Technical notes on peripheral refraction, peripheral eye length and retinal shape determination

To give an overview of the misconceptions and potential artefacts associated with measuring peripheral refractive error and eye length, the use of these measures to determine the retinal shape and their links to myopia development. Several issues were identified: the relationship between peripheral refractive error and myopia development, inferring the retinal shape from peripheral refraction or eye length patterns, artefacts and accuracy when measuring peripheral eye length using an optical biometer.


INTRODUC TION
There is continuing interest in the role of peripheral refraction in the development and progression of myopia. This has inspired spectacle and contact lens designs that decrease positive (hyperopic) relative peripheral refraction. 1 Although these designs have proven clinical benefits in slowing myopia progression, 2 much remains unknown about how they work. Consequently, there is a rapidly emerging field that studies peripheral refractive errors and the contributing biometric factors. However, as most clinical equipment being used was designed for use along either the ocular visual axis or the line of sight, there are some issues and artefacts of which researchers should be aware. This work discusses the misconceptions and potential artefacts associated with measuring the peripheral

O R I G I N A L A R T I C L E
Technical notes on peripheral refraction, peripheral eye length and retinal shape determination David A. Atchison 1 | Jos J. Rozema 2,3 refractive error or eye length, with the use of these measures to determine the retinal shape and with their links to myopia development.

Peripheral refraction and myopia onset
The link between peripheral hyperopia and myopization seems firmly established in the literature to the point where the former should almost inevitably lead to the latter. The first report indicating this link was from Hoogerheide and colleagues, 3 which received wider attention thirty years later. 4,5 The general understanding is that these works reported a relative hyperopic peripheral refraction along the horizontal visual field of young adult males that was associated with future myopia development. Closer reading showed inconsistencies in this interpretation as the peripheral refractive error was often measured well after people did, or did not, develop myopia. 6 Subsequent clinical studies in children showed that relative peripheral hyperopia does not manifest itself until after myopia development. [7][8][9] Moreover, no relative peripheral hyperopia can be observed along the vertical field meridian since emmetropes and myopes both have relative peripheral myopia instead. 10,11 It seems unlikely that peripheral refraction would act only as a driver for myopia development along the horizontal field and nearby meridians. These observations clearly show that peripheral hyperopia is a consequence rather than a cause of myopia and that any role it plays in progression is very limited at best. As the use of positive additions in the peripheral field obviously slows myopia progression, 2 these peripheral corrections must affect another visual quality parameter driving myopia.

Retinal shape inferred from peripheral refraction
There has also been a tendency in the literature to match the peripheral refractive error pattern with a particular retinal shape. Relative peripheral hyperopia, by which is meant that the peripheral refraction is more positive or less negative than the axial refraction, has been associated with a prolate retinal shape; relative peripheral myopia has been associated with an oblate retinal shape. The terms oblate and prolate refer to ellipses that steepen and flatten, respectively, away from their vertices, which in this context approximately corresponds to the line of sight. 12 For wider angles, the posterior globe is generally oblate for most emmetropic and myopic eyes, but this tends to reduce for larger amounts of myopia. 13,14 It is likely that optical coherence tomography and its algorithms will develop to improve the determination of the retinal shape at relatively central regions, corresponding to the range of angles over which peripheral refraction is usually measured.

Alignment error
Optical biometers such as the IOL Master (zeiss.com) and the Lenstar (haag-streit.com) may be used to determine peripheral eye lengths. Under these circumstances, it is common practice to ask volunteers to fixate on a peripheral target. Such targets can be placed on a wall a few metres away, or as part of a Badal system in which an arm rotates around an axis passing through the eye's centreof-rotation. 15,16 The target positions corresponding to the different angles are easily calculated. However, if the target is attached to the instrument, as was done, for example, by Ding and Gu 17 or Mutti et al., 18 then an artefact may occur when the device is moved laterally for corrective realignment.
To perform a measurement, a partial coherence interferometer passes a narrow beam into the eye that is normal to the anterior cornea. 19 Figure 1 shows this with the eye aligned along the instrument axis with the target at A. There is a peripheral target at B attached to the instrument, which is placed to subtend the required field angle θ measured at the anterior cornea. When the eye rotates to look at this target, the instrument must be realigned so that the beam is again normal to the anterior cornea. This means that the target at B, attached to the instrument, moves from B to C. When the eye rotates to look at C because the ocular centre-of-rotation is behind the centre of curvature of the anterior corneal surface, the eye rotates by an angle θ′ that is smaller than θ.
If the errors involved are appreciable, then the operator should either change the target location to get the intended angle or calculate a new angle. The theory for investigating errors in field angles and locations of targets is developed in the Methods section.

Retinal shape inferred from peripheral eye length
An artefact can occur when the relative peripheral eye length (RPEL), that is the peripheral length relative to the central length, is used to indicate changes in retinal shape

Key points
• There is considerable interest in the roles of peripheral imagery and peripheral optics in the development of myopia. • The application of devices, intended for on-axis measurement of refractive error and eye length, to the periphery may produce incorrect interpretation and artefacts. • Estimates of the artefacts and possible mitigations are provided.
indirectly. The inference is that a more negative RPEL indicates a more curved retina, thus providing additional information beyond an increase in eye length. 20 This can be used to specify how myopia treatments affect retinal shape as well as altering the rate of change of axial length (AL). Such claims should be treated with caution, as can be demonstrated with a simplistic model eye that assumes that light directed towards the centre of curvature of the anterior cornea does not deviate on its way to the retina ( Figure 2). The model considers the retina of an emmetropic eye with a radius of curvature r r (solid curve) and of a myopic eye with a radius of curvature r r + Δr r (dashed curve). The emmetropic on-axis and peripheral eye lengths are AL′ a and AL′ p , respectively, while those of the myopic eye are longer by ΔAL′ a and ΔAL′ p . In this instance, AL′ a and AL′ p are the same so that the RPELs are equal for both eyes, F I G U R E 1 Field angles θ and θ′ before and after eye rotation to look at targets off the instrument axis. See text for details.

F I G U R E 2
A demonstration that a flatter retina of a myopic eye might be considered to have the same retinal radius of curvature as a steeper retina of an emmetropic eye if using on-axis and peripheral eye lengths to infer retinal shape. See text for details.
leading to the incorrect conclusion that the two retinas have the same curvature. The error arises from the fact that the measurements are taken relative to the centre of curvature of the anterior cornea rather than to the centre of curvature of the retina.
If, instead, the retinas had the same radius of curvature, then the myopic RPEL would be shorter for the same field angle and the myopic retina would incorrectly seem steeper than the emmetropic retina. For a myopic retina with a smaller radius of curvature than the emmetropic retina, one would correctly conclude that the myopic retina is steeper than the emmetropic retina, but the difference might be overestimated.
The Methods section presents the theory for investigating such errors when using peripheral eye lengths to estimate changes in retina shape.

Calibration artefact
Rather than the AL, optical biometers determine the total optical path length (OPL), the sum of the distances the light travelled through the optical media multiplied by their respective refractive indices. Since especially the lens indices may differ between individuals and can currently not be measured in vivo, assumptions must be made about these values, as well as the relative thickness of the optical media, to estimate the AL from the OPL. In the IOL Master, this estimation is done using 21 : Although this calibrating equation serves well for on-axis AL measurements in clinical practice, it cannot be taken for granted that this would also be the case peripherally since the relative thicknesses of the optical media would be different. 22 In particular, the higher refractive indices of the crystalline lens may have an effect, as was reported for accommodating eyes both on-axis 23 and peripherally. 16 Errors using this estimation are investigated in the Results section.

Alignment error
Assume two targets at a plane located at a distance l from the anterior cornea, one on-axis target A, and an off-axis (peripheral) target B at a field angle θ measured at the anterior cornea ( Figure 3). The distance h between the targets is: As described in the Introduction, to perform a measurement a partial coherence interferometer passes a narrow beam into the eye that is normal to the anterior cornea. When the eye rotates to look at the peripheral target B, the instrument must be realigned so that the target has a small shift Δh when moving from B to C. The ocular rotation (or field angle) reduces to θ′ because the ocular centre-of-rotation is behind the centre of curvature of the anterior corneal surface. This gives an alternative equation where l + r + is the distance from the centre of curvature of the anterior corneal surface to the new on-axis target D, with r being the radius of curvature of the anterior cornea. Distance represents the backward shift of the corneal centre of curvature due to the rotation and is given by with l cor the distance from the centre of rotation to the anterior cornea. Since is generally small compared with r, the approximation is appropriate.
From the right-hand sides of Equations (2) and (3) follow the solution for ′ : From the geometry, it follows that realignment shifts the device and any targets attached to it by To correct the field angle from θ′ to , the eye should fixate on a new target E (Figure 3). Relative to the ocular centre-of-rotation, target E is located at a height h * * consists of h * , the distance between the realigned instrument position D′ and target E, and which is the distance between A and D. From this, it follows that the correct height is The ratios of the incorrect angle and height to the correct ones are then given by

Retinal shape inferred from peripheral eye length
Verkicharla et al. 24 developed methods of determining a conicoid retinal shape from eye length measurements based on the Le Grand schematic eye. As mentioned in the Introduction, an artefact can occur when the RPEL, that is the peripheral length relative to the central length, is used to indicate changes in retinal shape indirectly. This is explored using the Stage 1 approach to retinal shape determination by Verkicharla et al., 24 which assumes that the light is not refracted within the eye (Figure 4). The same terminology is adopted as before, and with distances from surfaces to positions on their left taken as negative. This is in contrast with the previous section, where the radius of curvature of the retina was taken as positive. The sag Δz 3 at the retina is given by: where r cv is the radius of curvature of the anterior cornea, θ is the field angle and AL ′ a is the axial length of the eye. The corresponding height at the retina y r is given by: where AL ′ p is the peripheral eye length. For a spherical retina of radius of curvature, r r , y r relates to Δ z 3 as: Filling Equations (12) and (13) into Equation (14) gives: which can be expanded as: and solved as a quadratic equation for AL ′ p . The RPEL, that is the peripheral eye length minus the axial eye length, is then Figure 4 shows the retinas of both emmetropic and myopic eyes with the same radius of curvature (solid and dashed curves, respectively). The eye lengths of the myopic eye are greater than those of the emmetropic eye by ΔAL′ a and ΔAL′ p . In the Results section, the simple approach described here will be used to determine differences in RPEL in emmetropic and myopic eyes with the same retinal shape. The approach is compared with exact raytracing.

F I G U R E 3
More detailed view of Figure 1 showing the parameters when an eye rotates to fixate peripheral targets attached to the instrument and there is compensatory movement of the instrument to maintain alignment.

Alignment error
To put the misalignments into numbers as applied to an instrument for measuring peripheral eye lengths, assume a target plane at a distance of 68 mm from the anterior cornea, corresponding with the distance found for the Lenstar, a centre-of-rotation distance 15 mm and anterior corneal radius of curvature of 8 mm. Figure 5 shows errors for angles and heights. For a field angle θ of 30°, Equations (5), (9) and (2) give θ′ = 27.0°, h* = 44.2 mm and h = 39.3 mm. From Equations (10) and (11), percentage errors of θ′ and h are −10% and −12%, respectively. To give numbers as applied to an instrument for measuring peripheral refraction, such as one of the Grand-Seiko autorefractors, we take the fixation targets to be placed in a plane 200 mm from the anterior cornea. Instead of using the radius of curvature of the cornea, we use the entrance pupil position from the anterior cornea as it is this to which the instrument aligns. Taking this as 3.0 mm inside the eye and as r, for a field angle θ of 30° we have θ′ = 29.4°, h* = 118.1 mm and h = 115.5 mm. Percentage errors of θ′ and h′ are both −2%. These are much smaller than for the Lenstar situation and would become smaller still when the target plane moves further from the eye (e.g., both −1% for 500 mm distance). Figure 6 shows solutions for the RPEL using the biometry values of the Le Grand schematic eye model 25 of r cv = 7.8 mm, r r = −12 mm and AL ′ a = 24.197 mm. This shows the typical pattern of RPEL becoming more negative as peripheral field angle increases. 26 These results show a great similarity to those provided by off-axis ray tracing of the Le Grand model that can be attributed to the ray paths being directed near the nodal points, resulting in only minor deviations of the ray, for example, only +1.07° for a 30° field angle. For larger angles, the differences between the RPELs by the simple approach in the Methods section and by ray tracing increase from −0.014 mm at 20° to −0.085 mm at 40°.

Retinal shape inferred from peripheral eye length
Increasing the AL by 0.5 mm without altering the retinal shape, corresponding to a refractive change of −1.32 D, causes the relative peripheral AL to decrease more with the increasing field angles (e.g., by 0.122 mm at 30°; Figure 7). This is a consequence of the position of the ray path intercept along the optical axis, which is 4.2 mm in front of the centre of retinal curvature.
Another factor that could influence the difference in RPELs is the ray height at the retina, which is greater for the myopic retina than for the emmetropic retina ( Figure 2). Plotting RPEL against retinal height y r , calculated using Equation (13), reduces the differences in RPELs between the eyes by nearly 50%, but does not fully eliminate the distortion (Figure 8).

Calibration artefact
To estimate the influence of the path lengths of rays passing through the Le Grand schematic eye, these were determined using ray tracing, converted to the total OPL using Haigis' indices 21 and used to estimate the AL using Equation (1). This shows that the equation underestimates the relative peripheral AL of the Le Grand model by small amounts, for example, by 0.015 mm at 30° (Figure 9). Repeating the analysis for an additional AL of 0.5 mm leads to a similar result, albeit with a larger difference between the model and the estimation (e.g., 0.019 mm at 30°), showing that AL also affects this artefact.

DISCUSSION
The Introduction identified five issues associated with measuring and interpreting peripheral refractive error and peripheral eye length, three of which led to theoretical evaluations.
The misinterpretation of the Hoogerheide et al. 3 data by themselves and later authors caused a widespread belief that peripheral hyperopia leads to myopia, although the reverse is far more likely. 6 But since optical treatments of peripheral refractive errors seem to have clear clinical benefits, there must be other aspects of peripheral visual image quality that direct myopization. Regardless, it is common in the literature to infer the retinal shape from peripheral refraction or eye length measures. Although the peripheral refraction strongly depends on the retinal shape, the former cannot be used to reliably interpret the latter as refractive errors are also affected by the cornea and lens. Related to this, the terms "oblate" and "prolate" are often used to refer to retinal shape. These give the impression that the main retinal change as myopia progresses is a flattening of the periphery, while the more important change is probably an overall steepening around the posterior pole.
The retinal shape can be determined qualitatively using changes in RPEL, keeping in mind that this may be susceptible to distortions such as sensitivity to AL changes, even for a fixed retinal shape. This may be improved by no longer plotting the retinal shape as a function of the field angle as shown in Figure 6, but rather as the distance from the eye's optical axis using Equation F I G U R E 6 Relative peripheral eye length as a function of field angle, using parameters of the Le Grand schematic eye. 25 This is given for a simple model without ray deviation in the eye and for raytracing. Results are shown for an emmetropic eye and for an eye with an additional length of 0.5 mm without changes to the retinal shape.
F I G U R E 7 Difference in relative peripheral eye length between an emmetropic eye and the same eye with an additional 0.5 mm of axial length without any changes to the retinal shape. These are derived from the results in Figure 6 for a simple model without ray deviation in the eye and for raytracing. The negative values mean that the myopic eye has more negative relative peripheral eye length than the emmetropic eye. (13). This reduces the errors involved in interpretation of retinal shape by about 50% for the parameters used in this investigation (Figure 8), but further investigations may help reduce this error further. Rather than running the risk of incorrectly interpreting differences in retinal shapes based on different degrees of myopia or the effects of optical treatment on retinal shape, peripheral eye lengths can be converted into estimations of retinal shape instead. 22,24 While clinical instruments are typically designed for onaxis measurements of eye length or refraction, there is nothing fundamentally wrong with the peripheral use of these instruments, but it is essential to be cautious when interpreting such results. One aspect is that peripheral measurements may require a realignment of the instrument that could alter the field angle being tested, especially for targets relatively close to the eye or attached to the instrument. For targets placed in a plane only a few centimetres from the eye, errors  can exceed 10% ( Figure 5). The equations listed above can help to determine these magnitudes so that targets can be placed at the correct positions for particular field angles.
Finally, the artefact resulting from the peripheral use of biometers with an on-axis calibration is modest and can likely be ignored (Figure 9). Even so, it is conceivable to recalibrate the instrument for peripheral use to minimise this issue.
The treatments of this paper have several limitations that should be acknowledged, particularly with regard to parameter values. First, the treatment for alignment assumes a fixed ocular centre-of-rotation. While there is probably no such position, estimates have been made in the literature. Over a ±11.9° field, Ohlendorf et al. 27 determined l cor to be 15.3 ± 1.5 mm horizontally and 12.5 ± 1.4 mm vertically. Both showed low positive correlations with central AL although not with refraction. Similar horizontal and vertical differences have been reported by Fry and Hill 28 (14.9 and 12.3 mm, respectively) and Grolman 29 (14.5 ± 0.55 and 13.1 ± 0.7 mm, respectively). Lower l cor values compared with the 15 mm used here will decrease the errors of h′ and θ′ relative to those reported above. Second, the alignment and peripheral eye length treatments assume spherical anterior corneas. In the Lenstar example with alignment, a change of ±0.5 mm in r due to asphericity or toricity would alter the error in θ by ±0.5%. And third, we assumed the correctness of the refractive indices and retinal shapes used in the Le Grand eye. We did not account for the gradient index of the lens but, given the broad shapes of the iso-indical surfaces and the minute effect, this is unlikely to have much influence on results in Figure 9.
In conclusion, this paper demonstrated several issues with peripheral measurements of refraction and eye length that should be considered when interpreting such data, for example, for myopia studies. A solution was proposed for the alignment issues, as well as a partial solution for the distortions to the retinal shape when derived from peripheral eye length measurements.