By

**Thomas Olsen, MD**

**Associate Professor of Ophthalmology**

**Aarhus University**

**Aarhus, Denmark**

While it may not be that difficult to calculate the IOL power for a normal eye, long and short eyes present a challenge as we push the boundaries for our models and measurement accuracy. Most formulas tend to break down in the extremes. Why is that? We shall try to identify some of the sources of error and discuss the means to overcome the issues.

## The axial length

Needless to say, an accurate axial length is crucial for an accurate IOL power calculation. A general rule is that an error in axial length of 1 mm is the equivalent of a 2.5 D refractive error in the spectacle plane. When optical biometry was introduced around year 2000 (Zeiss IOLMaster) it was a revolution. The wavelength of light is so much shorter than ultrasound that from a physical point of view, the technology has to potential to be so much higher. The precision of optical biometry is often stated to be around 0.02 mm (as compared to 0.2 mm with ultrasound). This translates into a 0.05 D refractive error, only. In other words, if the axial length measurement were the only source of error in the IOL power calculation, we would have no clinically significant error in our predictions! We all know this is not the reality and hence we have to identify further sources of error.

What is measured by optical biometry is the time - the ‘optical path’ - it takes light to travel through the media. To convert optical path length into geometrical path length we must know the index of refraction in the given media. The original calibration of the IOLMaster was based on the fundamental work of Wolfgang Haigis[1]. Based on a large database of (segmented) immersion axial length readings, he found the following relation:

Ax = (OPL/1.3549 – 1.3033) / 0.9571

where OPL = optical path length and Ax = immersion ultrasound axial length (= output reading of the IOLMaster). By rearranging we get the OPL as a function of the axial length:

OPL = (Ax * 0.9571 + 1.3033) * 1.3549

The OPL is the true physical reading of the biometer, equivalent to the transit time of light. The important question to ask is however, how this time-measurement is transformed into the geometrical length that we need for our calculations.

In a study by Olsen & Thorwest[1] with comparison of pre- and postoperative IOLMaster measurements, an apparent shortening of the postop axial length readings was observed, the mean value of which was 0.08 mm (+ 0.12 mm, SD). It is hard to believe this is due to a true shortening of the eye after surgery and a more likely explanation was therefore, that the index calibration of the phakic eye may need a modification. The dark horse is the crystalline lens of the phakic eye and Olsen showed that when the assumed index of the crystalline lens was changed from 1.407 to 1.429 – corresponding to a change in the overall group refractive index of the phakic eye from 1.3574 (Haigis model) to 1.3516 (Olsen proposed model) – the observed off-set was eliminated and the pre- and postoperative readings were consistent. The new calibration gives slightly lower readings in the long eyes and slightly higher readings in the short eyes Fig 1). In other words, this might explain (part of) the hyperopic error observed with many formulas in the long eyes (for which an axial length correction was proposed by Wang & Koch[2]) and vice versa for the short eyes. Another factor to be considered is the potential error in the translation of K-reading into power, which may skew the predictions in the long eyes (se section K-reading)

For this article, the experiment was repeated using the Lenstar LS900 with pre- and postoperative measurements on a series of 1573 routine cataract extractions. Again, although a very high correlation was found between pre- and postoperative readings (fig 2), a significant shortening of 0.06 mm (__+__ 0.057 D, SD) was observed (fig 3)

## Segmented axial length readings

In recent years several researchers have proposed that a segmented axial length (using an individual index of each compartment of the eye) may be better than the standard non-segmented reading (using an average index) (Li Wang et al [1]). From a theoretically point of view, this sounds plausible. However, what evidence can we provide showing it is better?

For this article the pre- and postoperative segmented axial length was calculated in 1573 routine cataract extractions and compared with the standard non-segmented (average index) readings. Again, a high correlation was found between the pre- and the postoperative readings (fig 4) and a small mean difference of – 0.05 mm (__+__ 0.070 D, SD) was observed between the pre and the postoperative readings (fig 5)

The systematical difference may be a question of adjusting the refractive indices used in the calculation of the segmented length. The variation, however, is noteworthy and is not adjustable by the indices. When we compare the variation between pre- and postop readings with the standard and the segmented technique, we found a highly significant difference (table 1).

In fig 6 a Bland-Altman plot is shown of the difference between the segmented and the non-segmented preoperative reading. As can be seen the segmented reading gives a shorter reading in the long eyes and a longer reading in the short eyes, much the same effect as the recalibration of the IOLMaster readings mentioned above (fig 1)

In conclusion, the segmented axial length reading may have the advantage over the standard (non-recalibrated) reading in that the extremes of the axial length are better represented, i.e. tending to countereffect the hyperopic error in long eyes and the myopic error in short eyes seen with most formulas. However, the larger variation of segmented readings between pre- and postoperative readings is of concern because it will add to the total error of IOL power calculation over the entire axial length range. Further studies are needed to elucidate the potential benefit of segmented axial length readings.

## ELP prediction

The basic scheme of a thin-lens formula is as follows:

(eq 1)

where P = IOL power of emmetropia, Ax = axial length, n = refractive index, K = Corneal power and ELP = **E**stimated **L**ens **P**lane. The IOL power is found according to vergence considerations by which the vergence anterior to the IOL (right part of the equation) is subtracted from the vergence posterior to the IOL (left part). In this scheme the IOL is regarded as a ‘thin’ lens where all refraction occurs at one single plane.

The ELP is never the true (measurable) position of the IOL for several reasons: Firstly, the IOL is not a ‘thin lens’ but rather a ‘thick lens’ with a finite thickness having 2 principal planes for the refraction according to Gaussian Optics. Secondly, it is obvious that for the ELP to be close to reality, the other parameters in the equation need also to be close to reality. If, for instance the corneal power is not the true physical value, there will be a bias in the equation skewing the ELP for the equation to predict the accurate power.

If we assume the other parameters of the above equation are known and correct, the ELP is the only parameter to be *predicted* by the formula. There are various ways to model the ELP and this is the engine room of the various ‘Formulas’ developed by different authors, each having their own way of calculating the ELP.

In practical life the ELP is often back-calculated from the actual outcome observed in a large database. If the database is sufficiently large the ELP is often correlated with other clinical variables like the axial length, the corneal power, the preoperative anterior chamber depth and maybe other metrics and in this way a multi-dependency is worked out. The aim is to increase the predictability of the ELP but it is important to stress that the ELP derived in this way is not a physical distance but rather a virtual distance which works only with the given formula. Other terms like the actual lens position (ALP) or physical lens position (PLP) are used to describe the physical distance eventually to be measured by clinical methods.

The accuracy of the ELP estimation is directly related to the accuracy of the IOL power calculation. For a normal eye 1 mm error in the ELP is the equivalent of 1.5 D error in the final refraction after IOL implantation. However, this effect is very much influenced by the IOL power and hence the axial length of the given eye. For a high powered IOL a shift in the ELP will have a large effect on the refraction while a low powered IOL will have only a minor effect on the refraction for the same shift. The exact magnitude of the effect will also be influenced by the corneal power and the starting position of the IOL.

To illustrate the variation a simulation study was performed on a case-mix database of 2970 routine cataract extractions performed at the University Clinic of Aarhus some years ago. The cases had a broad range of axial lengths from 18 mm to more than 36 mm and the IOL power ranged from – 5.0 D to + 40 D. In each case the refractive effect of shifting the predicted IOL position by 0.25 mm was calculated and multiplied by 4 to give the effect of a 1 mm ELP change. The result is shown in fig 7. As can be seen the effect almost doubles in a very short 18 mm eye as compared to a normal 23.5 mm eye. Hence, the question of the ELP prediction becomes especially important in the short eyes. The longer the eye the lower the effect. For a very long eye with a negative IOL power implant the effect changes sign so that an increase in the ELP will have a myopic effect.

## K-reading

What is the corneal power? Most clinicians will ask for the ‘K-reading’ neglecting the fact that the keratometer does not measure the power directly. What is measured is the size of the Purkinje I image reflected from the front surface of the cornea in a para-central ring of 3 mm or so and from this the radius of curvature is calculated. Once the curvature has been established, the ‘keratometric’ power is calculated according to a single surface refraction as

(eq 2)

where F = dioptric power of the cornea, r = radius of curvature in meters and n = index of refraction.

The keratometer index of refraction ‘n’ is often set to 1.3375 by convention. The reason seems to be from early days of instrument making where the exact corneal power was of less clinical interest than the astigmatism which can found as the difference between the flat and the steep meridian. For practical purposes the value of 1.3375 means that a corneal curvature of 7.5 mm would give a reading of 45 D so it was easy to check the calibration of the instrument. In 1909 Gullstrand wrote *“Diese Zahl wurde aus technischen Gründe gewählt, damit 45 Dptr einem Radius von 7,5 mm entsprechen zollte”* *.* (“This number was chosen for technical reasons, so that 45 D corresponded to a radius of 7.5 mm”) [1]

There is strong evidence that the diopter reading of the standard keratometer is higher than the true corneal power by almost a diopter. This conclusion can be drawn from theoretical considerations as well as from the clinical measurements of both corneal surfaces as can be done by modern Scheimpflug or OCT techniques [2]. This has important implications for IOL power calculation. Referring to the thin lens formula (eq 1) what happens if the ‘K’ is too high? To achieve the same IOL power, the ELP needs to increase maybe beyond reality giving the false impression of a deep location of the IOL. This is the reason why the ‘ACD’ stated on the IOL label is often listed as more than 5 mm, which was calculated according to the old Binkhorst formula. This value is about 0.5 mm higher than can be measured by clinical methods.

Does it matter, you may say? To compensate the error you can choose to adjust the mean (virtual) ELP in order to make your predictions accurate in the average case. This process - called optimization - is identical to adjusting the A-constant of the SRK/T formula. This may work for the normal eye, but it will not work in the very long eye: remember with a low powered IOL the refractive effect of a shift in ELP is minimal. Consider a long eye with an IOL power of zero. If the corneal power is off by 1 D, this will affect your predictions by the same amount (assuming you have the axial length right) and you cannot escape the error by adjusting the ELP. The result is a hyperopic error in the long eye. This mechanism may be behind some of the reported errors observed with the Holladay and the SRK/T formula in the very long eyes.