say that the regression approach is inaccurate as its accuracy is comparable with that of the theoretical approach in a normal dataset. The theoretical approach does, however, give more accurate predictions, as can be demonstrated in a significant number of observations under optimized conditions. The disadvantages of the regression approach include the need for a large series from which to derive the empirical constants and the limitations defined by the ‘normal’ population and the clinical environment. If the clinical environment changes, for example by using a more accurate device for axial length measurement or by using a more standardized surgical technique, the offset value and coefficients of the regression equation are likely to change. The regression formula will not work in extremes and hence inaccurate predictions are likely to occur in unusually long or short eyes, in steep or flat corneas and in combinations thereof. Although paraxial ray tracing still has an important role to play in the description of IOL optics, wavefront technology and exact ray tracing (Preussner et al. 2002) may in the future represent better ways of describing the optics of the pseudophakic eye, especially in cases of non-spherical postLASIK corneas, but also when dealing with aspheric IOLs. However, if we, as surgeons, are to treat the pseudophakic eye as a truly optical system, we need more information regarding the physical properties of the IOL to be implanted. The time when an A-constant and power to the nearest 0.5 D could be regarded as sufficient information on an implant to be placed in someone’s eye for the rest of his or her life is over. It would be extremely helpful if manufacturers would include other useful information on the IOL label, such as the IOL’s exact power, the exact curva-

tures of the front and back surfaces, thickness, index of refraction, and conic coefficients, etc. In the accurate prediction of the optical properties of the pseudophakic eye, whether the optic configuration changes with power, what the conic coefficients of the aspheric surfaces are, and whether the power varies by ± 0.5 D or ± 0.1 D from the labelled value matter greatly. The average refractive prediction accuracy that can be achieved with modern methodology (using optimized conditions and the latest generation ACD prediction algorithms) is < 0.5 D (absolute error). The standard deviation of the numerical error is < 0.6 D, which means that about 90% of cases fall within ± 1.0 D and 99.9% within ± 2.0 D of their targets, again assuming optimized conditions. Prediction is more accurate in long eyes and less accurate in short eyes. The calculation and selection of appropriate IOL power are among the most significant tools in refractive surgery today.

References Apple DJ (2006): Sir Harold Ridley and his Fight for Sight. Thorofare, NJ: Slack Inc. Aramberri J (2003): Intraocular lens power calculation after corneal refractive surgery: double-K method. J Cataract Refract Surg 29: 2063–2068. Bellucci R (2005): Multifocal intraocular lenses. Curr Opin Ophthalmol 16: 33–37. Bennett A & Rabbetts R (2000): Clinical Visual Optics. Oxford: Butterworth-Heinemann. Binkhorst RD (1975): The optical design of intraocular lens implants. Ophthalmic Surg 6: 17–31. Binkhorst RD (1979): Intraocular lens power calculation. Int Ophthalmol Clin 19: 237– 252. Chiam PJ, Chan JH, Aggarwal RK & Kasaby S (2006): ReSTOR intraocular lens implantation in cataract surgery: quality of vision. J Cataract Refract Surg 32: 1459–1463. Colenbrander MC (1973): Calculation of the power of an iris-clip lens for distant vision. Br J Ophthalmol 57: 735–740. Dick HB (2005): Accommodative intraocular lenses: current status. Curr Opin Ophthalmol 16: 8–26. Drexler W, Findl O, Menapace R, Rainer G, Vass C, Hitzenberger CK & Fercher AF (1998): Partial coherence interferometry: a novel approach to biometry in cataract surgery. Am J Ophthalmol 126: 524–534. Dubbelman M, Sicam VA & van der Heijde GL (2006): The shape of the anterior and

483

Acta Ophthalmologica Scandinavica 2007

posterior surface of the ageing human cornea. Vision Res 46: 993–1001. Dubbelman M, Weeber HA, van der Heijde RG & Volker-Dieben HJ (2002): Radius and asphericity of the posterior corneal surface determined by corrected Scheimpflug photography. Acta Ophthalmol Scand 80: 379–383. Dunne MC, Royston JM & Barnes DA (1992): Normal variations of the posterior corneal surface. Acta Ophthalmol (Copenh) 70: 255–261. Feiz V, Mannis MJ, Garcia-Ferrer F et al. (2001): Intraocular lens power calculation after laser in situ keratomileusis for myopia and hyperopia: a standardized approach. Cornea 20: 792–797. Feiz V, Moshirfar M, Mannis MJ, Reilly CD, Garcia-Ferrer F, Caspar JJ & Lim MC (2005): Nomogram-based intraocular lens power adjustment after myopic photorefractive keratectomy and LASIK: a new approach. Ophthalmology 112: 1381–1387. Ferrara G, Cennamo G, Marotta G & Loffredo E (2004): New formula to calculate corneal power after refractive surgery. J Refract Surg 20: 465–471. Findl O, Kriechbaum K, Sacu S et al. (2003): Influence of operator experience on the performance of ultrasound biometry compared to optical biometry before cataract surgery. J Cataract Refract Surg 29: 1950–1955. Fyodorov SN, Galin MA & Linksz A (1975): Calculation of the optical power of intraocular lenses. Invest Ophthalmol 14: 625– 628. Gimbel HV & Neuhann T (1991): Continuous curvilinear capsulorhexis. J Cataract Refract Surg 17: 110–111. Gullstrand A (1909): Die Dioptrik des Auges. In: Helmholz H (ed). Handbuch der Physiologischen Optik. Hamburg: L Voss 41– 375. Haigis W (2001): Pseudophakic correction factors for optical biometry. Graefes Arch Clin Exp Ophthalmol 239: 589–598. Haigis W (2003): Corneal power after refractive surgery for myopia: contact lens method. J Cataract Refract Surg 29: 1397–1411. Haigis W (2004): The Haigis formula. In: Shammas HJ (ed). Intraocular Lens Power Calculations. Thorofare, NJ: Slack Inc. 5– 57. Haigis W, Lege B, Miller N & Schneider B (2000): Comparison of immersion ultrasound biometry and partial coherence interferometry for intraocular lens calculation according to Haigis. Graefes Arch Clin Exp Ophthalmol 238: 765–773. Hoffer KJ (1993): The Hoffer Q formula: a comparison of theoretic and regression formulas. J Cataract Refract Surg 19: 700–712. Holladay JT (1989): Consultations in refractive surgery. [Letter.] J Cataract Refract Surg 5: 203. Holladay JT, Piers PA, van der Koranyi GMM & Norrby NE (2002): A new intraocular lens design to reduce spherical aberration of pseudophakic eyes. J Refract Surg 18: 683–691.

484

Holladay JT, Prager TC, Chandler TY, Musgrove KH, Lewis JW & Ruiz RS (1988): A three-part system for refining intraocular lens power calculations. J Cataract Refract Surg 14: 17–24. Jansson F & Kock E (1962): Determination of the velocity of ultrasound in the human lens and vitreous. Acta Ophthalmol (Copenh) 40: 420–433. Javitt J, Brauweiler HP, Jacobi KW et al. (2000): Cataract extraction with multifocal intraocular lens implantation: clinical, functional, and quality-of-life outcomes. Multicentre clinical trial in Germany and Austria. J Cataract Refract Surg 26: 1356– 1366. Joslin CE, Koster J & Tu EY (2005): Contact lens overrefraction variability in corneal power estimation after refractive surgery. J Cataract Refract Surg 31: 2287–2292. Kershner RM (1991): Sutureless one-handed intercapsular phacoemulsification. The keyhole technique. J Cataract Refract Surg 17: 719–725. Kiss B, Findl O, Menapace R et al. (2002): Refractive outcome of cataract surgery using partial coherence interferometry and ultrasound biometry: clinical feasibility study of a commercial prototype II. J Cataract Refract Surg 28: 230–234. Latkany RA, Chokshi AR, Speaker MG, Abramson J, Soloway BD & Yu G (2005): Intraocular lens calculations after refractive surgery. J Cataract Refract Surg 31: 562– 570. Norn M & Jensen OA (2004): Marius Tscherning (1854–1939): his life and work in optical physiology. Acta Ophthalmol Scand 82: 501–508. Norrby S (2004): Using the lens haptic plane concept and thick-lens ray tracing to calculate intraocular lens power. J Cataract Refract Surg 30: 1000–1005. Norrby S, Lydahl E, Koranyi G & Taube M (2005): Clinical application of the lens haptic plane concept with transformed axial lengths. J Cataract Refract Surg 31: 1338– 1344. Olsen T (1986a): On the calculation of power from curvature of the cornea. Br J Ophthalmol 70: 152–154. Olsen T (1986b): Prediction of intraocular lens position after cataract extraction. J Cataract Refract Surg 12: 376–379. Olsen T (1987a): Theoretical approach to intraocular lens calculation using Gaussian optics. J Cataract Refract Surg 13: 141– 145. Olsen T (1987b): Theoretical, computer-assisted prediction versus SRK prediction of postoperative refraction after intraocular lens implantation. J Cataract Refract Surg 13: 146–150. Olsen T (1988b): Measuring the power of an in situ intraocular lens with the keratometer. J Cataract Refract Surg 14: 64–67. Olsen T (1988a): Pre- and postoperative refraction after cataract extraction with implantation of standard power IOL. Br J Ophthalmol 72: 231–235.

Olsen T (1992): Sources of error in intraocular lens power calculation. J Cataract Refract Surg 18: 125–129. Olsen T (1993): On the Stiles)Crawford effect and ocular imagery. Acta Ophthalmol (Copenh) 71: 85–88. Olsen T (2006): Prediction of the effective postoperative (intraocular lens) anterior chamber depth. J Cataract Refract Surg 32: 419–424. Olsen T (2007): Improved accuracy of the intraocular lens power calculation with the Zeiss IOLMaster. Acta Ophthalmol (Copenh) 85: 84–87. Olsen T, Arnarsson A, Sasaki H, Sasaki K & Jonasson F (2007): On the ocular refractive components: the Reykjavik Eye Study. Acta Ophthalmol Scand Published OnlineEarly, doi: 10.1111/j.1600-0420.2006.00847.x Olsen T & Corydon L (1990): Contrast sensitivity in patients with a new type of multifocal intraocular lens. J Cataract Refract Surg 16: 42–46. Olsen T & Corydon L (1993): We don’t need fudge factors in IOL power calculation. Eur J Implant Refract Surg 5: 51–54. Olsen T, Corydon L & Gimbel H (1995): Intraocular lens power calculation with an improved anterior chamber depth prediction algorithm. J Cataract Refract Surg 21: 313–319. Olsen T, Dam-Johansen M, Bek T & Hjortdal JO (1996): Evaluating surgically induced astigmatism by Fourier analysis of corneal topography data. J Cataract Refract Surg 22: 318–323. Olsen T & Gimbel H (1993): Phacoemulsification, capsulorhexis and intraocular lens power prediction accuracy. J Cataract Refract Surg 19: 695–699. Olsen T & Olesen H (1993): IOL power mislabelling. Acta Ophthalmol (Copenh) 71: 99–102. Olsen T, Olesen H, Thim K & Corydon L (1990): Prediction of postoperative intraocular lens chamber depth. J Cataract Refract Surg 16: 587–590. Olsen T, Olesen H, Thim K & Corydon L (1992): Prediction of pseudophakic anterior chamber depth with the newer IOL calculation formulas. J Cataract Refract Surg 18: 280–285. Olsen T, Thim K & Corydon L (1991): Accuracy of the newer generation intraocular lens power calculation formulas in long and short eyes. J Cataract Refract Surg 17: 187–193. Olsen T & Thorwest M (2005): Calibration of axial length measurements with the Zeiss IOLMaster. J Cataract Refract Surg 31: 1345–1350. Olson RJ & Crandall AS (1998): Prospective randomized comparison of phacoemulsification cataract surgery with a 3.2-mm versus a 5.5-mm sutureless incision. Am J Ophthalmol 125: 612–620. Packer M, Fine IH, Hoffman RS, Coffman PG & Brown LK (2002): Immersion A-scan compared with partial coherence interferometry: outcomes analysis. J Cataract Refract Surg 28: 239–242.

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Packer M, Fine IH, Hoffman RS & Piers PA (2004): Improved functional vision with a modified prolate intraocular lens. J Cataract Refract Surg 30: 986–992. Preussner PR, Wahl J, Lahdo H, Dick B & Findl O (2002): Ray tracing for intraocular lens calculation. J Cataract Refract Surg 28: 1412–1419. Rosa N, Capasso L & Romano A (2002): A new method of calculating intraocular lens power after photorefractive keratectomy. J Refract Surg 18: 720–724. Sanders DR & Kraff MC (1980): Improvement of intraocular lens power calculation using empirical data. J Am Intraocul Implant Soc 6: 263–267. Sanders DR, Retzlaff J & Kraff MC (1988): Comparison of the SRK II formula and other second-generation formulas. J Cataract Refract Surg 14: 136–141. Sanders DR, Retzlaff JA, Kraff MC, Gimbel HV & Raanan MG (1990): Comparison of the SRK ⁄ T formula and other theoretical

and regression formulas. J Cataract Refract Surg 16: 341–346. Savini G, Barboni P & Zanini M (2006): Intraocular lens power calculation after myopic refractive surgery: theoretical comparison of different methods. Ophthalmology 113: 1271–1282. Shammas HJ, Shammas MC, Garabet A, Kim JH, Shammas A & LaBree L (2003): Correcting the corneal power measurements for intraocular lens power calculations after myopic laser in situ keratomileusis. Am J Ophthalmol 136: 426–432. Sorsby A (1956): Emmetropia and its aberrations. Trans Ophthalmol Soc U K 76: 167– 169. Speicher L (2001): Intraocular lens calculation status after corneal refractive surgery. Curr Opin Ophthalmol 12: 17–29. Stenstro¨m S (1946): Untersuchungen u¨ber die Variation und Kovariation der optischen Elemente des menschlichen Auges. Acta Ophthalmol (Copenh) 26 (Suppl): 1–121.

Vogel A, Dick HB & Krummenauer F (2001): Reproducibility of optical biometry using partial coherence interferometry: intraobserver and interobserver reliability. J Cataract Refract Surg 27: 1961–1968.

Received on September 22nd, 2006. Accepted on December 2nd, 2006. Correspondence: Thomas Olsen MD Associate Professor University Eye Clinic Aarhus Sygehus Nr Brogade 44 DK-8000 Aarhus C Denmark Tel: + 45 89 49 32 28 Fax: + 45 86 12 16 53 Email: [email protected]

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