Influence of Refractive Index of the Crystalline Lens on Optical Biometric Measurements for Cataract Patients

Abstract

Purpose

Measurement of the axial length (AL) of the eye and the crystalline lens thickness (LT) relies on information of the refractive index of the crystalline lens for converting optical path-length (OPL) to geometric size. The purpose of this study was to measure the lens refractive index (N_Lens) in vivo for patients with cataract and evaluate the influence of the N_Lens on AL and LT measurements.

Methods

Cataract eyes of 7 patients, aged 56 to 77 years old (mean = 64.7 years), were imaged with a combined whole-eye optical coherence tomography (OCT) and Scheimpflug Imaging (SI) system before surgery. The images were processed using a custom MatLab program to trace the central light-ray from the cornea to the retina and to solve the average lens refractive index (N_Lens). The OPLs of other ocular segments, except the lens, were converted to geometric lengths by using the refractive indices from a model eye. The AL and LT were calculated with the measured N_Lens and also a constant N_Lens (1.41) for comparison.

Results

For the patients with cataract, mean AL and LT were 23.897 ± 0.854 and 4.309 ± 0.438 mm, respectively, when the measured N_Lens was used for OPL conversion, and changed to 23.918 ± 0.875 mm and 4.330 ± 0.461 mm, respectively, when the N_Lens was set at 1.41. The lens N_Lens from individual eyes varied from 1.400 to 1.434 among the patients, and caused a substantial individual variation in the AL and LT measurements, as compared with those with a constant N_Lens in a range from −0.070 to 0.027 mm.

Conclusions

The Scheimpflug imaging-equipped whole-eye OCT system provides a useful technique to measure AL and LT with the N_Lens determined for the cataract eye in vivo. The N_Lens varies substantially from individual eye to individual eye among the patients with cataract, and plays a critical role for deriving accurate AL and LT measurements. During cataract surgery the accurate AL and LT measurements should be used for IOL power calculation.

Optical methods, such as optical coherence tomography (OCT) and partial coherence interferometry (PCI), are currently widely used in the areas of optometry, ophthalmology, and vision research to access the dimensional size of the eye. The technique measures the interference pattern of the light reflected from the ocular surfaces with a reference light. The distance on the interference pattern, however, is not a direct measure of the ocular distance, but a measure of optical path-lengths (OPLs), which is a product of the physical distance and the refractive index of the medium between the surfaces. To convert the OPL to physical distance, the OPL needs to be divided with the refractive index. In current optical biometry, the ocular refractive indices used for OPL conversion were conventionally adopted from Gullstrand's model eye.^1–4 The model eye's refractive indices of the cornea, aqueous, and vitreous, are homogeneous, relatively stable, and constant across individual eyes, and, thus, reliable for the OPL conversion. But the crystalline lens grows throughout the whole lifespan, with continuous changes in its size, shape, and internal structure (see reviews^5–9), which directly causes age-dependent change in lens refraction index (N_Lens). Therefore, for almost every eye, the model eye's N_Lens does not reflect the true value at most times. This dynamic property of the N_Lens imposes a need for an in vivo technique to conveniently access it at any time as required. However, accessing the N_Lens in vivo has long been a challenge in eye research. Fortunately, in vivo measurement of the N_Lens has recently succeeded by combining an anterior segment OCT and a Scheimpflug imaging (SI) system.¹⁰ By using the novel system, significant individual variation in the N_Lens was observed among young adults. The changing trends of the N_Lens with age and accommodation were also revealed.

The variations of N_Lens within and between individual eyes may cause measurement error of the axial length (AL) of the eye and lens thickness (LT) if the model eye's standard N_Lens was used for OPL conversion, because the model eye's N_Lens is not the true N_Lens at the time of measurement for individual eyes. Accuracy of the AL and LT measurements is important for cataract surgery planning. In cataract surgery, the natural cataract lens is removed and replaced with an artificial intra-ocular lens (IOL). If the AL is accurately obtained, along with an accurate estimate of the IOL position, the IOL power should be accurately determined with an optical model because the pseudophakic eye forms a simple two-lens component system. However, errors of IOL power calculation have been frequently reported in cataract surgery, especially for patients with too long or too short eyes.^11–13 The AL and LT measurements have been considered as two significant sources contributing to the calculation error of IOL power.^14,15 The LT is a factor because it is a parameter used to estimate the IOL position. Therefore, the impact of AL and LT measurements on errors in IOL power calculation might be associated with the miss-estimate of the N_Lens for individual patients due to the use of the model eye's value for lens OPL conversion.

It is apparent that the impact of individual variation in N_Lens on accuracy of AL and LT measurements should not be ignored in cataract surgery. The purpose of this study was to measure the variation in lens refractive index among a group of patients with cataract and to examine the influence of individual variation in N_Lens on AL and LT measurements.

Methods

Apparatus

A slit-lamp platform was used to mount the whole-eye OCT (WE-OCT) scanning probe, SI system and a Badal system. The operational principle is sketched in Figure 1, where the SI (channels highlighted in green), WE-OCT (in red), and Badal (in blue) systems were combined for measuring the N_Lens and the length of all ocular segments (including the corneal thickness, anterior chamber depth, LT, and vitreous chamber depth), hence, the AL of the human eye in vivo. The subject's pupil section is monitored with the SI system. The SI shares the illumination light with the WE-OCT, but with a faster scanning speed when the SI image was taken. A hot mirror is used to introduce a fixation target inside the Badal system to correct refractive error.

Figure 1.

Sketch of operational principle of the Scheimpflug imaging (SI) equipped whole-eye OCT (WE-OCT) system. The SI (channels highlighted in green) and WE-OCT (in red) were combined with a Badal (in blue) system. See the text for more details.

The WE-OCT system has been previously described in detail.¹⁶ Briefly, it included a superluminescent diode light source (SLD; InPhenix, IPSDD0808, Livermore, CA, USA) with a center wavelength of 840 nm and bandwidth of 50 nm, a spectrometer consisting of a line array complementary metal-oxide-semiconductor (CMOS) camera (Basler Sprint spL4096-140k; Basler AG, Ahrensburg, Germany), a collimating lens (f ¼ 50 mm; OZ Optics, Ottawa, Ontario, Canada), an 1800 line/mm transmission grating, and an image enlargement lens (f ¼ 240 mm; Schneider Optics, Hauppauge, NY, USA). A switchable reference arm with four mirrors was used to sequentially acquire four images. The scan depth of each single reference arm was 12.57 mm. The first three reference arms overlay the anterior chamber, allowing for measurement of the central corneal lens thickness (CLT), anterior chamber depth (ACD), and thickness of either the crystalline or intraocular lens (LT). The final mirror captured the vitreous chamber depth (VCD) and retina, which established overall OPL, and the 4 switchable reference arms obtained an equivalent scan depth of 37.71 mm in air. The measured axial resolution is 7.7 µm in the air, with a resolution of 5.6 µm for ocular biometry. The SI system included an SI imaging lens and a high-speed, high-resolution CCD (Raspberry Pi HQ camera module) connected to a computer through OBS Studio software.¹⁰ The SI system shares the light source of the WE-OCT system, and thus illuminated exactly the same section of the eye as in the WE-OCT. The SI imaging lens is set 45 degrees sideways from the optical axis of the WE-OCT. The axial resolution of the SI system was approximately 5.5 µm.

Subjects

Seven subjects participated in this study with an age range of 56 to 77 years, with a mean age equal to 64.7 ± 6.8 years. The refractive errors of the subjects ranged from hyperopic to myopic, the spherical equivalent from +2.00 diopter (D) to −7.125 D, and the astigmatism was less than 2.50 D. Subjects had no ocular abnormalities. Exclusion criteria for each eye in the study were as follows: macular degeneration, glaucoma or glaucoma suspect, prior ocular surgery, history of acute vision loss, history of systemic inflammatory disease, and cataracts that were ≥4 in clinical severity. The basic subjects’ information is listed in Table 1. This study followed the tenets of the Declaration of Helsinki and was approved by the Institutional Review Board of the New England College of Optometry. Informed consent was obtained from the participants after oral and written explanation of the nature and possible consequences of the study.

Table 1.

Subjects’ Basic Information

Subject Code	Age, Y	Sex	Rx, D	Cataract Type	Cataract Grade
SF1	65	F	0.5	NS, PSC	Trace NS, 2+ PSC
SF2	65	F	−1.38	NS	2+ NS
SF3	67	F	−1	NS, CS, PSC	1+ NS, 1+ CS, 1-2+ PSC
SF4	65	F	2	NS	2+ NS
SF5	56	F	−7.13	NS, PSC	1+ NS, 1+ PSC
SF6	77	F	1	NS, CC, PSC	2+ NS, 1+ CC, 3+ PSC
SF7	58	M	−6.75	NS	Trace NS

CC, cortical cataract; CS, cortical spoking; NS, nuclear sclerosis; PSC, posterior subcapsular cataract; Rx, spherical error.

Procedure

The subject was asked to sit in front of the system with the chin and forehead against the rest of the slit-lamp platform while the room light was turned off. At the start of each experiment, the subject was aligned with the apparatus and the Badal system was adjusted to the subject's far point. The subject's task was to fixate on a letter approximately 5 degrees nasally with the right eye while the left eye was patched. The pupil was dilated using 2 doses of 2.5% concentration of phenylephrine. As the subject aligned with the apparatus, both the SI and WE-OCT images appeared on the corresponding computer screens. By carefully adjusting the slit-lamp base, the system can be perfectly aligned to the corneal vertex of the subject's eye, enabling sharp reflection of the WE-OCT light. Once sharp corneal reflection appeared, image acquisition for SI and WE-OCT was performed. Three image acquisitions were repeated for the far-point visual condition. A typical measurement session lasts approximately 20 minutes. After a 15-minute break, the session was repeated. The experiment was completed within 2 hours of each visit.

Image Processes and Data Analysis

The image processes were previously explained in detail.¹⁰ Briefly, both SI and the WE-OCT images were processed with a customized MATLAB program with semi-segmentation to derive the morphometric dimensions of the eye, including the radius of anterior and posterior cornea, the corneal thickness the LT, radius of the anterior and posterior lens surfaces, and vitreous chamber depth. The radius was fitted using a conical curve. The WE-OCT images were processed in three steps¹⁰: (1) the positions of all interfaces in the original WE-OCT image are semi-segmented; (2) the surface extraction algorithm is used to produce a set of positions for well-defined interfaces; and (3) the refraction correction algorithm is applied using the Snell's law ray tracing algorithm. The SI image is first corrected for projection error, and a ray tracing program based on Snell's law¹⁰ is then used to correct the refractive distortion in the SI image. To derive N_Lens, both anterior segment OCT and SI images were separately processed with a constant N_Lens (1.40), and then the difference in the derived LTs between the two images was checked. If the difference in the LTs was greater than 0.001 mm, the images would be processed again with a new N_Lens, until the difference in LTs reached the minimum. With this error-trial procedure, both N_Lens and LT were solved. The individually measured N_Lens was then used to derive the AL from further processing of the WE-OCT images.

Three measurements were averaged to derive the mean estimate for each session, and were used to calculate the refractive index, LT, and AL estimates.

Results

Measurement of Refractive Index of the Lens for Individual Cataract Eyes

Figure 2 shows the WE-OCT and SI images with the image segmentation lines for two subjects, S1 (Figs. 2a, 2b) and S2 (Figs. 2c, 2d), respectively. From Figure 2, it can be seen that subject S2 had a deeper ACD and thinner LT than subject S1 in both WE-OCT (Figs. 2a, 2c) and SI images (Figs. 2b, 2d), whereas the AL is longer than that of the S1.

Figure 2.

WE-OCT image (a, c) and Scheimpflug imaging image (b, d) for subject S1 a and b and subject S2 c and d.

Figure 3 shows the N_Lens along the optical axis for the seven subjects displayed against their age. Each symbol represents the mean of three measurements, with the error bar depicting the standard error of the mean. The mean N_Lens was 1.416 ± 0.014. As shown in Figure 3, the N_Lens values varied substantially from subject to subject within a range from 1.400 to 1.434. The N_Lens was significantly correlated to age (r = −0.82, P = 0.025) and spherical refractive error (r = −0.79, P = 0.036).

Figure 3.

Average axial refractive index for the crystalline lens for seven patients with cataract, displayed against subject's age. Each symbol indicates mean of the three measurements with the error bar representing one standard deviation of the mean.

Measurement of the Lens Thickness for Patients With Cataract

The central LT for the 7 subjects was calculated with the measured N_Lens values, and the mean LT was 4.309 ± 0.438 mm. Table 2 lists the mean LT with standard deviation of the mean for the seven patients with cataract. Also presented in Table 2, are the mean LT (LT2) calculated with a constant N_Lens value of 1.41 for the 7 subjects. From Table 2, it can be seen that the LT2 calculated with measured N_Lens was different from that with constant N_Lens for 4 subjects, the LT2 was under or over-estimated within a range from −0.027 to +0.070 mm.

Table 2.

The Mean Values (and Standard Deviations) of Refractive Index of the Crystalline Lens (N_Lens), and the Measured Lens Thickness (LT) and AL of the Eye, and the Calculated Lens Thickness (LT2) and AL (AL2) for the Seven Subjects

Subject Code	NLens	LT (mm)	LT2 (mm)	P Value	AL (mm)	AL2 (mm)	P Value
SF1	1.4278 (0.007)	4.674 (0.016)	4.615 (0.015)	0.001****	23.740 (0.021)	23.682 (0.021)	0.028*
SF2	1.4119 (0.007)	4.935 (0.050)	4.928 (0.050)	0.878	22.572 (0.023)	22.565 (0.022)	0.735
SF3	1.4083 (0.005)	4.007 (0.009)	4.012 (0.009)	0.558	24.122 (0.015)	24.127 (0.015)	0.719
SF4	1.4019 (0.016)	3.995 (0.032)	4.018 (0.032)	0.425	23.692 (0.251)	23.715 (0.251)	0.916
SF5	1.4342 (0.004)	4.130 (0.005)	4.060 (0.005)	0.0001****	25.520 (0.396)	25.450 (0.396)	0.839
SF6	1.3998 (0.006)	3.770 (0.005)	3.798 (0.005)	0.003****	23.681 (0.010)	23.708 (0.010)	0.027*
SF7	1.4300 (0.007)	4.800 (0.016)	4.733 (0.015)	0.006***	24.095 (0.273)	24.028 (0.272)	0.778

The results of statistical analysis were listed by P values.

^* P < 0.05.

^** P < 0.025.

^*** P < 0.01.

^**** P < 0.005.

Measurement of the Axial Length of the Eye for Patients With Cataract

The AL for the 7 subjects was calculated with the measured N_Lens values, and the mean AL was 23.897 ± 0.853 mm. Table 2 also lists the mean AL for the seven patients with cataract with the standard deviation of the mean. Also presented in Table 2, are the mean AL (AL2) calculated with a constant N_Lens value of 1.41 for the 7 subjects. From Table 2, it can be seen that the AL2 calculated with measured N_Lens was significantly different from that with constant N_Lens for two subjects, the AL was under or over-estimated within a range from −0.027 to +0.070 mm.

Discussion

By using the SI-equipped WE-OCT system in this study, the N_Lens was measured for a group of patients with cataract, and the mean N_Lens was found to be 1.416, a value very close to, but slightly higher than theoretical value of 1.41, as suggested by Charman and Atchison for similar aged healthy eyes.¹⁷ This slight increase of mean N_Lens value (0.006), even though not significant statistically with the small sample size (P > 0.05), might be due to the cataract formation. Cataract-induced refractive change has been reported in the literature, and the change in ocular refraction was attributed to a change in lens refractive index.¹⁸ However, the influence of lens refractive index on lens refraction is not simple. The lens has a gradient distribution of refractive index, and a change in lens refractive power is caused by a change in equivalent lens refractive index, but not a change in average lens refractive index. The measured N_Lens in this study was the average N_Lens,^10,17 so, the slightly increased average N_Lens is not necessary to be associated with the cataract-induced refractive change. However, at least it provided an interesting hint for deeply investigating the optical mechanism of cataract-induced refraction change. Further study on this point is expected.

A substantial individual variation in the N_Lens was revealed among 7 patients with cataract, within a range from 1.400 to 1.434. Individual variation in lens refractive index has been reported previously from in vitro measurements, with the N_Lens changed from approximately 1.392 to 1.425 for isolated human lenses.^19–22 The study of in vivo measurement also revealed an individual variation in the N_Lens from 1.383 to 1.419 for healthy adults.¹⁰ The results in this study are consistent with the previous studies with aspect of the individual variation of the N_Lens, but not the mean N_Lens. The patients’ cataract might be one of the factors responsible for the difference in mean values.

Two approaches of OPL conversion are currently used in optical biometry for AL and LT measurements: (1) by using a single weighted mean refractive index of the whole-eye ocular media (N_om or N_we) from a model eye to divide the OPL of the whole-eye (OPL_we); and (2) by using the refractive index of each segment of the ocular components to divide the OPL of the corresponding segment for deriving the segment size first, and then, to sum all segment lengths together to derive the whole-eye AL,^23–25 The first approach is called the weighted mean ocular refractive index (WMORI), and the second approach is called segmented refractive index (SRI) or sum-of-segments. The LT calculation is provided with lens OPL conversion in the SRI.

In the WMORI approach, the model eye's mean N_we does not reflect the true mean N_we of the real eyes if the N_Lens and VCD were varied in the real eyes from the model eye's values. The difference in mean N_we between the model eye and real eyes would directly cause AL measurement error if the model eye's N_we was used to convert the real eye's OPL. According to a theoretical study, when the N_Lens was assumed to be varied from 1.388 to 1.428, combined with variation in VCD for high myopia and high hyperopia, the AL measurement was found to be −0.209 mm underestimated and 0.390 mm overestimated, with a range over a half of 1.00 mm.²⁶ This amount of AL measurement error corresponds to a refractive error of the eye over 1.5 D.

When the SRI approach was adopted in optical biometry, as used in this study, the method of separating OPL conversion by segment improved the AL measurement, with the measurement errors of AL and also LT in a range of −0.027 to +0.070 mm (see Table 2), which was smaller than the range calculated with the WMORI approach. The real range among the subjects might be even narrower or wider if the measurement errors and other uncertainties in the measurement were considered. With the SRI approach, the influence of VCD on AL measurement error was eliminated, and thus the influence of variation in N_Lens on OPL conversion was isolated. Therefore, the AL and/or LT measurement error in the SRI approach was much reduced as compared to the WMORI approach. However, without in vivo N_Lens measurement for individual eyes, the impact of variation of the N_Lens could not be solved even with the SRI approach. So, the SRI approach with the lens refractive index from the model eye provides an improvement in the biometric measurements, but, without the true N_Lens the AL and LT measurement errors could not be eliminated.

During cataract surgery planning, the IOL power needs to be determined prior to the operation, and it could be calculated with a simple formula if the optical thin-lens model was used,^11–13

$$\begin{array}{c}\hfill \mathrm{P}=\mathrm{n}/\left(\mathrm{AL}-\mathrm{ELP}\right)-1/\left(1/\mathrm{Kc}-\mathrm{ELP}/\mathrm{n}\right),\end{array}$$ (1)

where the n, Kc, and ELP represent the refractive index of aqueous and vitreous, corneal power, and the effective IOL lens position. If the IOL was not thin enough, a ray-tracing formula with the similar parameters could be used. In Equation 1, the refractive index n is relatively stable across age and individuals, and thus a constant from the model eye could be used for the calculation. With the current corneal imaging technique, such as SI, the Kc could be accurately measured, and thus directly used for the calculation. The accuracy of AL and LT measurements, therefore, become the critical factors controlling the accuracy of IOL power calculation. The LT is important because it provides an estimate of the ELP. Due to the individual variation of the N_lens and its linked errors in AL and LT measurements, as observed in this study, it is understandable that, for individual eyes, calculation error of the IOL power is unavoidable. Fortunately, as demonstrated in this study, in vivo measurement of the N_Lens were succeeded for individual patients with cataract. With the N_Lens determined in vivo for individual eyes, its linked errors of AL and LT measurements could be eliminated, and thus, a significant improvement in IOL power calculation would be expected.

Due to a relatively thicker lens in the old subjects, pupil dilation is required to image the posterior surface of the lens with the SI system. It is possible that the pupil dilation could influence LT and lens refractive index, and thus influence the measurements in this study. But the influence could be limited because the dilation effect of phenylephrine was relatively weak and the subjects were old with presbyopia. However, this is a topic that needs to be investigated for deriving accurate measurements in the future.

This study has the limitation of a small sample size, with only seven subjects. Therefore, the statistical results could not be interpreted extensively, especially on the relationship between the lens refractive index and the related influencing factors, such as the age and the refractive error. Future studies, with increased sample size, to explore the mechanism underlying the individual variation in lens refractive index among the patients with cataract as found in this study are needed. In addition, the exclusion criteria in this study were set to a clinical severity grade of greater than four. Our patients participating in this study were limited to those at a grade three or less, in order to ensure the reproducibility of the WE-OCT measurement. Future study is expected to extend the cataract to a higher severity grade.

Overall, the refractive index of crystalline lenses in the cataract eyes was measured in vivo for a group of patients with cataract using a SI-equipped WE-OCT. Substantial individual variations in the lens refractive index were revealed among the patients with cataract. The AL of the eye and LT were calculated with the measured refractive index of the lens, and also compared with those calculated with the lens refractive index from a model eye (1.41). The results indicated that lens refractive index plays an important role in deriving accurate AL and LT measurements. During cataract-surgery planning, the accurate AL and LT measurements with lens refractive index determined for individual eyes should be used for IOL power calculation.