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Abstract
We propose a new method of determining the optical axis (OA), pupillary axis (PA), and visual axis (VA) of the human eye by using dual-depth whole-eye optical coherence tomography (OCT). These axes, as well as the angles “α” between the OA and VA and “κ” between PA and VA, are important in many ophthalmologic applications, especially in refractive surgery. Whole-eye images are reconstructed based on simultaneously acquired images of the anterior segment and retina. The light from a light source is split into two orthogonal polarization components for imaging the anterior segment and retina, respectively. The OA and PA are identified based on their geometric definitions by using the anterior segment image only, while the VA is detected through accurate correlation between the two images. The feasibility of our approach was tested using a model eye and human subjects.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The human eye is an optical system composed of a cornea, an iris, a crystalline lens, and a retina. It is similar to a camera but not rotationally symmetric. The human eye has three axes: a visual axis (VA), an optical axis (OA), and a pupillary axis (PA). The VA is a path that extends from the fixation point to the fovea, and the OA and PA are related to the geometrical structure of the eye. The OA is a unique line joining the centers of curvature of the optimal surfaces, and the PA is a line perpendicular to the surface of the cornea and passes through the center of the pupil [1]. In terms of anatomy, the optical components of the human eye are aligned along the OA; however, the human views objects along the VA, which is tilted at an angle of approximately 5° with respect to the OA [2]. The angle between the VA and OA is “α,” while the angle between the VA and PA is “κ” [1]. Figure 1 presents a cross-sectional schematic of the human eye in which the three axes and two angles are indicated.
Fig. 1 Cross-sectional schematic of the human right eye (top view). α is the angle between the VA and OA. κ is the angle between the OA and PA.
Accurate measurement of angles and axes can be used to improve refractive surgeries such as cataract surgery and laser-assisted in situ keratomileusis (LASIK) surgery. These parameters allow for quantification of the position and shape of intraocular lens (IOL). Misalignment of the IOL and subsequent decentration of the entrance pupil can introduce a variety of optical aberrations (coma, astigmatism, achromatic and spherical aberration and etc.) and a degradation in visual acuity. The angle kappa is also clinically relevant as a large angle may lead to alignment errors during photoablation in laser refractive surgery or to lens de-centration in intraocular refractive surgery [3, 4].
Optical coherence tomography (OCT) is one of the most modern optical imaging technologies and provides cross-sectional images of biological tissues by measuring light backscattered from boundaries because of changes in the refractive index [5–7]. OCT is widely used in ophthalmology to diagnose ocular diseases and obtain quantitative information about the eye.
Recently, various OCT studies that image both the entire anterior segment and retinal area simultaneously (i.e. whole-eye OCT imaging) have been reported [8–12]. Dai et al. and Fan et al. combined two spectral-domain OCT (SD-OCT) systems for whole eye imaging [8, 9]. Ruggeri et al. developed a whole eye SD-OCT system which has three reference arms with different optical path lengths [10]. Grulkowski et al. obtained a whole-eye image using a single beam from a vertical-cavity surface emitting laser (VCSEL) with long coherence length [11]. Kim et al. presented a SD-OCT system employing dual illumination configuration with orthogonal polarization and interlaced detection by a single spectrometer and an optical switch [12]. However, these systems measure only the eye dimensions [9–12], without discussing eye axes and angles. Ortiz et al. showed that the OA and the PA could be identified only with the full anterior segment image, but the VA could not be measured in this way [13].
In this study, we developed a 1060 nm swept-source OCT (SS-OCT) system containing a single source that cannot only record dual OCT images (anterior segment and retina) simultaneously but also provide biometric data. Based on the developed system, we employed an OCT-based method to estimate the three axes in the eye, as well as the “α” and “κ” angles. To the best of our knowledge, this is the first study to use an OCT system for determining the three axes and two angles in the eye and providing a full in vivo quantification of the subjects.
2. Experimental setup and method
2.1 System configuration
Figure 2 depicts the schematic of the dual-depth SS-OCT system used to obtain cross-sectional whole-eye images; it employs two light beams with orthogonal polarization states from one source, similar to a previously described setup [12, 14, 15]. A commercial swept-source laser (HSL-10-100; Santec Corp., Aichi, Japan) centered at 1060 nm was employed in this system and operated at a sweep rate of 100 kHz. The output power of the source was 25 mW. Before reaching the fiber-based interferometers, the light was split by using a 99:1 fiber coupler (FC1; Thorlabs, USA). Then, 1% of the light was coupled through fiber Bragg grating (FBG; λc = 1063.57 nm, Δλ = 0.13 nm) to generate a stable-phase A-scan trigger signal, used in place of an A-scan trigger from the source, for removing fixed-pattern noise [16, 17]. The remaining 99% of the source light was separated and sent to two Mach–Zehnder interferometers (MZIs) to image the anterior segment and retina. Fiber couplers (FCs) 3 and 4 apportioned the light to the reference and sample arms. In the sample arm, the beam from the single source was divided into two orthogonally polarized beams by using a polarization beam splitter (PBS1). These orthogonally polarized beams illuminate the human eye by traveling along different paths [12, 15]. The horizontal (p-polarized) component, after being reflected by a 2D galvanometer mirror (GM), was collimated by two lenses (L1 and L3, with focal length f = 75 mm) and was focused on the retina by the optics of the eye. The vertical (s-polarized) component, after being reflected by the 2D GM, passed through L1, L2, and L3 and was focused onto the iris plane to image the entire anterior segment. For L2, a focal length of f = 150 mm was used to reduce the lateral scanning range for anterior segment imaging because at the image plane, after L1, 30 mm are scanned to obtain a large scanning range at the retina. Each beam entered the appropriate MZI after being reflected from the anterior segment or retina, and interference signals containing depth information were generated at FCs 5 and 6. In the reference arm, a frequency shifter (FS; Brimrose Corp., USA) was employed to extend the imaging depth and enable imaging of the entire anterior segment from the top surface of the cornea to the posterior surface of the crystalline lens [18]. It shifted the frequency of the reference light for the anterior segment to 300 MHz. High-speed balanced photodiode detectors (BPDs 1 and 2; Thorlabs, USA) were used to detect the interference signals. The bandwidth of BPDs 1 and 2 are 400 MHz and 1.0 GHz, respectively. A 12-bit and 1.8 GSPS digitizer (ATS9360; Alazar Technologies Inc., Quebec, Canada) was used to capture and digitize two interference signals by using channel A and B ports.
Fig. 2 Schematic of the dual-depth SS-OCT system, which consists of two interferometers. PD: photodiode detector, L1–L3: lenses, PBS: polarization beam splitter, GM: galvanometer, BPD: balanced photodiode detector, FC: fiber coupler, FS: frequency shifter.
To minimize eye motion, the image of the fixation target was delivered to the retina. The fixation target is composed of a laser diode delivering 660 nm beam to a single mode fiber. The diverging beam from the fiber is then collimated and passes through an iris which controls the target size and the amount of light delivered to the eye. The two beams used to image the anterior segment and retina and one beam from the fixation target were aligned coaxially to reduce the errors in the measurements of the eye axes and angles. The details are provided in Section 2.3 and 3.2.
2.2 Image-distortion correction
To determine the eye axes in the 2D OCT images accurately, the cross-sectional images should not contain distortions. To measure the various axes of the eye, which is to be determined in three-dimensional (3D) space, we recorded horizontal and vertical cross sections of the eye. The axes from each image are projections of the 3D axes onto two orthogonal planes. Because the back-focal plane of L1 is only conjugated to the horizontal axis galvanometer, only the horizontal scan is recorded in a telecentric configuration, while the vertical scan deviates from this configuration (non-telecentric scan). Thus, the cross-sectional image recorded with the vertical scan will be distorted. To correct for this distortion, we imaged the surface of a mirror that was located at different imaging depths. The depth was changed with a step size of 1mm. Then, we estimated the distance between the scanning pivot point and each mirror position to calculate the angle from the center A-line to all pixels. Finally, the distorted images that have the angular scanning pattern are polar-transformed to new coordinates without the distortion. The full details of this technique are explained in [19].
Also, OCT images can be distorted in two ways because of the refractive indices of the eye components: 1) the images may be enlarged in the axial direction, and 2) the scanning light may be refracted at the round corneal surface [19, 20]. The minor distortion that may have occurred at subsequent boundaries with small changes in refractive indices were not considered in our correction. To compensate for the distortions, we implemented the following image-processing algorithm. First, we fitted the outer surface of the cornea to a 6th polynomial equation after detecting the surface [13]. The incidence angle of the scanned beams at the outer surface of the cornea were calculated. The beam is incident on the eye parallel to the optical axis of the sample when performing telecentric scans. In case of the non-telecentric scans, we could calculate the incidence angle with the angular scanning pattern obtained in the method mentioned previously. Then, the refraction angles of the scanned beams were calculated at all pixels on the corneal surface based on the Fermat’s principle which describes the path taken by a light ray. According to this principle, each point from the object space corresponds to a unique incident point in the image space, so that the optical path difference is minimized (with the change of refractive index at the boundary being accounted for). The corresponding point in the image space must be found along the path of the refracted beam and therefore the light intensity of all the image points can be corrected through interpolation. An in-depth review of this technique can be found in [20].
To reconstruct the whole-eye image, we measured the distance in air between the zero-delay lines of the anterior segment and retina images and calculated the distance in the eye by using previously found refractive indices of the eye [21].
2.3 Image-processing methods for locating eye axes in 2D images
- (1) VA: The VA is a path that extends from the fixation point to the fovea. To identify the VA from the OCT image, two beams for imaging the anterior segment and retina and one beam from the fixation target are aligned coaxially, while the 2D GM is fixed at the center position. Under this condition, if a human volunteer gazes at the fixation target, three beams are simultaneously delivered to the fovea, which forms the beam path through the VA in either a telecentric (Fig. 3(a)) or non-telecentric (Fig. 3(b)) scanning methods. These two different scanning configurations are employed for transverse and sagittal imaging, respectively, which are described in Section 2.4. Therefore, the VA is a center A-line in an OCT image, since the 2D GM scans symmetrically from the center position. In reality, the fovea is located slightly off the center pixel due to several reasons, including poor subject gaze fixation, which will be discussed in the Discussion section. The decentration of the fovea in an OCT image is less than 200 μm in most cases.
- (2) OA: The OA of the eye can be estimated by using a “best-fit line” through the centers of curvature of the refractive surfaces [22]. We developed an algorithm that could be employed to find this best-fit line for the eye optics by using edge defects and circular fits of the anterior and posterior surfaces of the cornea and lens. The circular fitting process is based on the random sample consensus algorithm. In this process, at least three points that do not lie along a line are identified first. Then, the center and radius of the circle are set, and the best-fit circle is calculated. Finally, the best-fit line through the centers of the best-fit circles of all the optical components is defined as the OA of the eye (Fig. 3(c)).
- (3) PA: The PA of the eye is perpendicular to the line tangent to the cornea and passes through the center of the entrance pupil [23]. The outer surface of the cornea is fitted to a sixth-degree polynomial. Then, the equations of the lines that are normal to the outer surface are estimated. The PA is determined by identifying the line that passes through the center of the pupil (Fig. 3(d)).
Fig. 3 Conceptual diagrams of the algorithms used to define the VA, (a) using a telecentric and (b) non-telecentric scanning configuration; (c) the OA, which was determined by fitting a line through the centers of the best-fit circles of the anterior and posterior corneal surfaces (C1 and C2, respectively), and the centers of the best-fit circles of the anterior and posterior lens surfaces (L1 and L2, respectively); and (d) the PA, which is perpendicular to the cornea surface and passes through the center of the pupil.
2.4 Image processing in 3D images
The axes and angles acquired as described in Section 2.3 exist in the 2D domain. However, the axes in the real eye are in the 3D domain. Hence, the data obtained using the methods described in Section 2.3 are projections onto the transverse plane. To locate these axes in 3D space, we merged the projections of these axes onto the sagittal and transverse planes, which were acquired by performing beam scanning in two orthogonal directions. Specifically, the transverse plane image, which included the vertical line passing through the fovea that was defined as the VA, was obtained by performing horizontal scanning. Similarly, the sagittal plane image, which also contained the VA, was obtained by performing vertical scanning. After acquiring the representative images, we applied axis-finding algorithms to the images and located the three axes. To find the merged axes in 3D space, the plane images were matched using the VA as a reference. Figure 4 illustrates the process of summing two vectors, which could be two PAs or two OAs, in the transverse and sagittal planes [25].
Fig. 4 Schematic of axis summation in 3D space. The axis vectors (blue arrows) are the projections of an axis onto the transverse and sagittal planes, while the red axis (red arrow) is the sum of the axis vectors in 3D space.
2.5 Imaging protocol
We applied our technology in the left eyes of 10 subjects with a visual acuity of 0.6 and above. The subjects were in the mean age range of 28.60 ± 2.33 years. The Institutional Review Board at the Korea University (1040548-KU-IRB-16-150-A-2) approved this study for the normal eye. All measurements were performed according to the tenets of the declaration of Helsinki and informed consent were received from all subjects prior to measurement.
3. Results
3.1 System performance
The system performance was evaluated using a gold mirror and a −23.12 dB neutral-density filter at the sample position. In the sample arm, the optical power used to obtain the anterior segment and retina images were 1.8 mW and 0.5 mW, respectively, which satisfied the ANSI Z136.1 (2007) safety limits. The details regarding power safety are provided in Section 4.
Full anterior segment imaging requires high sampling rate up to 1.2 GSPS because we use a frequency shifter and collect 4096 sampling points per one A-line. In this way, we obtained an imaging depth of 15.17 mm and doubled the −10 dB depth range from 6 mm to 12 mm [18]. The axial resolution and the signal to noise ratio (SNR) were 10.58 μm and 92.33 dB at a 150 μm depth position in the air, respectively. The axial resolution is larger than the theoretical axial resolution of 4.96 μm possibly due to some errors during k-linear interpolation and dispersion mismatch between the sample and reference arms.
For retina imaging, the sampling rate and points were the same as used for the anterior segment. However, the imaging depth and −10 dB depth range were 7.59 mm and 6 mm because the frequency shifter is not necessary for retinal imaging. The axial resolution was also 10.58 μm and the SNR was 90.28 dB.
3.2 System validation
To validate the accuracy of our method described in Section 2.3, first, we measured the transverse beam profiles of multiple beams at various locations in the sample arm. Secondly, the angle “α” were measured using a model eye oriented in specific directions.
Even after accurate measurement of the VA, several other sources might introduce errors in this technique. Three beams, including the two beams from the OCT source and a beam from the fixation target, need to be well-aligned coaxially. To confirm this, we utilized a beam profiler (WinCamD-UCD15-1310; DataRay Inc., CA, USA) to carefully monitor how the three beam shapes were changed in the range of −20 mm to + 30 mm that spanned the axial eye dimension. Figure 5 shows the transverse beam profiles of all three beams superimposed at 10 mm intervals. Beam shapes and sizes of the two beams for the retinal imaging and from the fixation target were similar and we made sure that they were well aligned coaxially. The beam for the anterior segment imaging was focused in Fig. 5(c) and defocused at other locations. The three beams were found to be aligned well coaxially. However, we could not confirm their alignment at large distances because the beam for the anterior segment was rapidly defocused after reaching the focal position. A slight misalignment of these beams would result in measurement errors in angles “α” and “κ.” It is also possible that chromatic aberration in the crystalline lens may negatively affect our measurement since the beam wavelength from the fixation target is in the visible range of 660 nm, which is distant from the OCT source wavelength.
Fig. 5 Transverse profiles of the three beams, including the two beams from the OCT source and a beam from the fixation target, at various axial positions in the sample arm. The red arrow indicates an OCT beam for the retina and a beam from the fixation target which are similar in size and shape. The yellow arrow indicates an OCT beam for the anterior segment imaging. Beam profiles were measured at (a) −20 mm (b) −10 mm, (c) 0 mm, (d) + 10 mm, (e) + 20 mm, and (f) + 30 mm from the focal position. The negative distance implies a direction closer to the objective lens and a positive distance implies a distance farther from the objective lens.
We measured the OA and the angle “α” using a model eye. The anterior chamber of the model eye consists of contact lenses, an intraocular lens, and an acrylic chamber filled with water. The model eye was rotated in the range of −5.75° to 5.75° using a rotation stage with a step size of 1.15°. We assumed that the VA of the model eye was fixed as the vertical line in an OCT image from which the “α” angle was measured. The image distortion of the model eye was corrected in a manner similar to that in the real human eye.
Figure 6 shows the measured angle “α” of the model eye, which was rotated with a 1.15° step size. As expected, the anticipated and measured values were well matched, and the difference between adjacent measurements was just 1.15° ± 0.02°.
Fig. 6 Measured angle “α” of the model eye at various orientations and differences between adjacent measurements
3.3 Distortion correction and whole-eye image reconstruction
Figures 7(a) and 7(b) present the simultaneously obtained anterior segment and retina images, respectively, of a human volunteer’s left eye.
Fig. 7 Simultaneously obtained (a) anterior segment and (b) retina images of a human volunteer’s left eye, each of which is the average of 30 images. (c) Whole-eye image reconstructed using (a) and (b). The distortions caused by the refractive indices of the eye and refraction at the corneal surface are corrected in the whole-eye image.
The images shown in Figs. 7(a) and 7(b) were each obtained by averaging 30 images and consist of 2048 (axial) × 1000 (lateral) pixels, although they were cropped axially. The lateral scanning ranges of the anterior segment and retina images are 15 mm and 7.5 mm, respectively. Figure 7(c) is the whole-eye image that was reconstructed using Figs. 7(a) and 7(b). All the distortions caused by the refractive indices of the eye and refraction at the corneal surface are corrected in the whole-eye image.
3.4 Evaluation of three axes and two angles in two orthogonal planes
As demonstrated in Section 2.4, the VA, OA, PA, and the angles between them are first estimated in two orthogonal planes, which are later combined to reconstruct three-dimensional structures. Figures 8(a) and 8(b) present the 2D whole-eye images obtained in transverse and sagittal planes, respectively. In both figures, the VA, OA, and PA are labeled, and appear as green, red, and blue lines, respectively. The VAs are the vertical lines passing through the fovea, and the OAs are the best-fit lines obtained by performing linear regression using the centers of the best-fit circles of the surfaces of the optics in the anterior segments of the eye (anterior and posterior surfaces of the cornea and crystalline lens). The anterior segment and the retina images obtained in the transverse and sagittal planes consist of 2048 (axial) × 1000 (lateral) pixels. However, the retina images were cropped axially and the left and right side of the anterior segment images obtained in the sagittal plane were cropped due to the eyelids. All images are the average of 30 images. The acquisition time of 60 images (30 images for the anterior segment and the others for the retina) is 0.6 seconds.
Fig. 8 Three axes of the whole eye for 26-year-old volunteer (a) in the transverse plane and (b) in the sagittal plane.
3.5 Full biometry in three-dimensional whole-eye images
Figures 9(a) and 9(b) represent the side-view and top-view of the 3D whole-eye images of a 26-year-old volunteer of Fig. 8. We reconstructed and rendered Figs. 9(a) and 9(b) using Amira software (FEI Visualization Sciences Group, USA). The VA, OA, and PA in 3D were drawn by vector-summing two projected axes on the transverse and sagittal planes, as described in Section 2.4.
Fig. 9 3D whole-eye images of the left eye of a 26-year-old volunteer, containing the VA, the OA, and the PA. (a) side view (see Visualization 1) and (b) top view.
The average α and κ measured in the transverse plane were 5.60° ± 1.32° and 3.11° ± 1.79°, respectively. In the sagittal plane, the average α and κ were 4.36° ± 1.04° and 2.22° ± 1.26°. In full three-dimensional space, the average α and κ are measured as 7.21° ± 0.96° and 4.08° ± 1.66°, respectively. Figure 10 shows the box-whisker plot of α and κ in 3D space.
Fig. 10 Box-whisker plot of α and κ from 10 subjects. Box plot represents interquartile range (25%–75% percentiles) and whisker plot represents min and max values. In each box, the longitudinal bars and squares represent the median and the average values, respectively. All data are plotted as diamond shapes.
Also, we measured central corneal thickness (CCT), anterior chamber depth (ACD), lens thickness (LT), and axial eye length (AEL) of the left eye for 10 subjects. The average and standard deviation of CCT, ACD, LT, and AEL were 0.48 mm ± 0.05 mm, 2.96 mm ± 0.31, 3.69 mm ± 0.38, and 25.22 mm ± 0.87 mm, respectively (Fig. 11).
In literature, α is known to be approximately 5° and 2° in the transverse and sagittal planes, respectively [2]. However, κ varies by age, gender, race, axial eye length, and country [26]. In addition, κ is smaller in myopes and larger in hyperopes [26, 27]. Hwang et al. and Yeo et al. reported that Korean κ is approximately 3–4° [28, 29]. Our results demonstrate that the current approach produces a reasonable outcome with good agreement to previous reports.
To certify our data, we measured the repeatability of our system. We examined the left eye of each of the 3 subjects. Each eye was examined 10 times with a full 1-minute rest between examinations (i.e. subjects were asked to sit back and withdraw their head from the chin rest). Also, the position of the sample arm was re-initialized at every examination. In each examination, 30 images of the whole-eye were obtained and analyzed to measure full biometry. Also, we tried to ensure that the position and slope of the anterior segment and the retina were the same for high repeatability. We measured the repeatability of our measurements of α, κ, CCT, ACD, LT, and AEL, as shown in Table 1. α and κ were measured in the transverse plane. Outliers that may have been caused by obvious involuntary eye movement were excluded. Table 1 demonstrates that full biometry measured with the dual-depth whole-eye OCT system were shown to be repeatable and trustworthy.
Table 1. Repeatability of full biometry using dual-depth whole-eye OCT system
4. Discussion
We developed a new method of determining the OA, PA, VA, and values of α and κ using a whole-eye OCT system. Our results show high accuracy in measuring these values in a model and human eye. In addition, the system showed a good repeatability of the measured parameters in human subjects.
Whole-eye imaging techniques have been published previously by many groups. Most recently, Kim et al. reported on a dual-depth, whole-eye, spectral-domain (SD)-OCT method involving interlaced detection. This technique can be used to obtain high-resolution whole-eye images, but the anterior segment images acquired using this method exhibit weak signals at the top surface of the cornea and posterior surface of the crystalline lens because of the signal-to-noise ratio (SNR) fall-off that results from the intrinsic limitations of SD-OCT [12]. Therefore, in this study, we developed a dual-depth SS-OCT technique using an FS to overcome the limitation of SNR fall-off and extend the imaging depth. Furthermore, we utilized two digitizer channels to acquire and process interferograms from the anterior segment and retina simultaneously. This method enables a reliable measurement of the full biometry of the eye.
Since multiple beams are needed for the proposed technology, we have to ensure that the radiant exposure is within the safety limits. According to ANSI Z136.1 (2007), the sample beam used to image the anterior segment is an extended source, while that used to image the retina is a point source because of the angle subtended at the cornea. The maximum permissible exposures (MPEs) for 1060 nm extended and point sources were calculated to be 107.78 mW/cm2 and 5 mW/cm2, respectively, when we assumed that the exposure time was longer than 25 s. Thus, the MPEs for entering a 7 mm pupil (0.385 cm2) are 41 mW and 1.92 mW for extended and point sources, respectively, and the corresponding incident power/ANSI standard ratios are 0.044 and 0.260, respectively. Since the sum of these ratios is only 0.304, which is less than 1, the total power applied in this study is safe for the human eye and satisfies the ANSI standard [9, 12].
The VA is defined as a line extending from the fixation point to the fovea, which cannot be easily identified based on OCT image of just the anterior segment. On the other hand, OA and PA can be calculated using the image of the anterior segment. The accuracy of angles “α” and “κ” depends on the solid measurement of the VA using both the retinal and anterior segment. Therefore, it is crucial that a subject should gaze at the fixation target during measurement. In the case of poor fixation capabilities of the subject, the VA cannot be reliably determined.
In summary, we developed a dual-depth SS-OCT system for whole-eye imaging and employed it to measure the full biometry (the three axes and two angles) of the human eye. This complete quantitative biometric information can be compared before and after refractive surgeries such as cataract surgery and LASIK and can facilitate intraocular lens type definition and decentration measurement [3].
Funding
Ministry of Health and Welfare (MOHW) (HI13C1501); Ministry of Trade, Industry, and Energy (MOTIE) (10063364); NRF (National Research Foundation of Korea) Grant funded by the Korean Government (NRF-2016-Fostering Core Leaders of the Future Basic Science Program/Global Ph.D. Fellowship Program) (NRF-2016H1A2A1907062).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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