Standardizing sum-of-segments axial length using refractive index models

: Optical biometry uses interferometry to measure the axial length (AL) of the eye. Traditionally, one-variable regression formulas have converted the optical path length measured by a biometer to a geometric AL. An alternate calculation of axial length sums the individual segments of the eye (sum-of-segments AL). This calculation has been shown to improve predictions of some intraocular lens power formulas when used in place of traditional axial length. Sum-of-segments ALs are determined from 13 refractive index models. As measured in 1695 eyes, these yield diﬀerent ocular axial lengths. A path to standardization from these models is presented.


Introduction
There are two methods of determining axial length of human eyes: traditional axial length (AL) and sum-of-segments AL, also known as segmented AL, segmental AL, and segment-wise AL. Traditional axial length is the axial length displayed by the original IOLMaster (Carl Zeiss Meditec AG) where only the positions of the anterior cornea and retinal pigment epithelium were identified by the instrument. Traditional axial length was originally calibrated by Haigis et al. [1] to ultrasound biometry by mathematically moving the image-side reference position from the retinal pigment epithelium to the internal limiting membrane as was measured by immersion ultrasound. The traditional axial length is thus considered to represent the distance between the anterior cornea and the internal limiting membrane. Most biometers have since calibrated their units to produce similar values [2][3][4][5].
Sum-of-segments AL is determined by summing the segment lengths (cornea, aqueous, lens, and vitreous) of the eye. Using sum-of-segments AL improves many formulas [6][7].
Ocular biometers are interferometers in which an incoming radiation beam passes through a beam splitter. In the initial time-domain biometers, part of the beam travels through the eye, and the other beam part travels through air to a moveable mirror. After reflections, these beams travel to a sensor which detects the intensity of light formed by interference of the two beams. The intensity can be shown as a function of movement of the mirror, which gives an "air distance" equivalent to the optical path length within the eye. Spikes in the signal indicate where there are sharp changes in refractive index such as at media boundaries. Newer instruments use more efficient methods of interferometry, such as swept-source interferometry, but the concepts of air distance and optical path length still apply.
Actual, or geometric, lengths of segments in the eye (geometric lengths) are determined by applying assumed refractive indices RI λ at the wavelength λ used by an instrument, to each segment's optical path length OPL λ : After using Eq. (1) to determine the geometric length of each segment (central thicknesses of the cornea, aqueous depth, lens thickness and vitreous depth), the sum-of-segments AL is determined by simply taking the sum of geometric lengths of all segments. Because the internal limiting membrane of the retina can be identified only intermittently on ocular biometers like the Haag-Streit Lenstar LS900, the vitreous depth optical path length is actually the vitreo-retinal complex, measured between the lens capsule/vitreous interface and retina/retinal pigment epithelium interface. In this study, we determined a mean theoretical retinal thickness RT T , which when subtracted from the mean sum-of-segments AL gave a mean adjusted sum-of-segments AL equal to the mean traditional AL. We did not calculate traditional AL. It was the displayed value read directly off the Lenstar's automatically-generated report. The refractive indices of the ocular segments are not known. In this study we evaluate 13 refractive index models, most of which are based on specified refractive indices of well-known model eyes, and, in two cases, the chromatic dispersion of such eyes. This study attempts to answer two questions: How different are ocular length measurements due to the various refractive index models? Can we standardize these various ocular length measurements to give similar values to each other and to traditional AL?

General
This study conformed to ethics codes based on the tenets of the Declaration of Helsinki. An institutional review board (Lakeland Hospitals Niles and St. Joseph, Institutional Review Board #1) exempted the study from review. This research was compliant with the U.S. Health Insurance Portability and Accountability Act.
Consecutive eyes undergoing cataract surgery between March 2010 and December 2012 were included if they received uncomplicated in-the-bag placement of a hydrophobic acrylic intraocular lens (IOL) (AcrySof SN60WF, Alcon Laboratories, Inc.). Measurements were made with an optical biometer, the Lenstar LS900 (Haag-Streit AG), whose graphical user interface, EyeSuite, was used to identify eyes with highly discernable segment spikes. The database was described previously [8]. As mentioned above, traditional AL was from displayed values.
Two instruments were used with slightly different wavelengths of 820.5 nm and 831.81 nm. The software calculates refractive indices specific to each instrument.
Many model eyes have refractive indices specified at a visible wavelength. Chromatic dispersion equations are used to convert these into refractive indices at other wavelengths. The usual refractive indices, termed "phase" refractive indices n p , have to be converted to "group" refractive indices n g , which are applicable to wave packets such as those that occur with interferometers. For a wavelength λ, the two indices are related by: Section 2.2 gives information about 13 refractive index models. These models are Cornu Le Grand, Navarro, four methods of water-scaling, six models based on Atchison & Smith [9], and Lenstar. Cornu Le Grand and Navarro models use the chromatic dispersion of the Le Grand and Navarro model eyes, respectively. The water-scaling variants are based on the refractive indices of the Le Grand and Gullstrand eyes, taken to occur at either 555 nm or 589 nm, and then scaled according to the chromatic dispersion of water as done by Suheimat et al. [10]. The Atchison & Smith models use a Cauchy chromatic dispersion equation and scaling according to the model eye (Le Grand, Gullstrand), reference wavelength (555 nm, 589 nm) and which lens reference index is used by the Gullstrand variants. The Lenstar model is used by the EyeSuite software of the Lenstar LS900.
We obtained optical path lengths from the instrument. For every refractive index model, we applied refractive indices to each of the ocular segments using Eq. (1) to calculate the geometric segment lengths, and then summed these to determine the sum-of-segments AL. We adjusted the sum-of-segments AL by subtracting a mean theoretical retinal thickness (mean RT T ), to give adjusted sum-of-segments axial lengths. The mean RT T , which was a different constant for each refractive index model, was selected so that the mean adjusted sum-of-segments axial length for each refractive index model was equal to the mean traditional AL of the dataset.

Refractive index models
In some of the original works, wavelengths were specified in micrometers, and conversions have been made to make the equations applicable to nanometers.

Cornu Le Grand
Le Grand [11] used the Cornu chromatic dispersion equation to describe phase refractive index n p as a function of wavelength λ: where n ∞ , K and λ 0 are constants based on 589 nm for each ocular segment. Given Eq. (3),

Navarro
Navarro et al. [12] used the following equation for chromatic dispersion: where n * * , n F , n C and n * are specified refractive indices at wavelengths 365 nm, 486.1 nm, 656.3 nm and 1014 nm, respectively. a 1 (λ) through a 4 (λ) are equations of the form: where A 0 , A 1 , P and R form a set of constants distinct for each a i (λ), and λ 0 = 167.3. Two errors given by Navarro et al. were corrected by Atchison and Smith [9]. Given Eqs. (5) and (6), where These models are based on a particular visible wavelength being chosen to represent the refractive indices of model eyes. Common wavelengths used are 555 nm and 589 nm. Refractive index of water n λw at wavelength λ was obtained as a four-term Sellmeier chromatic dispersion formula where A 1 −A 4 and λ 2 1 −λ 2 4 are constants [13]. We used the constants for 20.0°C; maybe a higher temperature would have been better but results were not provided above 24.0°C.
The refractive indices were scaled according to refractive indices of the different media specified in Gullstrand exact and Le Grand full theoretical eyes at a reference wavelengthλ: where n p is the scaled refractive index of the media, nλ is the refractive index of the media at the reference wavelength and nλ w is the refractive index of water at the reference wavelength. The reference refractive index was given asλ = 555 nm. The Gullstrand eye lens has a two-shell structure, and the two refractive indices were replaced by the average axial refractive index 1.3994 along the lens. We note another estimate of average lens refractive index could have been obtained from Gullstrand's gradient refractive index modelling [14] as 1.4050 [15]. If a reference wavelength higher than 555 nm had been chosen, then refractive indices at any specified wavelength would increase.
We refer to the chromatic dispersions described above as D&M Gullstrand (555) and D&M Le Grand (555), with D&M referring to Daimon and Masumura [13] for the variation of refractive index of water with wavelength. As applied to the wavelength of 589 nm rather than 555 nm, the dispersions are referred to as D&M Gullstrand (589) and D&M Le Grand (589).
Given Eqs. (9) and (10), Using fittings derived from Le Grand and Navarro, Atchison & Smith [9] used the Cauchy chromatic dispersion equation: in conjunction with the Gullstrand number 1 eye, where A−D are constants. They gave two sets of coefficients for the lens, one set each for the high and low index (1.406 and 1.386, respectively, at reference wavelength 555 nm). We refer to these dispersion equations as A&S Cauchy (HL) and A&S Cauchy (LL), depending on whether high or low indices are being used for the lens.
Scaling based on Eq. (10) was used to convert the refractive indices for the Gullstrand lens to match the average index 1.3994 [10]: where n λ,1.3994 is the refractive index when refractive index at the reference wavelength is 1.3994 and n λ,1.406 is the refractive index when refractive index at the reference wavelength is 1.406. Scaling was also used to convert the refractive indices of the Gullstrand eye to those for the Le Grand eye 10]. Similar to Eq. (10) we have where n λL is the refractive index to be obtained for the Le Grand eye, n λG is the corresponding index for the Gullstrand eye, and nλ G and nλ L are the refractive indices at the reference wavelength of the Gullstrand and Le Grand eyes, respectively. We refer to these dispersions as A&S Gullstrand and A&S Le Grand. A&S Gullstrand is identical to A&S Cauchy except for the lens.
As for the water scaling variants, if a reference wavelength higher than 555 nm had been chosen, then refractive indices at any specified wavelength would increase. We term the A&S Gullstrand and A&S Le Grand variants at 589 nm as A&S Gullstrand (589) and A&S Le Grand (589). Given Eqs. (12) and (14),

Lenstar
The refractive indices used by the Lenstar biometer in the EyeSuite code were verified by Haag-Streit. The phase indices are given by the Cornu Eq. (3) and can be converted to group indices using Eq. (2) to become However, Haag-Streit used a slightly more complicated equation Refractive indices of cornea, aqueous depth, lens thickness, and vitreous depth were compared between Eqs. (16) and Eq. (17) . They gave results agreeing to 0.00011 -0.00018 for the different ocular media for each of the two wavelengths used by our Lenstar machines.

Results
This study comprised 1695 eyes of 1244 patients presenting for cataract surgery. They were all obtained from one private practice. The traditional axial length range was 20.84 to 29.51 mm with a mean of 23.76 ± 1.16 mm. Age was 71.2 ± 8.6 years with a range from 37.5 to 93.0 years. All eyes were from cataract patients. Sex was not identified in 20 eyes. Of the remainder, there were 976 females and 699 males. The preoperative manifest refraction was unavailable in 43 eyes. In the remaining 1652 eyes, the average spherical equivalent was −0.70 ± 2.74 D with a range from −15.75 D to +6.00 D.
The effect of including two eyes of one patient was negligible; to study this, we evaluated only 1 eye per patient, keeping only the first eye to be operated. The mean RT T values changed by ≤ 0.001 mm for all refractive index models. The effect of age was also negligible; RT T plotted against age gave R 2 values ≤ 0.01. Table 1 shows the phase and group refractive indices for the models at the wavelengths used by our Lenstar machines. These are arranged in order of increasing mean axial length estimation (also Table 2). There is a trend of decreasing index downwards in the table for the aqueous, lens and vitreous, and particularly for the lens.  Figure 1 shows Bland-Altman plots in which the differences in RT T, between axial lengths determined from the traditional AL and refractive index models are plotted as a function of the means of the two measures. The plots are for the two models showing the extreme results. The mean RT T for each model and its Bland-Altman regression equations are given in Table 2. The relationships between the model ALs and traditional ALs were similar except for the mean RT T values which varied between 0.11 mm and 0.29 mm (range across all eyes was 0.19 ± 0.01 mm). There was a minimum variation in individual RT T for any one eye of 0.15 mm and a maximum variation for any one eye of 0.22 mm. When the shortest adjusted sum-of-segments axial length was subtracted from the longest one for each eye, mean difference was only 0.01 ± 0.01 mm and the maximum difference for any eye was only 0.04 mm. To showcase the results of adjusting Table 2

Refractive index Model
Mean unadjusted sum-of-segments AL (mm) and SD (mm)  Table 3. Note the similarity in adjusted sum-of-segments AL between all models for both eyes; no model AL varied from another by more than 0.02 mm.  Table 4 shows component lengths for models. The major component contributors to variation in axial length were the lens and vitreous, with the lens accounting for two-thirds.

Table 3. Unadjusted and adjusted sums-of-segments AL (mm) for the shortest and longest eyes.
Refractive index models are in the same order as for Table 2.

Discussion
In the Introduction, we asked how different are ocular length measurements due to the various refractive index models, and whether these can be standardized to give similar values to each other and to traditional AL. We have presented 13 models which produce widely-varied ALs, with a range across eyes of 0.19 ± 0.01 mm. They can be made to agree to 0.01 ± 0.01 mm by a simple correction factor we have called mean RT T . However, the models look much different from traditional AL across the axial length spectrum, as seen by the slopes in Fig. 1. Although this effect has not been thoroughly studied, several vergence IOL power formula predictions (Barrett, Haigis, Hoffer Q, Holladay 1, SRK/T) have improved when sum-of-segments AL was substituted for traditional AL [6,7]. For IOL power formulas to give the same predictions with different instruments, these instruments need to agree with each other. When traditional AL was developed by Haigis et al., he included the formula he used [1]. This has become the standard to which instruments have been calibrated. Traditional AL is about the same on all instruments [2][3][4][5]. One instrument, the ARGOS biometer, gives only sum-of-segments AL. This has also been calibrated so that it gives similar average output to traditional AL [3].
Biometers capable of producing sum-of-segments AL include the Lenstar LS900 used in this study, Galilei G6 (Zeimer, Port, Switzerland) and ARGOS (Movu Inc, Santa Clara, CA). Zeimer has provided no information about its RI model(s). ARGOS measures with a wavelength between 1050 and 1060 nm and gives refractive indices without information to scale to another wavelength: corneal index is 1.374, aqueous index is 1.336, lens index is 1.41, and vitreous index is 1.336 at 1050 nm [16].
We present a method in this paper which can normalize sum-of-segments ALs. This is important because if one machine's AL does not equal the AL of others, it is possible that IOL power formulas will not only need to be optimized by IOL model and by AL method (traditional AL or sum-of-segments AL), but also by instrument. As can be seen in Fig. 1, it is important to note that while the RT T correction factor standardizes the varying refractive index models to each other, it does not cause the sum-of-segments measurements to be equivalent to traditional axial length values.
In a prior study, using the same method presented in this paper, we found that adjusted sum-ofsegments axial lengths yielded lens constants for most formulas which were interchangeable with those developed using traditional AL [6].
It is important to note that this work does not lead to increased accuracy of ocular measurements, but rather aims to improve consistency of measurements between instruments. We need an independent way to verify AL before we can say which RI model is most accurate.
Determining the most accurate RI model will likely be most helpful for yet-to-be-developed IOL power calculation formulas which incorporate the full axial length of the eye from the cornea to the retinal pigment epithelium, instead of traditional AL which measures from the cornea to the internal limiting membrane.
Not only do these different RI models result in a wide disparity in mean RT T values (Table 2), they result also in a wide disparity of lens thickness and vitreous depth values (Table 4). Currently there is no standard method to calibrate these segments.
The method of determining sum-of-segments ALs presented in this paper does not address how to standardize thickness of the lens or of any other segment. Some formulas for determining IOL power, such as the Olsen formula, use variables other than axial length, such as lens thickness and central corneal thickness [17]. If the lens thickness varies between biometers due to differences in assumed refractive indices, then formulas that rely on lens thickness as a preoperative variable will likely give different predictions. It would be helpful to standardize all segment refractive indices.
To give an approximate magnitude of axial length on prescribed intraocular lens power, we applied the basic SRK formula [18] where F is power of the IOL for emmetropia (D), A = A-constant for the IOL being used, L = axial length (mm), and K = average keratometry (D). The adjusted sum-of-segments AL for the longest eye in Table 3 for most models was 29.26 mm, 0.25 mm shorter than the traditional AL of 29.51 mm. This difference would calculate a difference in recommended IOL power of 0.63 D for this eye. These differences can be clinically important.
Research will benefit from full-disclosure by biometer manufacturers. We look forward to a time when all machines use the same RI models in order to produce the same measurements for all segments of the eye. The next-generation IOL power formulas will likely be more accurate if they are based on sum-of-segments axial lengths. This assumes we can determine which refractive index model is most accurate.
Despite using different RI models, when a biometer is adjusted (calibrated) well to traditional AL, the sum-of-segments axial lengths should be expected to be interchangeable with those of other biometers. In such cases, lens constants, formal "fudge" factors which every IOL power formula uses to calibrate its mean predictions to fit actual outcomes, should also be expected to be interchangeable between biometers.
We elected to exclude the ARGOS biometer's refractive index model from this study because no method has been presented to scale the indices from the 1050 nm wavelength used by the ARGOS biometer to the 820 nm wavelength used by the Lenstar. Similarly, we excluded the Liou and Brennan refractive index model from this study because it has inaccurate chromatic dispersion.
The Liou and Brennan eye [19] is an excellent and widely used model eye, with equations for its media of the form The dispersion should increase with index of the media, so that the lens should have the highest dispersion of all the ocular media. Because the coefficients are the same for all media in the model, dispersion is the same for all media. Furthermore, while refractive index should decrease with increase in wavelength, the phase index increases again after 757 nm. Mean RT T is 0.64 mm, much higher than all the refractive index models presented above.
To assist others in determining refractive indices, we have written an Excel spreadsheet which can be obtained at https://cookeformula.com/refractiveindices. We included these 13 refractive index models as well the ARGOS and Liou and Brennan refractive index models. The wavelength can be changed in cell A2. Phase and group refractive indices for the models appear in columns R and T, respectively.