1. Introduction

Optical imaging is used to study, diagnose, and manage ocular, neurological, and systemic diseases that affect the retina [1–6]. The advent of digital imaging and the rapidly evolving field of automated image segmentation have led to the development of retinal image metrics that describe anatomical and pathological features objectively and quantitatively [7–13]. When clinically useful, these metrics are referred to as imaging biomarkers [14–21]. The sensitivity with which such biomarkers can be compared across eyes and against normative datasets is limited, among other factors, by the accuracy of the scaling of retinal features in units of length.

After correction of image distortion due to the optics of the ophthalmoscope [22–25] and/or non-uniform sampling [26–30], the scaling of retinal images consists of two steps. First, the calibration of the field of view in terms of angle relative to the optical axis and exit pupil of the ophthalmoscope, which can be achieved by imaging an object of known size at a known distance, as described in the International Organization for Standardization standard 10940:2009 for fundus cameras [31]. The second step, and the topic of this work, is the estimation of a retinal magnification factor (RMF) to convert these angles to lengths along the retina. Because the retina is not perfectly spherical and the field of view of ophthalmoscopes is not measured with respect to the center of such an ideal sphere, the conversion of angle to length across the retina by a multiplicative factor is only valid over a small field of view. That is, RMF changes with retinal location, although for some applications, assuming a constant RMF within a retinal region surrounding the locus of fixation is acceptable.

RMF values that are identical for all eyes can be estimated using optical models of the eye known as schematic eyes, typically designed to be representative of an average adult eye [32,33]. The development of optical biometry based on low coherence interferometry [34–37] allowed the non-contact measurement of optical path lengths between the main ocular refracting surfaces, that is, the anterior and posterior surfaces of the cornea and lens, and the retina. The commercialization of this technology [38,39], together with measurements and/or assumptions about the refractive indices of the ocular media, namely the cornea, the aqueous humor, the crystalline lens, and the vitreous humor, led to the individualization of these schematic eyes. Paraxial ray tracing through these individualized models were then used to derive individualized RMF values [40–49]. The combination of these optical paths with anterior corneal surface topography allowed further refinement of the paraxial RMF calculations [48–52], assuming rotationally symmetric ocular surfaces (known not to be true [32,53–57]), homogeneous refractive index ocular media (not true for the crystalline lens [58–66]), and ignoring changes in vitreous refractive index with age [67–69]).

A new generation of commercial ocular optical biometers [70–73] provides optical path measurements that can be used to estimate the topography of the anterior and posterior surfaces of both the cornea and crystalline lens. In this work, we use this biometry data to generate more anatomically accurate four-surface eye models, assuming homogeneous refractive indices, and use these models to further individualize RMF paraxial and non-paraxial optics calculations. For completeness, we review a correction factor derived from paraxial optics using the iris as an aperture stop, which coarsely accounts for RMF changes when imaging the retina through spectacles, trial lenses, or contact lenses [74]. The proposed RMF calculation methods are compared in an adult subject cohort against those from widely adopted RMF definitions based on paraxial optics. This is followed by a discussion of how the magnification of retinal images can change with axial eye positioning, a simulation of how cycloplegia depth can impact RMF, and finally, elementary statistical analysis illustrating how the new ocular biometry data can be used to study the prevalence of axial and refractive myopia. After a summary, we conclude with a proposed standard for reporting ocular biometry in retinal imaging biomarker studies, irrespective of the adopted RMF calculation method, to facilitate the comparison of results across studies and normative databases.

2. Paraxial retinal magnification factors

Schematic eyes [33,75] are often chosen to model the relaxed or accommodating eye. Here, we only study relaxed schematic eyes because retinal imaging is often performed under pharmacologically induced paralysis of the ciliary muscle of the eye, which results in the temporary loss of accommodation (cycloplegia) and pupil dilation (mydriasis). With this in mind, we review next four existing and one proposed individualized RMF definitions based on rotationally symmetric schematic eyes and paraxial optics.

2.1 Posterior nodal distance of individualized Emsley reduced eye

The simplest schematic eyes, referred to as reduced eyes, model the eye as a single refracting surface that represents the anterior surface of the cornea followed by a single homogenous medium, as depicted on the top left panel of Fig. 1. An example RMF calculation using such schematic eye, involves the scaling of the posterior nodal distance (PND) in the Emsley reduced eye, individualized using the ocular axial length ($\textrm{AL}$) [40,45,46],

 

Fig. 1. Rotationally symmetric schematic eyes used for paraxial retinal magnification factor calculations with all ocular media assumed homogenous. The angles of the incident rays, in red, have been exaggerated for ease of visualization. The cyan surfaces and distances are estimated using ocular biometry. In Li’s schematic eye, the posterior corneal surface (yellow) has a radius proportional to that of the anterior surface (blue).

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2.2 Posterior nodal distance scaling of individualized Gullstrand-Emsley eye

Eye models consisting of three refractive surfaces representing the anterior corneal surface, as well as the anterior and posterior surfaces of the crystalline lens (top right panel of Fig. 1), are referred to as simplified schematic eyes. Chui et.al. [41,42] use one such individualized model, namely the Gullstrand-Emsley simplified schematic eye, to propose the following RMF definition based on the scaling of the posterior nodal distance,

2.3 Posterior principal distance scaling of individualized Bennett-Rabbetts eye

An alternative RMF definition [9,44,47], uses the posterior principal distance (PPD) of the individualized simplified schematic eye by Bennett et. al. [9,43,44] (see bottom left panel of Fig. 1),

2.4 Posterior nodal distance scaling of individualized Li eye

Li et al. [52] proposed a more refined RMF paraxial calculation based on an individualized schematic eye with four surfaces, incorporating measurements of axial length, anterior corneal curvature, and the anatomical anterior chamber depth ($\textrm{AC}{\textrm{D}_{\textrm{anatomical}}}$), defined as the distance from the apex of the posterior corneal surface to the apex of the anterior crystalline lens surface. In this method [48,49], the posterior nodal distance of the individualized schematic eye is found by calculating the elements of the following ray transfer matrix [78],

Table 1. Optical parameters used in Li’s four-surface schematic eye.

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2.5 Paraxial ray tracing through schematic eye with four individualized surfaces

Most RMF definitions, including those above, are based on cardinal points of the schematic eyes (i.e., nodal points or principal points) that are convenient idealizations. In practice, however, there is no obvious or simple method for aligning the ophthalmoscope with respect to these idealized points, which do not actually exist due to the decentration and lack of rotational symmetry of the refractive ocular surfaces. Therefore, we set out to develop RMF definitions that better model how retinal images are captured, which is by aligning the ophthalmoscope with respect to the eye to minimize vignetting (i.e., by trying to overlap the exit pupil of the ophthalmoscope with the entrance pupil of the eye).

To incorporate this, we calculate RMF values assuming that the exit pupil of the ophthalmoscope overlaps with and is perfectly centered with respect to the entrance pupil of the eye, and by tracing chief rays, which by definition, pass through the center of the entrance and exit pupils of the eye and ophthalmoscope. This requires the incorporation of an aperture stop in the schematic eye, something mentioned in a transverse chromatic aberration study [82] that predicted the reduction of the RMF, when compared to the RMF estimated using nodal points that we show later. The use of chief rays is in opposition to RMF calculations based on the tracing of rays through nodal points or principal points. We also propose further individualization of the schematic eyes by taking advantage of the latest generation of optical ocular biometers to estimate the corneal and lens refractive surfaces as it will be shown later.

In our paraxial RMF calculation, we assume a rotationally symmetric eye model consisting of four refracting surfaces separated by distances calculated as the optical path length (OPL) between them divided by the group refractive index of the ocular media at 840 nm [83]. The optical axis is defined as the line connecting the biometer’s fixation target as seen by the subject and the apex of the anterior cornea. The radius of curvature of each refracting surface is calculated by fitting a sphere to the refractive index-corrected surface using only the Zernike defocus term. For reference, the analytic formula for paraxial ray tracing of a chief ray at 1-degree field of view through a four-surface schematic eye is provided in Appendix A (Eq. (S1), $\delta z = 0$).

3. Retinal magnification factor derived from real ray tracing

In this section, we describe the use of ocular surface biometry in combination with real ray tracing to first create individualized model eyes with four refractive surfaces and ocular media with homogeneous refractive indices, and then, to calculate RMF values along various directions (meridians) but always at the fixation locus [84]. This RMF should be more faithful to the anatomy of the eye because it considers the non-spherical nature of the refractive ocular surfaces, their tilts, and decentering, all of which are ignored in paraxial RMF calculations.

3.1 Optical ocular biometry

Ocular biometry data for this study were captured using a Hyper-parallel Optical Coherence Tomographer (HP-OCT; Cylite, Notting Hill, Victoria, Australia), which uses a light source with 25 nm bandwidth centered at 840 nm. This biometer [71], illuminates the eye with multiple parallel narrow beams (beamlets), each of which can be thought of as an optical ray. The light backscattered at various depths within the eye is made to interfere with light from a coherent (reference) beam, and the resulting beamlet interferograms are captured with a 2-dimensional camera. These spectra are then processed to create a 3-dimensional (3D) image of the anterior segment of the eye and estimate the axial length of the eye. The voxel values in the resulting 3D image represent the reflectivity of ocular structures along the path of each beamlet. The pre-release version of the analysis software of the HP-OCT used here automatically segments the anterior and posterior surfaces of the cornea and crystalline lens from this 3D volume and fits each of these surfaces using the 28 lowest-order Zernike polynomials defined per the Optical Society of America proposed standard for reporting the wavefront aberration of the eye [85], and the American National Standard Institutes [ANSI Z80.28-2004] [86] over a data fitting radius of 2 mm. This analysis software allows the exporting of a grid of points across each of these surfaces as sets of coordinate triads $({{x_0},{y_0},{z_{OPL}}} )$, in which the first two numbers are the Cartesian coordinates of the beamlets illuminating the eye before they impinge on the cornea, and the third number is the optical path length traveled by the beamlet from a reference plane that is perpendicular to the beamlets before reaching the eye. We refer to this data, as “uncorrected” because these coordinates do not account for the beamlet change in angle due to refraction at each ocular surface.

3.2 Real ray tracing for estimation of ocular surfaces

To estimate the ocular surfaces, we performed real ray tracing through the four surfaces segmented by the HP-OCT software, assuming ocular media with homogeneous refractive indices. This assumption is reasonable for the cornea and the aqueous humor that fills the anterior chamber of the eye [87,88], but it is less so for the natural crystalline lens, which is known to have a refractive index that changes across its volume [32,60,89]. The ray tracing is a repetition of two steps, the calculation of ray direction cosines after refraction at each surface, and the ray propagation along the refracted ray direction for a distance equal to the optical path length (measured by the biometer) divided by the medium group refractive index.

The change in direction cosines of a ray after a refractive surface can be calculated using the vector form of Snell’s law [90],

 

Fig. 2. Unit vectors along a ray incident on a curved surface with normal unit vector $\bar{{\boldsymbol N}}$ before (N) and after refraction (${{\boldsymbol N}^\mathrm{\ast }}$), related by Snell’s law shown in Eq. (7).

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Solving for $\gamma $ [90], yields

The normal unit vector $\bar{{\boldsymbol N}}$ at a point (${x_i},{\; }{y_i},{\; }{z_i}$) where the ray intersects the surface, is given by the normalized gradient $\nabla F/|{\nabla F} |$ [91], where $F({x,y,z} )= f({x,y} )- z = 0$ is the implicit form of the surface function. The explicit surface functions $f({x,y} )$ were obtained from least-squares fitting the surface data from the ocular biometry, using the 28 lowest order Zernike polynomials, which include polynomials up to 6^th order.

After the directions of the refracted rays are calculated, we find the following optical surface simply by moving along the ${{\boldsymbol N}^{\boldsymbol \ast }}$ direction by a distance equal to the $\textrm{OPL}$ divided by the group refractive index of the medium [83]

Here, we use the Cauchy’s equation fitted to the Le Grand’s schematic eye [80] for calculating the group refractive index and refractive index at various wavelengths (Table 2) [92].

Table 2. Group refractive index (${{\boldsymbol n}_{\boldsymbol g}}$) and refractive indices (${\boldsymbol n}$) at various wavelengths.

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The resulting $({{x_j},{\; }{y_j},{\; }{z_j}} ){\; }$points represent a refractive index-corrected surface (i.e., accounting for Snell’s law at each ocular surface), that is then least-squares fitted with Zernike polynomial to calculate the direction of the ray after refraction across this surface. These refraction and propagation calculations allow the sequential estimation of the refractive index-corrected posterior corneal surface, anterior crystalline lens surface and posterior crystalline lens surface. This approach was chosen because it provides the analytical solution to Snell’s law, avoiding iterative algorithms (e.g., section III of Ref. [91]).

The ray tracing calculations were implemented in mathematical software Matlab (Mathworks, Natick, MA, USA) and validated using CODE V, with the differences in $({x,\; y,\; z} )$ coordinates between the two calculations being on the order of tens of nanometers.

3.3 Estimating the retinal magnification factor

Once the refractive surfaces have been estimated as described above, the RMF for each eye was calculated by tracing chief ray pairs with 1-degree full field of view (FOV) using the refractive index at the desired wavelength. The distance between the intersection of these rays at the retina is then defined as the RMF. The optical direction cosines can be calculated as below, where $\alpha ,\; \beta $ are the half field of view (HFOV),

Since we do not initially have the coordinates of the intersection of these chief rays with the anterior corneal surface, we need to find these by first tracing a ray with the desired HFOV through an initial point on the corneal surface and then propagating it through both corneal surfaces. We then translate the initial ray before the cornea without changing its angle iteratively until the ray passes through the center of the aperture stop, which we assume coincides with the anterior surface of the crystalline lens, using the minimization routine fminsearch in Matlab. Importantly, because we do not have the optical path lengths traveled by these rays between the surfaces, we find the ray-surface intersection point as described in section II of Ref. [91]. This approach, implemented in Matlab, was validated against CODE V.

Once we have these two chief rays entering the eye on opposing sides of the optical axis with a 1-degree FOV, we continue to trace them through the crystalline lens accounting for refraction at its surfaces, propagation through the vitreous, until the retina is reached. Then, we calculate the RMF as the distance between their intersection with a planar retina an axial length away from the corneal apex. The use of a plane, rather than curved, retina was adopted because it is a reasonable approximation for such a small field of view (the difference between arclength and tangent is ∼0.015%).

Unlike paraxial model eyes, the use of tilted and decentered refractive ocular surfaces, can result in RMF variation with retinal meridian. To explore the magnitude of such variation, we traced four ray pairs at 45 deg increments rotated about the optical axis, with their HFOVs defined in Eq. (14), where the first two angle pairs are depicted in Fig. 3.

This resulted in 8 points on the planar retina which we least-squares fitted to an ellipse, from which we estimated the minimum and maximum RMF based on the minor and major axes.

 

Fig. 3. Vertical (left) and horizontal (right) crossed sections of proposed individualized four-surface non-rotationally symmetric schematic eye, in which all surfaces can be tilted and decentered relative to the optical axis.

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4. Lens correction factor

When the sphere equivalent of a subject’s refractive error is outside the focus range of an ophthalmoscope, one is not able to capture in-focus retinal images. This limited range can be expanded through the use of spectacles, contact lenses, or optometric trial lenses in front of the eye while capturing retinal images. In these situations, the RMF should be scaled by a factor M that can be coarsely estimated using paraxial optics modeling of the spectacle or trial lens as a thin lens with power $\phi $ (in diopters), placed at a (positive) distance d from the entrance pupil plane of the eye in meters (i.e., the image of the iris as seen through the cornea and anterior chamber of the eye) as depicted in Fig. 4 [93],

This paraxial formula is only valid when the distance between the eye and the ophthalmoscope is adjusted as necessary to make the entrance pupil of the eye overlap with the exit pupil of the ophthalmoscope-lens combination. This can be facilitated by setting the ophthalmoscope to its largest field of view before changing the distance between the subject and the ophthalmoscope to minimize vignetting, i.e., find the subject-ophthalmoscope distance that results in the most uniform image intensity across the field of view, and particularly, at the edges. When this condition is met, the formula above tells us that a negative spectacle/trial/contact lens will decrease the retinal magnification factor and result in larger retinal features in the ophthalmoscope’s images.

 

Fig. 4. Paraxial ray tracing notation used for calculating the change in retinal magnification factor due to the use of spectacle, trial or contact lens, noting that the distance d in Eq. (15) is that between the lens and the entrance pupil of the eye.

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The precision of this corrective magnification factor depends on the measurement of the distance d, which is not trivial. The distance between the lens and the corneal apex could be measured, for example, using a ruler and a camera with a telecentric lens perpendicular to the optical axis of the ophthalmoscope. The distance between the corneal apex and the entrance pupil plane of the eye could be estimated from ocular biometry and paraxial ray tracing as shown next. When modeling the cornea as a single optical surface, the distance between the cornea apex and the entrance pupil of the eye ( ${d_{EP1}}$) can be approximated as,

A more refined estimation of the RMF correction factor due to the use of lenses can be obtained using real ray tracing software, incorporating the real (as opposed to thin) lens parameters [94,95].

Importantly, and in addition to the magnification factor M, one should keep in mind that not all ophthalmoscopes have focus correction that preserve the RMF (i.e., not all ophthalmoscopes are telecentric) [96]. When this is the case, change in field of view with focus must be experimentally measured and accounted for.

5. Human subjects

A cohort of 22 subjects ages 16 to 63 (average age 39, standard deviation 13) with natural crystalline lenses were recruited for this study, which followed the tenets of the Declaration of Helsinki and a protocol approved by the institutional review board of Stanford University. Informed written consent was obtained after an explanation of the possible consequences of the study. One drop of 2.5% phenylephrine and one drop of 1% tropicamide were used to dilate the pupil and induce cycloplegia before capturing biometry data using the HP-OCT device without any correction of refractive error. That is, no spectacles, contact lenses or optometric trial lenses were used while capturing the biometry data.

6. Results

The biometry data was used to calculate ocular surface radii and separation for 34 eyes with spectacle prescription sphere equivalent ranging between -14.50 and +0.25D (Appendix B).

6.1 Comparison of RMF calculation methods

RMFs values were calculated using all the methods described above based on the refractive index-corrected biometry, with the resulting values plotted in Fig. 5 and listed in Appendix B. In this plot, RMF values estimated using real ray tracing are represented as vertical bars that extend between the minimum and maximum RMF values across all retinal meridians, but always at the fixation locus. These bars are small, because the maximum relative RMF change between the major and minor axes in our dataset is only 2.2%.

 

Fig. 5. RMF values calculated using the methods described in the main text in a cohort of 34 human eyes.

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By definition, the $\textrm{RM}{\textrm{F}_{\textrm{PND} - \textrm{E}}}$, $\textrm{RM}{\textrm{F}_{\textrm{PND} - \textrm{GE}}}$ and $\textrm{RM}{\textrm{F}_{\textrm{PPD} - \textrm{BR}}}$ calculations are proportional to the axial length of the eye, and hence, their correlation with axial length is exactly one. Interestingly, despite incorporating additional biometry data, all other RMF calculation methods show ≥0.996 correlation with axial length. The least-squares linear fits of all methods intersect around the axial length of a typical adult human eye (∼24 mm), which is reasonable, as schematic eyes are conceived to model typical eyes. Interestingly, the three linear fits with distinctly higher slopes correspond to the RMF calculations based on the scaling of the posterior nodal distance in model eyes with either three or four surfaces. Here it is important to note that the methods that depart the most from ours are those that use the scaling of posterior nodal or principal distances, rather than tracing chief rays, which is what more closely resembles how retinal images are captured, as we discuss in the next section. To further test this point, the RMF values for our four-surface paraxial model eye were also calculated using the PND scaling (yellow symbols and line in Fig. 5) instead of chief rays (see Fig. 6), resulting in values consistent with those from other PND scaling methods.

 

Fig. 6. Four-surface paraxial individualized schematic eye diagrams showing the ray used to calculate retinal magnification factors using posterior nodal distance (left) scaling and chief ray tracing (right).

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The RMF values Fig. 5 were calculated for 555 nm light. When comparing RMF values calculated using our real ray tracing method for 555 and 790 nm light, we found that the mean RMF difference is ∼0.1% and never larger than 0.5%.

Let us now consider the RMF values calculated using real ray tracing in the non-spherical non-rotationally symmetric model eye as the best estimates of the true (and unknown) RMF values. Then, we can use the average and/or maximum (across all eyes) of the absolute value of the RMF differences as an error (see Table 3), to rank the other methods. These numbers show that although the differences between methods are relatively small on average (0.1-2%), for highly myopic eyes (and we posit hyperopic eyes too) can be substantial (∼7.5% for -14.5D).

Table 3. Relative and Maximum norm errors between RMF definitions and the non-paraxial individualized RMF calculation method.

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The small difference (0.05% on average, 0.12% maximum) between our paraxial and real chief ray tracing RMF calculations using individualized four-surface eyes cannot be explained by the paraxial approximation, because for the small ray angles used in these RMF calculations (${1^\textrm{o}}$), the difference between angle and tangent or sine are ∼0.015%. Also, the presence of astigmatism does not seem to be the main source of difference, as the correlation of the cylinder in the subject’s refractive error with the relative change in RMF across meridians, that is, $({\textrm{RM}{\textrm{F}_{\textrm{max}}}-{-}\textrm{RM}{\textrm{F}_{\textrm{min}}}} )/[{({\textrm{RM}{\textrm{F}_{\textrm{max}}} + \textrm{RM}{\textrm{F}_{\textrm{min}}}} )/2} ]$, is < 0.1 ($p\sim 0.65$). Therefore, we posit that this small difference between our RMF calculation methods originates from the tilt, decenter and non-spherical nature of the surfaces.

6.2 RMF changes with axial positioning

Let us now explore the impact of change in distance between the eye and the ophthalmoscope, often referred to as vertex distance, because it can impact both the accuracy and the precision (repeatability) of retina image scaling. To illustrate the magnitude of this retinal scale change, we used our four-surface individualized paraxial and real ray tracing model eyes on five eyes, selected to sample our axial length and spectacle prescription ranges.

The axial shift of the eye relative to an imaginary ophthalmoscope was simulated as follows. First, we calculated the coordinates of the intersection of a ray at a plane tangent to the corneal apex at the ideal (zero shift) position (i.e., the exit pupil of the ophthalmoscope overlaps with the entrance pupil of the eye). Then, we moved the eye along the optical axis a desired distance from its ideal position to get the shifted ray coordinates at the plane tangent to the cornea at its apex. Finally, we trace this ray with the same incoming ray direction through these new corneal coordinates using both the paraxial approximation and real ray tracing to calculate the RMF, as described earlier in Section 2.5 & 3.

The paraxial calculation yields the analytical expression shown in Appendix A Eq. (S1). This lengthy formula can be thought of in terms of the ray height at the retina $({y_{\textrm{retina}}}$) in units of millimeters as a linear function of the ray height at the anterior cornea ($y{\mathrm{^{\prime}}_{ac}}$) after an axial shift $\delta z$,

Real ray tracing was performed using CODE V to calculate the RMF for various axial shifts of the eye at 820 nm, which is the wavelength used by the scanning laser ophthalmoscope for capturing the test images shown below (Heidelberg Engineering, Heidelberg GmbH, Germany). As with the plots in Fig. 7, we see good agreement between the paraxial and non-paraxial predictions of RMF change with axial positioning of the eye (see Table 4).

 

Fig. 7. Predicted RMF change with axial eye shift using non-paraxial ray tracing through a non-symmetric 4-surface model eyes (x and y axis, symbols) and paraxial ray tracing through a 4-surface paraxial schematic eye (dashed lines) in five eyes.

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Table 4. RMF relative change with ${\pm} 10$ mm of axial shift between the eye and the ophthalmoscope’s pupils, along the x and y axis in the real ray tracing calculation, as well as the paraxial calculations. The slope and constant in Eq. (19) for paraxial calculations are also listed.

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To illustrate that this change in RMF is both real and predicted by the proposed ray tracing, we captured images in the left eye of subject ADS_00302 at three different axial positions with and without a contact lens (-14.5D sphere equivalent) with the same Spectralis focus. These images are shown in Fig. 8 with annotations indicating absolute and relative distance changes between a blood vessel fork and the intersection of a blood vessel with the optic disc rim. The change of RMF in these images (∼13%) agrees reasonably well with the ray tracing (∼12%), given the ±2 mm error in our estimation of the ophthalmoscope’s axial shift. Importantly, the RMF change with axial positioning in the same eye is reduced to ∼1% when the subject was wearing contact lenses, as the images on the bottom row of Fig. 8 show. It should also be noticed, that when wearing the -14.5D power contact lens at an estimated d of 3.9 mm, the RMF should be multiplied by an ${M_{\textrm{spectacle}}}$ which according to Eq. (15) and (17) corresponds to a predicted change in magnification of ∼6% which is consistent with the 9% estimated from the images.

 

Fig. 8. Magnification changes with axial eye shift in retinal images from subject ADS_00302 (OS) with and without contact lens (-14.5D sphere equivalent). The diagonal dashed line joins the same two retinal features to show the vertical and horizontal scale changes in both absolute (arbitrary) and relative units.

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6.3 RMF changes with cycloplegic depth

Cycloplegia is most often induced in clinical and research settings by using cyclopentolate and tropicamide eye drops. Cyclopentolate induces “deeper” cycloplegia than tropicamide [97,98], which results in increased radii of the crystalline lens surfaces. In order to estimate the impact of such differences in cycloplegia depth, we simulated the use of cyclopentolate drops by using our biometry data, collected using a combination of phenylephrine and tropicamide drops, and increased the radii of the crystalline lens anterior and posterior surfaces by 0.59 and -0.16 mm, respectively, based on published values [97]. When comparing the RMF values calculated with phenylephrine-tropicamide and simulated cyclopentolate cycloplegia, we find that the RMF values are always larger for the simulated cyclopentolate by 0.08% on average, and always smaller than 0.13%. The refractive error differences due to the cycloplegia depth achieved by cyclopentolate and tropicamide (∼0.9 D) are comparable to the residual accommodation after using these drops [99], and thus, the impact of residual accommodation on RMF should be comparable too.

6.4 Axial vs. refractive myopia

The ability to obtain refractive index-corrected ocular surface radii and separation data from the new generation of optical ocular biometers allows the study of axial and refractive myopia with unprecedented detail. That is, whether the refractive error in myopic eyes is due to smaller radii of curvature of the ocular surfaces (refractive myopia) or increased distance between the crystalline lens and the retina (axial myopia) [100]. To demonstrate this potential, we plotted and calculated the correlations and significance values between the ocular surface radii and their separations in relation to the sphere equivalent of the subjects’ refractive error and the measured axial length. These results could be affected by the fact that biometry data from the left and right eyes are not independent. Hence, because of this and the very small number of eyes studied here, we think of the findings as informative, rather than authoritative.

The plots and data shown in Fig. 9 and Fig. 10, show weak correlation and statistical significance between surfaces radius and refractive error sphere equivalent, as well as axial length, whether considering all eyes in the study, or the higher myopes, defined as having sphere equivalent smaller than -4D. In contrast, vitreous thickness shows a strong correlation ($r = $ -0.92) and significance ($p \le {10^{ - 13}}$) with refractive error when considering all eyes in our data set, and when considering only the higher myopes ($r = $ -0.95; $p \le {10^{ - 5}}$) as shown in Fig. 10. Finally, in the highly myopic eyes we see that the aqueous thickness shows a statistically significant correlation with axial length ($r = $ 0.85) but not with refractive error ($r = $ -0.68), suggesting that although the aqueous thickness increases with eye size, it does not affect the refractive error (Fig. 10). Hence, the elongation of the eye is dominated by increased vitreous thickness, which is the strongest determinant of refractive error.

 

Fig. 9. Exploring the refractive myopia hypothesis in 34 eyes (left plots) and 9 myopic eyes with sphere equivalent ≤ 4D (right plots) through the correlation of the radii of the ocular surfaces against the spectacle prescription spherical equivalent and the ocular axial length.

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Fig. 10. Exploring the axial myopia hypothesis in 34 eyes (left plots) and 9 myopic eyes with sphere equivalent ≤ 4D (right plots) through the correlation of the thickness of the ocular media against the spectacle prescription spherical equivalent and the ocular axial length.

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7. Conclusions

Here we proposed and evaluated two methods for the calculation of retinal magnification factors at the fixation locus using chief ray tracing through models of the human eye with four refractive surfaces and homogeneous ocular media. The surfaces in these models were individualized using optical path length data captured with a new generation of ocular biometer and corrected for refraction at each surface. The proposed RMF calculation methods could be extended to other retinal regions, provided retina curvature can be measured.

RMF differences between paraxial ray tracing through rotationally symmetric eyes and real ray tracing through non-rotationally symmetric eyes averaged across meridians are within 0.1%, on average. RMF values calculated using real ray tracing varied by as much as 2.2% with retinal meridian due to the tilting and decentering of ocular surfaces.

The proposed model with four individualized refractive surfaces describes the anatomy of each eye more accurately than previous models. Also, the tracing of chief rays represents how retinal pictures are captured better than ray tracing through nodal or principal points, changing RMF values by as much as 5%. These RMF differences are doubled in imaging biomarkers derived from retinal area, such as cell or capillary densities.

The tracing of chief rays predicts RMF change with axial displacement of the eye relative to the ophthalmoscope which is approximately proportional to axial shift and refractive error. This is a known, but to the best of our knowledge not documented phenomenon, and that as we illustrated, can be easily tested. This change in scaling with axial positioning highlights that subject alignment when capturing retinal images is critical for both cross-sectional and longitudinal retinal imaging studies, as well as longitudinal imaging for clinical care.

We used the individualized four-surface model eyes and published data on biometry differences between cycloplegia induced with cyclopentolate and tropicamide to estimate the magnitude of RMF differences, finding these to be no larger than 0.13% in our subject cohort. Finally, we found highly statistically significant correlations between vitreous thickness with refractive error, and weak correlations with ocular surface radii, suggesting that axial myopia dominates over refractive myopia in our subject cohort.

In summary, the new ocular biometry data allow the development of more anatomically faithful individualized eye models, testing eye growth theories, and RMF calculation methods. We hope that the proposed RMF calculation methods proposed here lead to more accurate retinal scaling and improved biomarker sensitivity.

8. Proposed ocular biometry reporting

Unfortunately, most retinal imaging studies do not use ocular biometry for retinal image scaling [101]. Moreover, in surveying the literature for this study, it became apparent that, irrespective of the choice of RMF calculation method and type of biometry data available, retinal imaging biomarkers, individual biometric information and/or adequate RMF calculation methodology are rarely published. This represents a barrier to the comparison and/or aggregation of retinal imaging biomarker data originating from different investigative teams, sometimes even within the same institution. We therefore propose to address this problem by encouraging the inclusion of tables with individual biometry data as supplemental material in peer-reviewed publications and clinical trial reports (e.g., see Tables 5 & 6 in Appendix B), together with RMF calculation method descriptions with sufficient technical detail to enable reproducibility, or the citation of an open-access manuscript with such complete description.

Appendix A. RMF paraxial individualized due to axial shifting of the eye

Retinal magnification factor in µm derived from paraxial ray tracing through a schematic eye with four refractive surfaces separated by media with homogeneous refractive indices.

$$\textrm{RMF} = {10^3}\tan ({\textrm{HFOV}} )\left[ {g\delta z + \frac{{fg}}{{f({{n_c} - 1} )/{r_{ac}} + {t_a}({{n_a} - {n_c}} )/({{n_a}{r_{pc}}} )- 1}} + e} \right]$$

(S1)$$\begin{aligned} a & = {t_v}({{n_l} - {n_v}} )/({{n_v}{r_{pl}}} ) \\ b & = {t_l}({a + 1} )/{n_l} \\ c & = ({{t_v}/{n_v} + b} )({{n_a} - {n_l}} )/{r_{al}} \\ d & = ({{n_a} - {n_c}} )[{{t_v}/{n_v} + b + {t_a}({a + c + 1} )/{n_a}} ]/{r_{pc}} \\ e & = {t_v}/{n_v} + b + {t_c}({a + c + 1 - d} )/{n_c} + {t_a}({a + c + 1} )/{n_a} \\ f & = {t_a}/{n_a} - {t_c}[{{t_a}({{n_a} - {n_c}} )/({{n_a}{r_{pc}}} )- 1} ]/{n_c} \\ g & = c - d - e({{n_c} - 1} )/{r_{ac}} + a + 1 \end{aligned}$$

where $\textrm{HFOV}$ is the half-field of view, $\delta z$ is the axial shifting of the eye relative to the reference position defined as the anterior lens surface apex, ${r_{\textrm{ac}}},\; {r_{\textrm{pc}}},\; {r_{\textrm{al}}},\; {r_{\textrm{pl}}}$ are the radii of curvature of the anterior cornea, posterior cornea, anterior lens, and posterior lens, respectively; ${t_c},{t_\textrm{a}},{t_\textrm{l}},\; {t_\textrm{v}}$ are the center thickness of cornea, aqueous, lens and vitreous respectively, all in units of millimeters; ${n_\textrm{c}},{n_\textrm{a}},{n_\textrm{l}},{n_\textrm{v}}$ are the refractive index of cornea, aqueous, lens and vitreous, respectively.

Appendix B. Refractive index-corrected biometry data and RMF values

We propose that in addition to RMF calculation method and individual biomarker data, retinal imaging studies include individual ocular biometry data and RMF values, to facilitate the re-use of data across studies and meta-analyses. Table 5 provides an example of such biometry data report. Values that cannot be measured (irrespective of reason) could be replaced with assumed values. Table 6 below shows an example of RMF value report, in which we chose to report values calculated using multiple methods. With the exception of the rightmost column (which requires 28 Zernike polynomial amplitudes for each eye), all other RMF values can be calculated using the ocular biometry data included in Table 5.

Table 5. Ocular biometry refractive index-corrected data measured at fixation locus.

View Table | View all tables in this article

Table 6. Retinal magnification factors (RMFs) in $\mu \textrm{m}$/deg for posterior nodal distance scaling in Emsley’s reduced model eye (PND-E), posterior principal distance (PPD) scaling in Bennett-Rabbetts’ model eye (PPD-BR), PND in Gullstrand-Emsley’s model eye (PND-GE), PND in Li’s model eye (PND-L), PND in our paraxial model eye (PND-L) and chief ray tracing in our paraxial and non-paraxial model eyes (PI and NPI).

View Table | View all tables in this article

Funding

National Eye Institute (P30EY026877, R01EY027301, R01EY030361, R01EY031360, R01EY032147, R01EY032669); Stanford Center for Optic Disc Drusen at the Byers Eye Institute; Research to Prevent Blindness.

Acknowledgments

We would like to thank Zhenzhi Xia for initiating the literature review, Aubrey Hargrave for biometry data collection, and Amir Akhavanrezayat for assistance with retinal imaging. We are also grateful to (in alphabetical order) Rigmor Baraas, Phillip Bedggood, Andrew Bower, Steve A. Burns, Melanie Campbell, Toco Y. Chui, Hafeez Dhalla, Adam Dubis, Amani Fawzi, Angelos Kalitzeos, Zhuolin Liu, Andrew Metha, Jason Porter, Ethan Rossi, and Johnny Tam for comments and suggestions that greatly improved the manuscript.

Disclosures

X. Huang and A. Dubra declare no conflicts of interest. T. Anderson has financial interest in Cylite Pty. Ltd.

Data availability

Software and 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.

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