Motion-corrected Gaussian, mean, and parametric curvature mapping of the posterior eye with pupil-tracked whole-eye OCT

Abstract

Optical coherence tomography (OCT) is a promising alternative to magnetic resonance imaging (MRI) for accessible, high-resolution measurement of posterior eye curvature, including for applications such as exploration of local eye shape deformations in pathologic myopia. However, OCT is a point-scanning interferometric imaging method that spatially encodes the effects of patient motion during acquisition into an acquired volume. These motion artifacts reduce accuracy and increase intra-acquisition variation in measured curvature, reducing the utility of OCT for monitoring posterior eye curvatures longitudinally. We present a method utilizing 3D pupil tracking and sparse orthogonal scanning for post-acquisition correction of motion artifacts in posterior eye OCT volumes and their corresponding curvature maps. We imaged six subjects exhibiting moderate (−6.0 to −3.0 D) to high (<−6.0 D) myopia and demonstrate a reduction in intra-session curvature variability when using our pupil tracking motion correction scheme. We also propose alternative metrics for representing 2-dimensional posterior eye curvature data to researchers and clinicians.

1. Introduction

Since its invention [1], optical coherence tomography (OCT) has become a critical imaging modality in ophthalmic research and clinical practice. OCT has found success in ophthalmology due to its non-invasiveness and its ability to capture high-resolution 3D volumes of retinal [2,3] and anterior eye [4] structures.

In pathologic myopia (PM), distinct local deformations, such as staphyloma [5], may appear in the posterior pole due to local biomechanical changes associated with axial elongation [6]. Both global axial elongation and local deformations have been studied using partial coherence interferometry (PCI) [7–10] and high-resolution 3D magnetic resonance imaging (MRI). [11–14]. However, PCI requires manual repositioning for spatial sampling, thus has a low sampling rate, while MRI is often expensive and logistically unsuited for large-population, outpatient longitudinal studies.

Recent work has explored the use of OCT for measuring local posterior deformations in myopic eyes [15–24]. Compared to MRI, OCT has the distinct advantages of higher accessibility, lower cost, faster acquisition, and resolution an order of magnitude higher than high-resolution 3D MRI imaging protocols [13,14]. It should be noted that standard OCT images of the retina do not reflect the true ocular shape due to optical distortions inherent to retinal imaging with OCT. These distortions necessitate the use of analytical or numerical ray-tracing models to recover the true ocular shape [25,26]. Furthermore, this distortion correction method is dependent on a variety of factors, including patient biometry (corneal radii, axial length) and eye position relative to the imaging system. As a result, patient motion that occurs during acquisition can impart complex artifacts in the final distortion-corrected shape of the posterior eye. McNabb et al. [15] utilized simultaneous imaging of both the anterior and posterior eye sections, in conjunction with a radial scanning protocol, to track the position of the patient relative to the system throughout scanning. However, this method assumes no lateral decentration of the eye relative to the system and limited analysis to 1-dimensional meridonial curvatures. Tan et al. [17] minimized motion artifacts by reducing acquisition time with a high A-scan rate swept source. However, even with this improvement in acquisition time, motion artifacts had to be removed in post-processing with low-pass filtering polynomial fits. Bayhaqi et al. [27] utilized a contact handheld probe with a short acquisition time in order to capture quantitative curvature with minimal motion artifacts. However, this specialty probe removes the non-contact and non-invasive advantages of conventional OCT. Additionally, its nature as a handheld probe introduces another potential source of motion through operator tremor.

Due to the limited resolution of 3D MRI, studies of ocular shape have been limited to subjective descriptions of local deformation and shape [11] or quantitative analysis of the bulk shapes present within the eye [14] and posterior pole [12]. The higher resolution of OCT creates opportunities for evaluation of more subtle, local curvatures in the posterior eye, bridging the gap between subjective observation of local deformations and quantitative analysis of eye shape. This was first explored with sparse radial 1-dimensional curvature analysis by McNabb et al. [15,16]. A 2-dimensional representation of curvature using Gaussian curvature was then proposed by Tan et al. [17,18]. However, Gaussian curvature does not distinguish between concavity/convexity and is potentially unintuitive to clinicians, limiting the utility and translatability of this metric.

Numerous works have explored the clinical use of OCT to study the correlation between posterior curvatures and myopia. Miyake et al. [28] were able to associate the presence of staphylomas with the mean and variation of posterior curvatures. This study’s analysis was, however, based on raw OCT images, not distortion corrected images that better reflect the actual shape of the eye. Other groups have used OCT-derived Gaussian curvature to identify posterior staphyloma in high myopes [19] and to study posterior eye shape anisometropes [20]. Meridonial posterior curvature (1-dimensional) was also utilized to study myopic shift in children [21]. Dai et al. [29] examined the effect of retinal OCT image magnification on common OCT-Angiography (OCT-A) metrics. Recent studies have identified correlations between myopia, posterior curvature, and OCT-A measurements such as deep capillary plexus (DCP) density [22], choroidal thickness, and chorodial vascular volume [23]. Hu et al. [30] also utilized a whole-eye OCT system with a long imaging depth in order to accurately measure axial length and dewarp the retina to find the true fundus size. However, these aforementioned studies utilized a simpler analytic model [25] that assumes a common pivot point for all incoming rays. Furthermore, although many of these studies compensated for motion in their volumes with various methods, they did not model the effect of motion within their ray-tracing models. Zhao et al. [24] utilized a commercial OCT-A device with built-in software for posterior shape correction to study the correlation between posterior curvature and choroidal vascular volume and thickness. Xu et al. [31] utilized the same device to correlate common myopia biomarkers with posterior Gaussian curvature. However, there is little public information about the algorithm used for ocular distortion correction in this device, making it difficult to independently evaluate the accuracy of these measurements.

In this work, pupil tracking was used to implement a more complete model of the optical distortion to accurately correct motion artifacts without excessive low-pass filtering. Additionally, a sparse orthogonal scanning protocol was used to remove residual motion artifacts along the slow-axis of the OCT volume. Furthermore, we also explore the use of the principal curvatures as an additional metric to more fully represent the ocular shape to clinicians and researchers in an intuitive manner using a parametric color map with direct comparisons to both Gaussian and mean curvature maps.

2. Methods

2.1. Wide-field whole eye OCT system

2.1.1. System design

We used a modified version of our previously published polarization multiplexed, wide-field whole-eye OCT system with a retinal FOV of 55°and an anterior eye field-of-view (FOV) of 14×14 mm [15,16]. This system formerly utilized the coherence revival phenomenon [32] to achieve simultaneous wide-field imaging of both the anterior eye and retina. In this version of the system, the imaging path was switched between the retinal and anterior eye side every B-scan using a galvanometer switch (ScannerMAX). The anterior eye side used a half-waveplate (HWP) (Thorlabs) to rotate the input linear polarization to its orthogonal direction and rotate back on return. The path of the input retinal and anterior eye light is then controlled downstream by polarizing beam splitters (PBS), creating a telecentric imaging plane at the front of the eye through the anterior eye path and an imaging plane at the retina for the retinal path. The retinal imaging path also included an electrically tunable lens (ETL) (Optotune) for Diopter control and a retroreflector (Edmund Optics) on a motorized linear stage (National Aperture, Inc.) for eye axial length control.

The updated OCT engine used a custom 400 kHz VCSEL swept-source (Axsun Technologies) with a center wavelength of 1055 nm and a bandwidth of 87 nm and an imaging depth of 6.6 mm in air. A single mode fiber in a paddle polarization controller was used to launch linear polarized light into a custom polarization-maintaining fiber (PMF) transmissive Mach-Zehnder interferometer (Opticonx) to stabilize the input and output polarizations of the system sample arm. This removed the need for active adjustment of polarization controllers within the engine during image alignment and acquisition (Fig. 1(A)). Finally, the OCT interferogram was detected by a balanced photodetector (PDB482C-AC, Thorlabs) and digitized at 12-bits at up to 4 GSamples/s (ATS-9373, AlazarTech).

Fig. 1.

A) Whole eye OCT system schematic. Polarization was used to separate imaging paths. A galvanometer switches between the anterior eye and retinal imaging path every B-scan. B) Galvanometer control diagram for bidirectional whole eye acquisition and orthogonal scanning. C) Galvanometer scanning paths overlaid on maximum intensity projections of OCT volumes.

A stereo pair of pupil tracking cameras (BFS-U3-13Y3M-C, Teledyne FLIR) was mounted on each side of the OCT scanhead. Both cameras were calibrated to one another, located the pupil centroid through binary morphology segmentation at 120 Hz, and triangulated the pupil’s 3D position throughout OCT acquisition [33,34]. The pupil camera FOV was illuminated by four evenly spaced 940 nm LED’s mounted around the OCT scanhead. This near-infrared wavelength was chosen to minimize variation across subject iris colors and maintain pupil-iris contrast while not overlapping with the OCT spectrum [35,36]. Short-pass filters were used to filter OCT illumination at the camera sensors.

OCT image acquisition and processing was performed via a custom application written in Python 3.12 utilizing an open source high performance C++ library [37]. This application allowed for custom scan pattern creation and control of 3 galvanometers: x and y scanning, and the anterior/posterior path switch. There were separate processing pipelines to handle data acquired from the retina and anterior eye to parallelize processing and to allow for real-time display of both fields-of-view. The application also captured the relative timing of the B-scans within any given volume as metadata for future motion correction processing.

2.1.2. Scan pattern

Primary volumes were acquired in a custom raster scan pattern with dimensions of 2 volumes × 1500 A-scans × 200 B-scans. Anterior eye and retina volumes were interleaved every other B-scan, with anterior eye B-scans being acquired on the galvanometer flyback in order to save acquisition time. There were 50 inactive scans between each B-scan to allow the galvanometer switch to fully transition between the anterior eye and retinal imaging paths. The voltage offset caused by galvanometer flyback was measured and calibrated before dewarping volumes. To further address motion artifacts, we utilized a sparse unidirectional scan in the retinal imaging plane of 7 B-scans × 1500 A-scans with an orthogonal fast-axis scanning direction, acquired before each primary volume acquisition (Fig. 1(B)-(C)). The fast-axis of the orthogonal scan is oriented along the slow-axis of the primary volume, allowing for a relatively motion-free cross-section of the subject’s retina along the direction where motion artifacts are most prevalent [38,39]. A complete acquisition of both the sparse orthogonal retinal volume and the primary volumes for anterior eye and retina took approximately 1.9 seconds.

2.2. Calibration

2.2.1. Eye-to-system distance and imaging plane separation

To accurately retrieve retinal curvature, there are two necessary parameters: 1) the eye-to-system distance and 2) optical path length separation between the anterior eye and retinal imaging planes.

To record the eye-to-system distance, a reflective USAF 1951 test chart (Edmund Optics) was imaged at the focus of the anterior eye imaging plane. As a preliminary step, the axial pixel pitch ( $\mu$ m/pixel) was measured. We then adjusted the position of the retroreflector in the reference arm with a manual linear stage (0.001" resolution) until the last surface of the sample arm objective was within the anterior eye field of view. These two images were then used to find the distance between the terminal lens of the sample arm and a reference axial position in the anterior eye imaging field of view.

To find the separation between the anterior eye and retinal imaging planes, the USAF target remained in place at the focus of the anterior eye field of view. The optical path length of the retinal imaging path was then adjusted using the motorized linear stage until the target was also within the retinal imaging field of view. Using this measurement, the optical path length separation between the two planes is known for any given motorized linear stage position, giving the value $OP{L}_{Offset}$ .

2.2.2. Retinal Field-of-View

In order to measure the output optical angle of the retinal path for different galvanometer voltages, we placed a monochrome CMOS camera (Teledyne FLIR) on a manual translation stage (0.001" resolution) near the pivot point of the output. The output beam, mapped to a 15×15 grid of galvanometer mirror positions across the FOV, was detected by the camera at 9 different axial positions spaced 0.005" apart. The beam intensity centroid was found in each image, while vignetted beams were automatically excluded with a threshold operation. For each output galvanometer voltage, the output optical angle was then found by fitting a linear function to the centroid position across all 9 axial positions. Output optical angle was then fit to input galvanometer voltages with a 2-dimensional 4th order polynomial. Mechanical galvanometer angle was then extrapolated from the output optical angle using the angular magnification between the galvanometer surface and output in our system’s OpticStudio model.

2.2.3. Anterior eye volume fan distortion

In order to measure the anterior eye volume fan distortion [40], a reflective dot grid distortion target (Edmund Optics, 0.50 mm pitch) was imaged at two axial positions in the anterior eye field of view. Dot centroids were segmented using binary morphology and fitted to first and second order radial distortion coefficients. Additionally, distortion in the axial direction was modeled by fitting the surface of the distortion target to a 2-dimensional second order polynomial. The real-world pixel coordinates in each axial plane were then extrapolated to give a full 3-dimensional grid of anterior eye volume coordinates in air. This calibration was later validated by evaluating the radius of curvature of our phantom’s lens in section 3.1

2.3. Pupil tracking

As an initial step, the pupil camera pair was calibrated using chessboard calibration targets [41] with OpenCV stereo calibration [42]. The pupil centroid was detected and triangulated using previously published custom C++ software [34,43,44].

In order to find the transformation between the pupil tracking camera coordinate frame and the anterior eye OCT FOV, a 3D printed pupil phantom was simultaneously pupil tracked and imaged by the OCT system. First, an offset was added to the pupil camera coordinate frame so that its origin was at the lateral center of the anterior eye imaging FOV and the anterior eye focus axially. The pupil phantom was then translated across the depth of the imaging volume by a motorized linear stage (Zaber Inc.). By laterally centering the phantom pupil in the OCT FOV across imaging depths, the optic axis of the OCT system was aligned with the stage translation axis. The phantom was then tracked as it was translated sinusoidally by the linear stage. The pitch and yaw of the transformation matrix was then determined by minimizing the standard deviations in the lateral directions of the transformed pupil tracking data ( Supplement 1 ^{(4.1MB, pdf)} Fig. S1).

To synchronize OCT and pupil tracking data, the times of each pupil tracking frame and start of each B-scan were recorded using the system clock. Pupil tracking data was median filtered to remove frames where a pupil was not located, then Gaussian filtered to reduce jitter. Each B-scan was then assigned the patient position from the pupil tracking frame closest to its acquisition time.

2.4. Ocular measurements

Following imaging with pupil tracking, anterior and retinal volumes are segmented. The anterior segmentation is used to generate an individualized eye model, which is then used in conjunction with pupil tracking data to model the patient’s imaging session. The retinal segmentation is then dewarped to real-world coordinates using raytracing information. Artifacts are removed with a sparse orthogonal scan correction method. The surfaces are then smoothed, fit to a cubic spline, and used to generate curvature maps. A full flowchart for this processing pipeline is shown in Supplement 1 ^{(4.1MB, pdf)} Fig. S2

2.4.1. Segmentation of anterior eye and retinal volumes

The retinal pigment epithelium (RPE) of each retinal volume was automatically segmented using a MIRAGE foundational model [45] fine-tuned on previous whole-eye data. Segmentations were then manually reviewed and corrected as needed. The corneal epithelium was segmented using a semi-manual method based on graph-cut theory (Fig. 6) [15,46]. A 25 pixel thick slab was created above and below the RPE segmentation line and averaged axially. This mean intensity map was then used to identify the centroid of the optic nerve head (ONH) through thresholding and binary morphology. A circular area with a diameter of 2.5 mm [47] around the ONH centroid was then used as a mask to remove the ONH and its surrounding area from final curvature maps.

Fig. 6.

A) (left) 6 $\times$ averaged B-scan of central 6 mm of cornea with example epithelium segmentation and (right) 6 $\times$ averaged B-scan of retina with example RPE segmentation. Volumes are displayed in their original rectilinear format. B) Distortion corrected B-scans with corresponding curvature map. Saddle and elevated curvature areas are highlighted in inferior section.

2.4.2. Individualized eye model

An individualized eye model for each subject was created in OpticStudio with anterior corneal curvature, corneal thickness, and anterior chamber depth measured from anterior eye volumes. anterior eye volumes were motion corrected by translating B-scans according to pupil tracking data. The anterior corneal curvature was quantified using the best-fit sphere for the segmented surface in a 6 mm diameter centered around the apex of the cornea. Following the Gullstrand model eye, the posterior cornea radius-to-anterior cornea radius ratio was assumed to be 0.88 [48]. We utilized the Polans eye model gradient lens [49] to model the eye’s crystalline lens.

2.4.3. Posterior eye shape dewarping

OCT volumes are typically displayed in a rectilinear format wherein each A-scan is of a consistent optical path length (OPL) and lacking angular direction. Imaging through the patient’s eye results in a differential between the optical path length and linear dimensions as well as imparting angle relative to the central optical axis. This transform from a rectilinear array to linear dimensions results in a curved volume, similar to the shape of a curvilinear ultrasound probe’s FOV [25] (Fig. 2(C)). To find the true shape of this fan volume, knowledge of the patient’s eye biometry and position relative to the system is required (see 2.4.2). With a ray-trace model [15,25] of the subject’s eye and imaging session, OCT pixel coordinates can be transformed into real-world coordinates.

Fig. 2.

A) OpticStudio model of nominal system. Individual A-scans are simulated by tilting x and y galvanometers. Inset: Closeup of Polans eye model with simulated retina DC line. B) Effect of subject lateral and axial shifts on a single A-scan trajectory. Lateral shift tilts the volume, while axial drift imparts motion artifacts in the volume and affects the curvature of the retina image. C) Volume dewarping process utilizing segmented volume and outputs from raytracing, demonstrated on a 3D printed phantom. D) Example of in vivo registration of orthogonal and primary slow-axis scans. E) Deviation between orthogonal and primary slow-axis scans. Deviations are approximately in the range of tens of microns, close to the accuracy level of pupil tracking [34]

In OpticStudio, we modeled the DC line, which is the surface of zero optical path length difference between the reference and sample arms, of the retinal imaging volume using a spherical surface after the eye model’s lens (Fig. 2(A)). The distance from the lens to this surface is modeled using the parameters found in 2.4.2 with the following equation for in vivo eyes:

$$L_{DC}=\frac{OP{L}_{Offset}-2L_{Stage}-L_{Apex}-n_{g-cornea}{L}_{Cornea}-OP{L}_{Polans}}{n_{g-vitreous}}-L_{ACD}$$ (1)

and the following equation for our eye phantom:

$$L_{DC}=OP{L}_{Offset}-2L_{Stage}-L_{Apex}-OP{L}_{Lens}$$ (2)

where $OPL$ refers to an optical path length ( $n_{group}\ast L$ ) and $L$ refers to a physical distance ( Supplement 1 ^{(4.1MB, pdf)} Fig. S3). $L_{stage}$ refers to the position of the motorized stage in the retinal path for axial length adjustment, which changes the calibrated offset between the anterior eye and retinal volumes. A factor of 2 is included because it utilizes a retroreflector, resulting in a double pass in the sample arm. The group optical indices $n_{g-vitreous}=1.344$ and $n_{g-cornea}=1.388$ were used for the vitreous and cornea, respectively [50,51]. $OP{L}_{Polans}$ was found using the on-axis chief ray going into the Polans model in OpticStudio. $OP{L}_{Lens}$ was calculated using the nominal parameters for the Thorlabs AC127-025-C achromatic doublet used in the phantom.

The radius of the DC line surface was optimized to minimize OPL differences across the imaging field of view. Using a MATLAB ZOS-API script, the coordinates on the DC line surface, chief ray OPL, and chief ray trajectory were found for each A-scan position. For each B-scan, the eye model was translated relative to the system according to the pupil tracking camera output described in section 2.3 (Fig. 2(B)), imparting the effect of motion on the simulated DC line shape and ray trajectory. Downsampling in the fast-axis direction, we simulated 200 B-scans × 500 A-scans for the raster scan and 5 B-scans × 500 A-scans for the sparse orthogonal scan. Segmentations were likewise downsampled to the same dimensions for processing.

Once the DC line shape and A-scan trajectory were retrieved through raytracing, the volume was dewarped to real-world coordinates, allowing for measurement of true RPE shape. For each A-scan, the location of the RPE is found with:

$$\begin{array}{ll}\underset{¯}{{\mathbf{\text{X}}}_{\mathbf{\text{RPE}}}}(m,n)& =\underset{¯}{{\mathbf{\text{X}}}_{\mathbf{\text{DC}}}}(m,n)+\\ & \underset{¯}{\text{θ}}(m,n)(d(m,n)+\frac{OPL(m,n)-max\{OPL(m,n)\}+\gamma (m,n)}{n_{g-medium}})\end{array}$$ (3)

Where $(m,n)$ are the fast-axis and slow-axis indices, $\underset{¯}{{\mathbf{\text{X}}}_{\mathbf{\text{RPE}}}}$ is the real-world coordinates of the RPE, $\underset{¯}{{\mathbf{\text{X}}}_{\mathbf{\text{DC}}}}$ is the real-world coordinates of the DC line surface, $\underset{¯}{\text{θ}}$ is the vector of the directional cosines (trajectory) of the A-scan, $d$ is the distance from the top of the rectilinear image to the segmented RPE, $OPL$ is the optical path length retrieved from OpticStudio, and $\gamma$ is an additional calibration factor described in Section 3.1. Inclusion of the bias-subtracted $OPL$ in this formula allows for correction of small deviations of the spherical surface modeled in OpticStudio from the true equal path length DC line. This process extends the DC line surface along the A-scan trajectories to the true RPE surface (Fig. 2(C)).

2.4.4. Orthogonal alignment of B-scans

Pupil tracking-based motion correction through our raytracing model greatly mitigates motion artifacts in final curvature maps. One notable exception, however, is the presence of striping across the slow-axis of our scan, which is caused by high-frequency motion in the axial direction and error in pupil tracking (Fig. 5(A)). To address these residual motion artifacts, we utilized a sparse orthogonal volume to realign B-scans along the slow-axis.

Fig. 5.

A) Locally deformed phantom $K_G$ , $H$ , and parametric curvature maps under stationary conditions, moving conditions with different levels of motion compensation, and microCT reference. Each additional OCT motion compensation method brings the measured local curvature closer to the microCT reference. B) Photograph of phantom and comparison of measured $K_G$ , $H$ , $P_1$ , and $P_2$ of local ROI under stationary conditions, moving conditions with different levels of motion compensation, and microCT reference. Edema refers to the inward deformation, staphyloma refers to the outward deformation, bulk refers to the entire phantom excluding the locally deformed regions.

As in the primary volume, the sparse orthogonal volume can be synced to pupil tracking, raytraced, and dewarped to real-world coordinates. The real-world coordinates of each volume were used to laterally register the volumes to each other. After registration, the most central fast-axis scan from the primary volume was used to align the sparse orthogonal volumes. If this central B-scan contained the ONH, the next closest B-scan that did not contain the ONH was used instead. The axial deviation between the primary scan and orthogonal scans were calculated, then interpolated across the entire primary volume. Deviations were then removed from the primary volume, aligning the slow-axis of the primary volume to the fast-axis of the orthogonal volume (Fig. 2(D)-(E)).

2.4.5. Representation of 2-dimensional curvature

In order to quantify local 2-dimensional curvatures, we utilized the principal curvatures ( $P_1$ and $P_2$ ) to generate Gaussian curvature ( $K_G$ ) [17,18] and mean curvature ( $H$ ) maps. Additionally, we introduce parametric curvature maps that utilize a 2D colormap that encodes the local shape as described by the individual values of $P_1$ and $P_2$ (Fig. 3(D)). The color scale used for this parametric colormap is informed by colormaps frequently used clinically (such as in corneal topography) for readability and interpretability for clinicians. For example, normal nonpathologic curvatures are displayed in green. Local curvatures that indicate pathology are similarly highlighted in red, orange, and yellow for steep areas, cyan for flattened regions, and dark blues and purples for areas with negative curvatures.

Fig. 3.

A) Comparison of polynomial fitting and spline smoothing on a nominal digital phantom with local deformations. B) $K_G$ of sphere and saddle surfaces. C) $H$ of sphere and saddle surfaces. $H$ turns negative when the orientation of a surface is flipped. D) Parametric color map of sphere and saddle surfaces. E) Comparison of 2-dimensional principal curvature color map to Gaussian and mean curvature color maps on digital phantoms.

$K_G$ ( $m{m}^{-2}$ ) is the product of the principal curvatures ( $m{m}^{-1}$ ), $P_1$ and $P_2$ , which are the maximum and minimum curvatures at a given point, where by definition, $P_1>P_2$ . Thus, a positive $K_G$ indicates a surface that is curved in the same orientation across all directions at a given point. Conversely, a negative $K_G$ indicates a surface that curves in opposite orientations in its principal directions, such as a saddle point (Fig. 3(B)). As a result, the sign of $K_G$ does not provide information on whether a surface is concave or convex, only how 1-dimensional curvatures on the surface are related to each other.

$H$ ( $m{m}^{-1})$ , on the other hand, is the mean of the principal curvatures at a given point. $H$ is thus an extrinsic measure of 2-dimensional curvature, giving information about how a surface is oriented in space. Notably, in the case of a saddle-shaped surface, $H$ can potentially give the same value as a flat surface (Fig. 3(C)). Thus, $K_G$ and $H$ provide complimentary information about the overall topology of a surface as well its orientation in space. In order to represent $K_G$ and $H$ , we utilized custom color maps based on previous work [16–18]. These color maps are meant to highlight areas of concern/pathology in distinct colors based on corneal topography maps.

Although $K_G$ and $H$ are useful metrics for quantifying curvature, the meaning of each may not be immediately apparent or intuitive to clinicians, especially when the information is split over two separate surface maps. To address this, we developed a parametric curvature map with a 2D color map based on the principal curvatures $P_1$ and $P_2$ (Fig. 3(D)) . This color map overcomes the weaknesses of $K_G$ and $H$ displays, allowing for a clear delineation between concave/convex surfaces while also conveying the presence of saddle shapes and other structures with anisotropic curvatures. Although the underlying math behind calculating local curvatures differs [52], an analog set of local Dioptric powers, $K_1$ and $K_2$ , already exists in corneal topography. To make our color map more intuitive for clinicians, the chosen color scale is likewise based on corneal topography curvature maps. Normal curvatures are displayed in green to indicate a lack of pathology. Different forms of atypical curvature are highlighted in distinct colors. As a surface flattens, the normal green transitions to cyan and eventually to white for a perfectly flat surface. Convex surfaces are shown in purple, saddle shapes are displayed in indigo/dark blue, and overly concave surfaces are yellow-to-orange depending on severity. As a surface with positive curvatures becomes more asymmetric ( $P_2$ decreasing), the colors desaturate to indicate flattening along one axis.

In order to calculate $K_G$ , $H$ , $P_1$ , and $P_2$ , $C^2$ continuity is required. Previous works [16–18] have utilized 7th or 9th order polynomial fitting to achieve this condition. However, as shown in a simulation utilizing a nominal digital phantom, the presence of local deformations can cause major artifacts across the field of view when polynomial fitting (Fig. 3(A)). In this work, we propose the use of cubic spline smoothing as an alternative to polynomial fitting. Cubic spline smoothing fits a cubic spline to data with an additional smoothing factor $p$ that reduces the adherence of the fit to noisy data ( $p$ values close to 1 have little smoothing). This Gaussian and cubic spline smoothing is necessary due to the high level of noise that the discretization of OCT data introduces to the second derivative of the surface. Additionally, this smoothing helps remove residual submicron artifacts from pupil tracking and the orthogonal alignment process. We utilized a smoothing factor of p=0.99, which experimentally had a residual fitting error smaller than our axial resolution.

The orthogonally aligned retina surface is first interpolated to an isotropic linear grid. The surface is Gaussian filtered, then fit to a smoothed cubic spline using the MATLAB function csaps. The results are cropped to 90% of the full FOV to remove edge effects from the spline smoothing. $K_G$ , $H$ , $P_1$ , and $P_2$ are then calculated using the MATLAB function surfature [53]. When compared to polynomial fitting (Fig. 3(A)), this method introduces significantly less artifacts and low-pass filtering.

We demonstrate the parametric color map on two digital phantoms: one with both inward and outward deformation, and one with a large saddle point with 2 outward deformations on either side (Fig. 3(E)). These digital phantoms were chosen to demonstrate the gamut of shapes that can be represented with the parametric color map, many of which are clinically relevant. Areas of elevated concave curvatures may indicate conditions such as staphylomas, which are known to be prevalent in pathologically myopic eyes. [5]. Convexity may indicate inward deformations such as an edema, tumors, dome-shaped macula [54], or elevated intracranial pressure against the posterior pole. Large local saddle shapes are unlikely to occur, but small saddles may appear near or in-between other areas of local elevated curvature.

The inwardly and outwardly deformed phantom clearly demonstrates the same advantages as $H$ , highlighting the orientation of each deformation. In the saddle phantom with outward deformations, the saddle shape is clearly highlighted, similar to the $K_G$ map. Notably, this phantom has highly non-isotropic curvatures within and around the outward deformations. The presence of these anisotropic areas is obscured by both $K_G$ and $H$ maps, which display a "normal" value due to being the product/mean of an overly high $P_1$ and overly low $P_2$ value. However, this anisotropy is visible in the parametric color map, demonstrating another one of its advantages.

2.5. Imaging protocol

2.5.1. Phantom imaging

A custom phantom eye model was used to validate the accuracy and precision of local curvature measurements. The phantom consisted of resin 3D printed "retinas" located at the back focal point of a 25 mm focal length achromat (AC127-025-C, Thorlabs) using a 3D printed spacer. Retina phantoms were imaged by MicroCT (Nikon XT H 225 ST) (Voxel size $=16\mu m$ ) to establish a ground truth reference shape to compare to OCT results ( Supplement 1 ^{(4.1MB, pdf)} Fig. S4). A spherical phantom and a locally deformed phantom, each with a nominal bulk radius of 13 mm, were used to validate OCT measurements of overall curvature and local curvature. The locally deformed phantom utilized the same design as the inwardly and outwardly deformed digital phantom described in Section 2.4.5, which included a simulated inward deformation (such as an edema) and outward deformation (such as a staphyloma), each with a nominal radius of curvature of 9 mm (Fig. 3(E)). An FDM 3D printed "pupil" is located in front of the lens to allow for pupil tracking.

The spherical phantom was imaged by the whole-eye OCT system at 6 axial positions spaced 1 mm apart. Each volume was dewarped and rigidly registered to the microCT reference using fiducials (a hole and a slot) in the retina’s surface. The mean values of $P_1$ and $P_2$ across the surface, as well as the closest fit sphere radius ( $R_{fit}$ ), were used to compare OCT measurements to the microCT reference. Additionally, spheres were fit over the central 5 mm diameter of the anterior surface of the lens (nominal $R_{ant}=12mm$ ) in order to validate radius of curvature measurements in the anterior eye imaging plane. Deviations in $P_1$ , $P_2$ , and $R_{fit}$ from the reference were found in this step due to differences between the nominal whole eye OCT OpticStudio model and the as-built system. These deviations were corrected using an additional calibration factor $\gamma$ , which adjusted the shape of the DC-line in a fashion similar to fan-distortion correction.

To validate our system’s ability to measure local curvatures, the locally deformed phantom was imaged under two conditions: stationary and axial sinusoidal motion ( $f=2Hz,A=1.5mm$ ) with a motorized stage. Volumes were then dewarped and registered to the MicroCT reference. The local $K_G$ , $H$ , $P_1$ , and $P_2$ values of each deformation were compared between the OCT volumes and reference within a 1 mm radius region of interest. The measured bulk curvature of the phantom was also validated by excluding a larger area around each deformation, removing the influence of the deformations and their surrounding fillets. To validate the effectiveness of our motion compensation scheme, this analysis was performed on the moving phantom data 1) without any motion compensation, 2) with pupil tracking motion compensation, and 3) with both pupil tracking motion compensation and orthogonal scan alignment (Fig. 5).

2.5.2. In vivo imaging

All human subject research was approved by the Duke University Medical Center Institutional Review Board. Prior to imaging, informed consent was received for all subjects after explanation of the possible consequences and nature of the study. All portions of this research followed the tenants of the Declaration of Helsinki.

We recruited subjects with moderate ( $-6.0$ to $-3.0$ D) to high ( $<-6.0$ D) levels of myopia to have both eyes imaged by the whole-eye OCT system. Six subjects (n = 12 eyes) were enrolled: spherical equivalent refraction from $-4.8D$ to $-10.8D$ , axial length (AL) from $24.4mm$ to $29.2mm$ , ages 25 to 61 years, 4 males:2 females. Optical power prior to entering the eye was measured to be approximately 1.8 mW for the anterior eye channel and 2.0 mW for the retina channel, conforming to the safe power limits for a scanning beam under the ANSI Standard for Light Hazard Protection for Ophthalmic Instruments (ANSI Z80.36-2016). Throughout imaging, patients were stabilized by chin and forehead rests and fixated using a red LED (Fig. 1(A)).

To measure repeatability when using our motion compensation methods, each eye was imaged in triplicate. Each retina volume was dewarped 1) without any motion compensation, 2) with pupil tracking motion compensation, and 3) with both pupil tracking motion compensation and orthogonal scan alignment. Additionally, triplicate volumes were registered to each other using a manually selected control point located at a vessel bifurcation. For comparison, each of the 3 volumes were then interpolated to the same set of isotropic, gridded coordinates, and cropped to the same FOV based on the minimum FOV covered by the 3 volumes. For each eye, the standard deviation in $K_G$ , $H$ , $P_1$ , and $P_2$ measurements across all 3 trials was then compared for each method using the following equation:

$$\sigma =\sqrt{\frac{\sum _{\mathcal{l}}^{L=3}\sum _{m,n}^{M,N}[K_{\mathcal{l}}(m,n)-SampleMean(K_{\mathcal{l}}(m,n)){]}^2}{LMN}}$$ (4)

3. Results

3.1. Spherical phantom results

We measured the spherical retinal phantom at multiple axial positions to evaluate our method’s ability to measure bulk radius of curvature of the posterior eye (see Section 2.5.1). The reference microCT surface had a mean $P_1$ value across the surface of $0.07923m{m}^{-1}$ (equivalent radius of $R=12.62mm$ ). Without calibration, the dewarped OCT surface had a mean $P_1$ of $0.08139\pm 0.00054m{m}^{-1}$ ( $R=12.29\pm 0.08mm$ ). After calibration, $P_1$ was $0.08011\pm 0.0005m{m}^{-1}$ ( $R=12.48\pm 0.08mm$ ). Conversely, calibration had a relatively minor affect on mean $P_2$ values. The reference microCT surface had a mean $P_2$ value of $0.07416m{m}^{-1}$ (equivalent $R=13.48mm$ ). The uncalibrated OCT surface had a mean $P_2$ of $0.07492\pm 0.00033m{m}^{-1}$ ( $R=13.35\pm 0.06mm$ ), while the calibrated OCT surface had a mean $P_2$ of $0.07441\pm 0.00030m{m}^{-1}$ ( $R=13.44\pm 0.05mm$ )). The values for $R_{fit}$ fell between the equivalent radii for $P_1$ and $P_2$ , with values of $12.97mm$ , $12.69\pm 0.07mm$ , $12.91\pm 0.06mm$ for the microCT reference, uncalibrated OCT, and calibrated OCT (Fig. 4(C)).

Fig. 4.

A) Sag map and cross sections of spherical phantom. B) Example parametric curvature maps of dewarped OCT, dewarped OCT with additional calibration factor $\gamma$ , and reference MicroCT. C) Average $P_1$ value, average $P_2$ value, and closest fit radius $R_{fit}$ across the axial imaging range.

Over the full axial range of 5 mm, the uncalibrated surface had average $P_1$ and $P_2$ errors of $2.73\mathrm{\%}$ and $1.01\mathrm{\%}$ , respectively. After calibration, these errors reduced to $1.12\mathrm{\%}$ and $0.43\mathrm{\%}$ . Notably, over the first 3 mm of the imaging range, which are more clinically relevant due to the limited imaging depth of the system, these errors reduce even further to $0.80\mathrm{\%}$ and $0.29\mathrm{\%}$ . These results suggest that our additional calibration is primarily correcting an anisotropic effect similar to astigmatism or fan distortion in the anterior eye.

Across the 6 axial positions, the closest fit radius of the front lens surface was $R_{ant}=12.06\pm 0.1mm$ . These measurements were close to the nominal value of $R_{ant}=12$ , indicating accurate radius of curvature measurement along the entire anterior eye imaging volume.

3.2. Locally deformed phantom results

We imaged a phantom with local convex and concave deformations under stationary and moving conditions in order to evaluate our method’s ability to accurately measure local curvatures (see Section 2.5.1) . In Fig. 5, the measured curvature of our locally deformed phantom is compared qualitatively under different levels of simulated motion and motion compensation. The phantom imaged under stationary conditions both qualitatively resembles the microCT reference and quantitatively exhibits local $K_G$ , $H$ , $P_1$ , and $P_2$ values close to the reference. The addition of motion throughout acquisition encoded additional error into the curvature measurement, yielding highly varying, inaccurate results. Processing volumes with motion compensation through pupil tracking, and subsequently orthogonal scan alignment, brought local curvature measurements in the moving phantom closer to both the stationary condition as well as the microCT reference, demonstrating the effectiveness of our motion compensation method.

3.3. In vivo results

3.3.1. Posterior eye curvature repeatability

An example of dewarped B-scans is shown along with corresponding curvature map in Fig. 6. Comparison of triplicate data in each eye was performed with no motion compensation, pupil tracked motion compensation, and pupil tracking with orthogonal alignment. An example of this full process is shown with Subject 01 OD (-8.4 D, AL = 27.7 mm) in Fig. 7. In $K_G$ , $H$ , and parametric maps with no motion compensation, a high level of variability is apparent across all 3 trials. The addition of pupil tracking compensation removes the majority of this variability, but leaves some minor striping artifacts along the slow axis of the primary scan. Adding orthogonal alignment on top of pupil tracking motion correction attenuates these striping artifacts, creating consistent measurements of $K_G$ , $H$ , $P_1$ , and $P_2$ across all 3 trials. For visualization, $\mathrm{\Delta }{K}_G$ is defined as the difference between an individual map and the average of all 3 trials.

Fig. 7.

A) Subject 1 OD $K_G$ , $\mathrm{\Delta }{K}_G$ , $H$ , and parametric maps across 3 trials using different levels of motion compensation. B) Averaged $K_G$ , $H$ , and parametric maps across all 3 trials with pupil tracking and orthogonal alignment motion compensation.

The averaged, motion-compensated parametric color maps of our entire dataset (N=12) are shown in Fig. 8 ( $K_G$ and $H$ maps available in Supplement 1 ^{(4.1MB, pdf)} Fig. S5). Areas of elevated curvature can be identified temporally in Subject 01 OD and Subject 04 OS. Large areas of elevated curvature are visible in the inferior direction in Subject 03 OD/OS and Subject 05 OD/OS.

Fig. 8.

Averaged, motion-compensated parametric maps of all N=12 eyes.

The variability of curvature measurements across our 3 acquisitions can be quantified by examining the deviation of each curvature map from the average curvature map (Fig. 7). The sample standard deviations (Eq. (4)) of each eye under different motion compensation methods are shown in Fig. 9. The addition of pupil tracking motion compensation improves repeatability in $K_G$ across all patients. Theaddition of orthogonal compensation also improves repeatability in every patient except for Subject 04. We suspect that this failure of orthogonal alignment in Subject 04 resulted from fixation changes between the sparse orthogonal and primary volumes, leading to striping across the slow-axis of the primary volume in some trials ( Supplement 1 ^{(4.1MB, pdf)} Fig. S6). These trends are the same for $H$ , $P_1$ , and $P_2$ , but with much more modest changes in repeatability with the addition of motion compensation.

Fig. 9.

Triplicate (N=3) repeatability results across all subjects with different levels of motion compensation for A) $K_G$ , B) $H$ , C) $P_1$ , and D).

4. Discussion

Our previous iteration of this system [15] utilized cohrence revival coupled polarization multiplexing for completely simultaneous imaging of the anterior eye and retina. In this work, we have instead opted for a galvanometer mirror-based switch that switches imaging channels with each B-scan (forward sweep anterior, return sweep posterior). This approach provides advantages over coherence revival in signal-to-noise ratio and choice of swept source. Without time-division switching, the polarizations of each backscattered signal are orthogonal to each other at the detector, requiring the reference arm polarization to be circular or linear oriented at $\pm {45}^{\circ }$ to allow for interference with both channels. This misalignment between the sample and reference arm polarizations at the detector degrades the signal. Additionally, the coherence revival signal has a lower visibility compared to the base signal [55], further reducing the signal-to-noise ratio. Coherence revival can also cause artifacts to appear when an optical surface in the sample arm is located at a coherence revival peak. Imaging with coherence revival also necessitates a source with multiple longitudinal modes (such as an external cavity laser), precluding the use of VCSELs, which provide numerous advantages such as higher sweep rates and long instantaneous coherence lengths. The primary drawback of our approach is the limited switching frequency of the galvanometer mirror, which may be damaged when going above a certain B-scan repetition rate.

Previous works in MRI and PCI have primarily focused on qualitative observations of local deformations [11] or a closest fit sphere/ellipsoid to the entire posterior eye [13,14] due to their limited resolutions. The increased resolution of OCT has presented opportunities for measurement of local posterior curvatures, starting with 1-dimensional meridonial curvatures [15,16] and eventually evolving to 2-dimensional Gaussian curvature [17–20,24]. In this work, we presented an alternative parametric color map based on the principal curvatures from differential geometry. The gamut of this color map can represent all potential surface shapes. Clinically relevant surface shapes, such as convex shapes, flattened areas, healthy concave areas, and concave areas with elevated curvature, are all highlighted in distinct colors along a linear color scale for easy interpretation by clinicians. An analog to this concept exists in corneal topography with local $K_1$ and $K_2$ Dioptric powers [52].

To our knowledge, this work offers the most complete model and validation of posterior curvature measurement with OCT up to this point. Although previous works have validated and calibrated their systems with 3D-printed phantom retinas [15,17,27], none have independently validated the shape of the 3D-printed phantom, which may cause errors due to the limited spatial resolution of 3D-printing or shrinkage during the curing process. Calibration with the microCT (Voxel size $=16\mu m$ ) validated phantom provides a higher fidelity reference compared to previous works. Additionally, these previous works did not directly evaluate the effect of motion on their phantom volumes and the corresponding effectiveness of potential motion-compensation techniques. We further validated the effectiveness of our methods through an in vivo repeatability study, demonstrating quantitative improvements in the repeatability of motion-compensated posterior curvature measurements. While previous works validated their in vivo OCT measurements with MRI, CT, or ultrasound images of the whole eye, the spatial resolution of these other modalities limits their effectiveness in estimating the subtle local curvature changes demonstrated here.

Many other approaches exist for post-processing motion correction of OCT volumes. Swanson et al. [2] used cross-correlation to axially register B-scans. Other hardware-intensive approaches register OCT volumes to reference images generated by other modalities [56]. Kraus et al. [38,39] utilized a single modality where two raster-scanned OCT volumes with orthogonal fast axes are used to estimate and correct for motion. Although these registration-based techniques are effective for correcting visual defects in the volume, they do not account for the distortion in retinal volume shape caused by motion of the optics of the eye relative to the imaging system. For example, cross-correlating B-scans for axial registration will tend to flatten the true shape of the volume. Registration with two full orthogonal volumes will remove slow-axis artifacts, but will not correct for apparent changes in B-scan level curvature or tilting arising from patient motion. Properly evaluating the effects of motion using our ray-tracing model is therefore vital to our application of measuring true posterior shape. Our approach is not without drawbacks, however. The addition of pupil camera tracking hardware adds cost and complexity to the system. Our system currently tracks linear position, not gaze angle, which we mitigated through the use of a fixation target but it does prevent the correction of certain types of involuntary motion artifacts.

One limitation of our method was our simplified spherical corneal model, which assumed a uniform radius with no conic coefficients and a fixed posterior-to-anterior (P/A) ratio of 0.88. To account for the true shape of the human cornea, some eye models use a biconic shape [57], with two principal radii and two conic coefficients along the principal axes. Fitting to a uniform sphere instead will typically yield a value close to the average of these principal radii, causing a relative stretch/compression in the retinal image which will cause a decrease/amplification of the estimated curvature of posterior features. However, the error in curvature caused by this simplified spherical surface is negligible. The fixed P/A ratio of 0.88 from the Gullstrand model is an overestimation [48] and ignores population variation correlated to age, refractive surgery, and other factors [58,59]. This deviation from real posterior ratio will similarly affect retrieved curvature, but the effect will be relatively minor because the refraction of the chief ray caused by the posterior corneal surface is negligible compared to the anterior surface. A more accurate model may be possible by fitting to and modeling a biconic surface, using a P/A ratio informed by factors such as age [58,59], or directly retrieving the posterior corneal curvature by performing refraction correction on the anterior volumes [60].

Since $K_G$ is the product of the maximum ( $P_1$ ) and minimum ( $P_2$ ) curvature, its distribution becomes bimodal in the presence of motion, which creates valley and peak shapes along the slow axis. Conversely, the influence of motion on $H$ , $P_1$ , and $P_2$ is attenuated because the presence of waviness along the slow axis direction will only heavily influence one principal curvature at a time. As a result, the benefits of motion correction are likewise amplified in $K_G$ and attenuated in $H$ , $P_1$ , and $P_2$ .

Several subjects exhibited areas of elevated curvature (Fig. 8). These areas of elevated curvature suggest local outpouching similar to that found in previous MRI studies [11]. These outpouching features appear across the power range of our dataset, but broad generalizations are limited by the current limited sample size. Centrally located flattened areas appear in Subject 01 OD/OS and Subject 06 OD, which share similarities to the qualitative "barrel" shape or the quantitative "blunt" eye shape identified by Morita et al. [11,12]. A larger and more diverse subject population will be necessary to elucidate any existing correlations between these localized features and pathology.

Study of local posterior shape and its relevance to various pathologies is still a nascent field, and it is still unclear how it will be most effectively applied in clinical settings. The ability to directly compare quantitative values between measurements in between patient visits and against population values indicates that there is potential for this approach to assist in both longitudinal monitoring and early diagnosis of pathologies. Further work will be required to establish population-level biomarkers for pathologies in cross-sectional studies and to study the pathogenesis and effectiveness of therapies in longitudinal studies.

5. Conclusion

We have demonstrated an accurate, repeatable technique for motion-compensated retrieval of posterior eye shape with whole eye OCT. We have also presented an alternate curvature calculation method that better preserves the high lateral resolution of OCT, along with new representations of local 2-dimensional curvatures that are more intuitive to eye care providers. This work enables and supports future research on high-resolution, local posterior eye shape and its potential implications for pathologic myopia.

Funding

National Institutes of Health https://ror.org/01cwqze88 ( R01-EY024312, P30-EY005722); Research to Prevent Blindness https://ror.org/04drjs621 ( Unrestricted Grant).

Disclosures

McNabb RP: Johnson & Johnson Innovative Medicine(F), Leica Microsystems(P) Kuo AN: Leica Microsystems(P)

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 ^{(4.1MB, pdf)} for supporting content.