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An optical coherence tomography system has been developed that was designed specifically for imaging the isolated crystalline lens. Cross-sectional OCT images were recorded on 40 lenses from 32 human donors with an age range of 6 – 82 years. A method has been developed to measure the axial thickness and average refractive index of the lens from a single recorded image. The measured average group refractive index at the measurement wavelength of 825 nm was converted to the average phase refractive index at 589 nm using lens dispersion data from the literature. The average refractive index for all lenses measured was 1.408 ± 0.005 which agrees well with recent MRI measurements of the lens index gradient. A linear regression of the data resulted in a statistically significant decrease in the average refractive index with age, but a simple linear model was insufficient to explain the age dependence. The results presented here suggest that the peak refractive index in the nucleus is closer to 1.420, rather than the previously accepted value of 1.406.
Accommodation is the process by which the eye changes its optical power to focus on near objects such as when reading. The power increase during accommodation is primarily due to a change in the shape of the lens. As we age, the eye gradually loses its ability to change focus for near work eventually leading to presbyopia, the total loss of near focus. Normally, this condition is easily treated with the aid of spectacles but recent interest has focused on developing surgical treatments to restore accommodation, including the implantation of accommodating intraocular lenses and lens refilling procedures designed to replace traditional cataract surgery. All surgical methods designed to restore accommodation require a thorough understanding of the mechanism of accommodation in order to be effective and many details of the optomechanical response of the lens have yet to be quantified.
One property of the crystalline lens that is important for both accommodation and presbyopia is its refractive index. It is well known that the lens has a non-uniform refractive index distribution in the form of a gradient that increases from the lens surface toward the nucleus. The index gradient contributes to the total optical power of the lens, that is, the power of the lens is greater than it would be if the lens were homogeneous and had a uniform refractive index equal to its peak value in the nucleus. Furthermore, there is evidence that the index gradient may contribute to the amplitude of accommodation (Ho et al., 2001; Garner and Smith, 1997; Pierscionek, 1990), but the exact form of the gradient and how it changes with age is still not fully understood.
Accurate measurements of the lens index gradient are of interest to anyone concerned with optical modeling of eye, more so in recent years with the development of finite-element models of the lens which depend heavily on accurate data. Most measurements however, have done little to improve on the early refractometer measurements used by Gullstrand in his “exact schematic eye” where the lens surface index is 1.386 and the “equivalent core lens” is 1.406 (Southall, 1962). These values for the central lens (nucleus) index and the surface (cortex) index have largely become accepted values in the literature (Atchison and Smith, 2000), however, the value of 1.406 that Gullstrand uses for the central index is a calculated value that fits measured data of the lens power and surface curvatures in a schematic eye model, and it is not a direct measurement of the actual central index. Nevertheless, several studies have reported measured values of the central index that are very close to Gullstrand′s “equivalent core lens” (Sivak and Mandelman, 1982; Pierscionek and Chan, 1989; Pierscionek, 1997). In another report by Pierscionek and Augusteyn (1993), the refractive index gradient was measured by laser ray tracing resulting in a central index of 1.418. This value is similar to the accepted value for the equivalent refractive index, neq = 1.42 (Atchison and Smith, 2000), which is a constant value for the refractive index of the lens based on known surface curvatures and optical power. Since the index gradient in the lens contributes to the total power of the lens, it follows that the equivalent index must be larger than the maximum value of the central index.
Many of the studies reporting refractive index measurements rely on destructive techniques or suffer from artifacts due to tissue handling and storage. Given the variability in reported results, nondestructive measurement methods may help clear up ambiguity among the reported values. Magnetic resonance imaging (MRI) was used recently to map the refractive index distribution in the isolated crystalline lens (Jones et al., 2005; Moffat et al., 2002) and in vivo (Jones et al., 2007). From the index maps, sagittal profiles were extracted and the mean central refractive index was found to be nc = 1.418 for 20 isolated lenses (Jones et al., 2005), and nc = 1.420 for in vivo measurements on 44 subjects (Jones et al., 2007), both of which are higher than the previously accepted value of ≈ 1.406. Theses studies represent the first nondestructive measurements of the index gradient that were not based on any model assumptions of the lens. However, because MRI measures the transverse relaxation rate of water protons in a protein solution, careful calibration must be performed to translate this information into quantitative measurements of the local index. It is in this respect that additional nondestructive measurements of the refractive index using other techniques will help validate existing data as well as provide new insight into the nature of the index gradient.
Optical Coherence Tomography (OCT) is a noninvasive imaging technique that records cross-sectional images of transparent tissues. OCT is based on low-coherence interferometry and thus can be used to measure the optical thickness of a sample. Since the optical thickness is the product of the refractive index n and the geometric thickness t, OCT can provide accurate measurements of the refractive index provided the geometric thickness is known. OCT has been used previously to investigate the refractive index gradient of zebrafish lenses (Rao et al., 2006; Verma et al., 2007), however both studies rely on geometric or model assumptions of the lens index that limit their applicability to quantitative measurements of the human lens index. In this study, we report the development of an OCT system designed to image the isolated human crystalline lens and describe a method for simultaneously measuring the thickness and refractive index that is independent of any model assumptions of the gradient. OCT images are recorded on isolated human crystalline lenses and from the image data, we are able to measure the refractive index directly. The results presented will be compared to those from other reports.
All eyes were obtained and used in compliance with the guidelines of the Declaration of Helsinki for research involving the use of human tissue. OCT scans were recorded on 40 intact lenses from 32 human eyes with an age range of 6 – 82 years. Whole globes were obtained from several eye banks between 1 and 4 days postmortem and the globes were delivered in sealed chambers with saline soaked gauze. The lens was extracted within 4 hours of receiving the globe by sectioning the cornea, removing the iris, and cutting the zonules.
The lens was placed in a sample chamber filled with Dulbecco’s Modified Eagle Medium with Ham’s F-12 (DMEM) solution to preserve hydration (Augusteyn et al., 2006). The sample chamber (Figure 1) was terminated by optical windows that are anti-reflection (AR) coated on one surface which is oriented towards the air side of the chamber to reduce the power of unwanted reflections. The windows permit a clear view of the lens for the scanning beam and serve as reference planes in the OCT image for the refractive index measurements. An aluminum ring of thickness 5.0 mm is used as a spacer between the anterior and posterior windows and the lens is placed on a rubber O-ring within the spacer. The diameter of the O-ring is chosen so that the lens equator rests along the circumference of the O-ring. After the lens is placed in the chamber, the anterior window is placed on the spacer to close the sample chamber and it is aligned for scanning by translating the lens such that a specular reflection is seen from the anterior surface of the lens. It is at this location that the lens axis is aligned with the scan axis. In the case of a tilted lens, it was repositioned to assure that it was normal to the beam.
The OCT system is a time-domain system that acquires 20 A-lines/s. The axial scanlength of the system is 10 mm in air (≈ 7.5 mm in solution) and was designed so that an entire human crystalline lens could be imaged in a single scan. The light source is a superluminescent diode (SLD) with a nearly Gaussian spectrum having a specified center wavelength of 825 nm, a bandwidth of 25 nm, and an output power of 6mW (SLD-38-HP, Superlum, Cork, Ireland). The system resolution is 12 µm and the sensitivity is 85 dB.
The beam delivery system uses a telecentric scanner that minimizes image distortion by producing a flat scan field. Due to the long scan depth required to image the entire crystalline lens, the depth of focus of the delivery system was chosen to nearly match the axial scanlength of the OCT system (≈ 10 mm) and the beam diameter is 60 µm. The delivery system was aligned so that the midpoint of the interferometer scandepth coincides with the beam waist of the focused beam. Images are recorded with 5000 points/line and 500 lines/image and the lateral scan length is 20 mm.
In a typical configuration, OCT records a series of adjacent axial scans (A-scans) that are used to construct a cross-sectional image of the lens. Each of the A-scans in an image is the depth-dependent intensity of reflections or back-scattering along the beam’s path. The distance between peaks in an A-scan is the optical thickness which is the product of the group refractive index (ng) and the geometric thickness (t) of the medium. Normally, one assumes a value for the group refractive index of the sample in order to measure the geometric thickness, however if the optical thickness of the empty sample chamber (without the crystalline lens) is known, both the group refractive index and the geometric thickness of the lens can be measured simultaneously from a single recorded image (Sorin and Gray, 1992; Tearney et al., 1995).
Figure 2 shows a schematic diagram of a typical OCT scan and the parameters used to perform the calculation. All distances measured from the OCT image data are optical distances (z). The lens optical thickness τ = zp – za is measured from the central A-line (S) in the image. The optical displacement δ = zw – z0 is determined by measuring the change in position of the apparent posterior window in the central scan line, with that of the real position of the posterior window in the reference scan line (R), located in the periphery of the image. The geometric thickness and group refractive index of the lens are then calculated with the following formulas derived in Appendix A:
Because the lens index is nonuniform, care needs to be taken when determining what refractive index is measured by the OCT system. Individual A-scans in an OCT image are fundamentally a measurement of the optical path length (OPL) seen by the beam. In a medium with a nonuniform refractive index distribution such as the crystalline lens, the optical path length is:
The refractive index that is measured by the system is the ratio of the optical path length to the geometric path length, OPL/t. Substituting this into the above relation, we find that the measured group index is
From the mean value theorem, one can recognize that the measured refractive index is simply the average value of the index distribution seen by the beam. Recalling that the OCT system uses a broadband source and therefore records the group refractive index (ng), the average group refractive index of the lens at the measurement wavelength of λ = 825 nm will be referred to as g.
Since the OCT system uses a broadband source, the refractive index that is seen is the group refractive index ng, rather than phase refractive index n that would be seen if the light source were monochromatic. In addition, the central wavelength of the light source is in the near-infrared at λ = 825 nm. When comparing refractive index measurements, it is more convenient to work with a common wavelength, so the measured index values were converted to a wavelength of 589 nm using lens dispersion data from the literature (Atchison and Smith, 2005). First, the measured group index was converted to the phase index using the following relation (Saleh and Teich, 1991):
Here, ng(λ0) is the group refractive index at the measurement wavelength of λ0 = 825 nm, n(λ0) is the phase refractive index, and dn/dλ = −1.8 × 10−5 nm−1 is the slope of the dispersion curve of the crystalline lens at λ0 calculated from the Cauchy equation coefficients in Table 5 of Atchison and Smith (2005).
The phase index at 825 nm was converted to 589 nm using a constant scaling formula derived by Atchison and Smith (2005):
Here, λ1 = 825 nm is our measurement wavelength and λ2 = 589 nm is the wavelength at which we are interested in calculating the new index. The refractive index (n) in the scaling formula is calculated from the lens dispersion curve. These are used to calculate the average refractive index (λ2) based on the measured average index (λ1).
OCT scans of three isolated human lenses are seen in Figure 3. In each of the images, the entire lens is visible, including a clear view of the anterior surface, the posterior surfaces and the lens equator. The semi-circular structures near the lens equator are the cross-sectional views of the O-ring supporting the lens. The bright white lines anterior and posterior to the lens are the sample chamber windows which also produce weaker multiple reflections that are seen as faint horizontal lines in the image. In each of the images, the posterior window surface appears distorted beneath the lens due to the increased optical path length through the lens. The apparent location of the anterior surface of the posterior window is used to measure the axial refractive index.
The beginning of cortical cataracts are seen in 3A and 3B. In Fig. 3B the lens capsule has been damaged near the equator, leading to a capsule separation that can be seen near the anterior pole. Of the 32 lenses scanned, 6 suffered from capsule separations, as previously reported by Augusteyn et al. (2006). For these lenses, the optical thickness was measured from the images by marking the distance from the anterior to the posterior cortical surface, ignoring any space between the cortex and anterior or posterior capsule. The minimum and maximum distances between the cortex and capsule found for the lenses with capsular separations were 170 µm and 470 µm, respectively. In Fig. 3C, a clear internal boundary can be seen near the entire circumference of the lens, which may be due to a separation within the lens cortex. It is important to note that none of tissue preparation artifacts appeared to affect the refractive index measurement when the lens thickness was taken from the cortical boundaries.
The thickness and group refractive index of the lenses was measured from the recorded OCT images according to the description in section 2.3. Figure 4 shows two A-lines of data taken from a typical OCT scan of an isolated human lens. The upper trace is the sample scan from the central A-line in the image. In it, the lens anterior and posterior surface boundaries are marked za and zp, and the posterior chamber window is marked zw. The lens optical thickness τ = zp–za is measured by reading the cursor locations of the anterior and posterior surface reflections. The bottom trace is the reference scan from a line of data in the periphery of the OCT image that contains the reflections from the sample chamber windows and is unobstructed by the lens. The optical displacement of the posterior sample chamber window δ = zw – z0 is measured by reading the cursor locations of the real window location z0 in the reference scan and the apparent window location zw in the sample scan. The geometric thickness t and average group refractive index g at the measurement wavelength (λ = 825 nm) are then calculated according to Eq. (1 and Eq. 2).
The measured lens thickness and refractive index data are listed in Table 1. For each of the lenses scanned, the average group refractive index g(λ1) is converted to the average phase index (λ1) using Eq. (5). The average phase index at λ1 = 825 nm was then converted to the average refractive index at λ2 = 589 nm using Eq. (6). The average and standard deviation was calculated for all lenses scanned.
The measured average index values for the eyes scanned are plotted as a function of age in Fig (5). A linear regression of the data resulted in the relation = 1.415 – 0.000159 × Age (years) (R2 = 0.32, P < 0.001). The slope of the regression line (−0.000159 year−1) and the significance (P < 0.001) suggest that there is a statistically significant decrease in the average refractive index with age, but the low regression coefficient (0.32) suggests that a simple linear relationship does not explain the age dependence sufficiently, and that a different model is more appropriate.
The axial lens thickness as a function of age is plotted in Fig. (6). A linear regression of the data gave the relation t = 4.19 mm + 0.0064×Age (mm/year) (R2 = 0.10, P < 0.05). The increase in lens thickness with age is consistent with models of lens growth, however a simple linear model is not sufficient to explain the age dependence. For younger lenses (< 60 years), the thickness appears constant with a high degree of scatter. For older lenses (> 60 years), there appears to be a sharp increase in the thickness.
Accurate refractive index measurements of the crystalline lens are important for the development of surgical treatments for presbyopia. Numerical modeling of the lens is being used more commonly to study the dynamics of accommodation and the effects of aging during presbyopia and these modeling methods rely on accurate measurements of the static optical properties of the lens. The OCT images of the isolated lenses were used to calculate the average value of the refractive index profile along the optic axis. These measurements resulted in a mean value of = 1.408±0.005 for all of the eyes scanned in the age range 6 – 82 years. To the best of our knowledge, there is only one published report with measurements of the group refractive index of the crystalline lens with which we can compare our results. Drexler et al. (1998) used multiple wavelength low-coherence interferometry to measure the group dispersion (Δng/Δλ) of the crystalline lens and found g = 1.4055 at λ = 855 nm which corresponds to an average phase index of = 1.397 at λ = 589 nm (see sec. 2.3). However, their method of calculating the average group index (g) from the measured optical thickness involved using the dispersion of water instead of the crystalline lens, which can introduce significant errors.
The precision of the group refractive index measurement is limited by the axial resolution of the OCT system. Assuming we can reliably locate the peak intensity from a given surface within ±1 µm, we can estimate our group index measurement precision to ±0.0002. However, the conversion from group to phase index introduces two possible sources of error from the uncertainty in the central wavelength of our light source and in the value for the dispersion of the crystalline lens (see Eq. 5). Our light source has a nearly Gaussian output spectrum with a specified central wavelength of 825 nm. We measured the spectrum with a spectrometer and fit the data with a Gaussian function which calculated a central wavelength of 828 nm. Assuming an error of 5 nm in the determination of the central wavelength, this only impacts the calculated index by < 0.0001, which is below our group index measurement precision. The value for the dispersion of the lens (dn/dλ) that was used in the calculation represents the best estimate available from a review of published literature (Atchison and Smith, 2005). However, there is significant variation in published lens index data, due in part to the complexity of the lens index gradient. Assuming the published value is accurate within 20 %, the error in the phase index calculation is within ±0.001. If however, the dispersion is off by as much as 50 %, then error in the phase index may be as much as ±0.004.
The average refractive index measured with OCT was compared to the results of the MRI measurements from Jones et al. (2005) by integrating the sagittal index profiles and dividing by the axial thickness of the section to calculate the average value of the index profile. This results in a mean value of = 1.411 ± 0.006 for 21 lenses over the age range of 7 – 82 years. No significant age dependence was found (P ≈ 1.0) due to the large variability. However, there is good agreement between the OCT and MRI measurements; the mean average index over similar age ranges is well within the standard deviation of each technique. This would appear to validate the technique of using MRI to measure the refractive index distribution of the crystalline lens. Furthermore, the results from both the OCT and MRI measurements suggest that the peak refractive index in the nucleus is closer to nc = 1.42 as suggested by Jones et al. (2005) and Augusteyn et al. (2008), rather than the more commonly accepted value of ≈ 1.406 since the average value of the index profile (≈ 1.41) must be lower than the peak value.
A statistically significant decrease in the average index with age was found (−0.000159 year−1, P < 0.001). However, the age related changes in the average index are small compared to the precision of the measurement technique. If the slope of the regression line is assumed to represent the rate of change, over an entire life span (≈ 80 years), the total change in the average refractive index will only be ≈ 0.01. The precision of the refractive index measurement is limited by the axial resolution of the OCT system and the accuracy of the lens dispersion data, leading to an estimated precision of ±0.001. A very large sample size evenly distributed over the entire age range would therefore be required to determine a better model of the age dependence.
The increase in lens thickness with age seen in Fig. (6) is consistent with previous measurements of the isolated lens thickness, including the MRI measurements by Jones et al. (2005), which reported an age dependent thickness of t = 4.77 mm + 0.0042 × Age(mm/year). The lens thickness growth rate measured with OCT (0.0064 mm/year) is similar to the growth rate measured from the MRI imaging (0.0042 mm/year). It is also consistent with previous photographic measurements of the isolated lens by Rosen et al. (2006) who found a growth rate of (0.0123 mm/year), and by Glasser and Campbell (1999) who could not establish a statistically significant growth rate, but did note an increasing trend in the lens thickness with age.
One concern when performing measurements on ex vivo tissue is whether its properties have been affected by storage and preservation. Isolated crystalline lenses have a tendency to swell, or absorb water when stored too long before experimentation and fluid tends to pool in ”discreet lakes” located near the capsule and around the axis (Augusteyn et al., 2006). Indeed, of the 40 lenses measured in this study, 6 had visible capsule separations that clearly had fluid trapped between the capsule and the cortex. This raises a concern over whether the lens swelling had any effect on the measured refractive index. For each lens measured, the optical thickness that is used to calculated the refractive index was always chosen from the anterior cortical boundary to the posterior cortical boundary, neglecting the capsule and the fluid surrounding the lens cortex. This assures that the refractive index that is measured is due to the lens cell mass and is not influenced by possible tissue swelling. Since the capsule is believed to have a refractive index that is nearly equal to the surrounding aqueous or vitreous (Smith, 2003), the capsule’s contribution to the optical thickness of the lens in negligible. Furthermore, fluid trapped in the capsular space has the same refractive index as the surrounding medium, therefore it does not contribute to the measured optical thickness of the lens, and as such, has no measurable effect on the refractive index.
The question remains of how measurements of the average refractive index relate to the development of presbyopia. It appears that as the human lens ages, the refractive index forms a distinct plateau in the central region (Jones et al., 2005; Augusteyn et al., 2008). Early in life there is a gradual change in the refractive index from the cortex to the nucleus. As new layers of protein are laid down in the cortex, the refractive index in the nucleus increases due to tissue compaction (Augusteyn, 2007) until the peak refractive index is reached in the center of the lens. At that point, the refractive index in the surrounding layers increases until a large, optically homogeneous region of constant index is formed. This central region with constant index (plateau) grows faster than the overall lens thickness until the age of about 60 years (Augusteyn et al., 2008). As the refractive index in the cortical layers is increasing during the plateau formation, the overall lens thickness is increasing as well, so the net result is that the average value of the index profile remains nearly constant through the age of about 60 years. This can be seen in the data in Fig. 5. After the age of about 60, the index plateau has been fully formed but the total lens thickness continues to increase, thereby decreasing the average value of the refractive index profile as the age increases. This suggests that some form of a biphasic model is more appropriate to model the age dependence of the average index.
Other studies of the crystalline lens support this picture as well. In a study of 100 isolated human lenses with an age range of 6–94 years, Borja et al. (2008) measured the surface powers and equivalent refractive index using shadowphotography and lens power measurements. It was concluded that the contribution of the index gradient to the total lens power decreases until the age of about 60 years, at which point it is practically constant. This is consistent with the index plateau model since around the age of 60 years the plateau has fully developed and occupies the majority of the axial lens thickness, leaving only a small region of lower index cortical material. Viewed another way, the young lens could be thought of a single ‘thick’ lens (the nucleus) with high index surrounded by several ‘thin’ lenses (cortical layers) of decreasing index. Each of the thin lenses adds refractive power, but with increasing age, there are fewer refracting thin lenses to contribute to the total lens power. It is interesting to note that the relatively recent evidence supporting a refractive index plateau is consistent with Gullstrand’s original observations that a reflex from the lens nucleus can only be observed beyond the age of about 30 years since a sharp increase in the index gradient is required to create the reflex.
In this study, we report the development of an OCT system capable of imaging the crystalline lens with sufficient quality to visualize the formation of cataracts, a clear nuclear region, capsular separations, as well as other internal structures. Individual A-lines in the OCT images were used to calculate the axial average refractive index of 40 human lenses. From the results presented here, the following conclusions can be drawn:
The authors are grateful to Robert C. Augusteyn (Institute for Eye Research, Sydney, Australia) for many helpful discussions and James M. Pope for providing the MRI data used in the analysis. Thanks to Esdras Arrieta and Adriana Amelinckx (Bascom Palmer Eye Institute, Miami, FL) for surgical support, and Rakhi Jain (Advanced Medical Optics, Santa Ana, CA) for providing donor eyes. Donor human eyes were provided by the Florida Lions Eye Bank, Lions Eye Bank of Oregon, Lions Medical Eye Bank (Nor-folk, VA), Lions Eye Institute for Transplantation and Research Inc. (Tampa, FL), Illinois Eye Bank, Alabama Eye Bank, Old Dominion Eye Foundation Inc. (Richmond, VA), North Carolina Eye Bank, Utah Lions Eye Bank, and the North West Lions Eye Bank (Seattle, WA). This research was supported in part by a Ruth L. Kirschstein National Research Service Award (NIH F32 EY15630, S. Uhlhorn), the Australian Federal Government’s Co-operative Research Centre Scheme through the Vision CRC, NIH grants F31 EY15395 (D. Borja), R01 EY014225, and P30 EY014801, an unrestricted grant from Research to Prevent Blindness, and the Henri and Flore Lesieur Foundation (J.-M. Parel).
Referring to the geometry and nomenclature of Figure 2, the lens with group refractive index ng is immersed in a solution with index n0. All of the distances (z) measured from the image data are optical distances and the top of the image is z = 0. A reference line (R) provides the location of the undistorted anterior surface of the posterior window z0. The sample scan line (S) in the image includes the crystalline lens and the anterior surface of the distorted posterior window at location zw. The anterior and posterior surfaces of the lens are za and zp and the geometric thickness of the lens is t. The geometric distance between the anterior lens surface and the top of the image (z = 0) is ta and from the posterior lens surface to the posterior window is tp.
Since the geometric thickness of the sample chamber is fixed for all lines in the image, the geometric thickness of the reference scan line tR = z0/n0 is equal to the geometric thickness of the sample scan line tS, giving
The geometric distances in the sample scan lines are
Substituting ta and tp into Eq. (A.1) we calculate the lens geometric thickness,
Substituting ta, tp, and t into Eq. (A.1) we calculate the group refractive index of the crystalline lens,
If we define the optical thickness of the crystalline lens as τ = zp – za and the optical distance between the real and apparent locations of the reference window as δ = zw – z0 and we substitute these into Eq. (A.2 and Eq. A.3) we arrive at the following simplified expressions for the lens thickness and refractive index:
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