Potential measurement error from vessel reflex and multiple light paths in dual‐wavelength retinal oximetry

This study aims to characterize the dependence of measured retinal arterial and venous saturation on vessel diameter and central reflex in retinal oximetry, with an ultimate goal of identifying potential causes and suggesting approaches to improve measurement accuracy.

previously, and a simple correction has been incorporated into oximetry software (Hammer et al., 2008).Since then, several studies have reported influences from other retinal structures or hemodynamic factors on retinal saturation.One study noted ambiguity in interpreting a finding that oxygen saturation is correlated with nerve fibre layer thickness since the cause could be physiological or artifactual (Mohan et al., 2016).Another study reported that venous saturation is elevated in smaller peripheral veins and falls along the direction of flow toward the optic disc (Paul et al., 2013).Blood flow velocity, known to vary with vessel size, was shown to affect retinal vessel saturation, causing measured saturation to decrease at higher velocities (Jeppesen & Bek, 2019).These reports offer credible new insight into retinal oxygen utilization; however, to verify these observations and advance clinical applications accordingly, measurement artefacts affecting oximetry need to be recognized and understood.At issue are the current methodologies in dual-wavelength oximetry and our understanding of the light-vessel interactions involved with saturation measurements.
Retinal vessel oximetry was first introduced in 1960 by Hickam et al. (1959) and was later presented in a comprehensive study of retinal blood oxygen saturation (Hickam et al., 1963;Hickam & Frayser, 1966).Photographic film was used to record optical density changes associated with the oxygen saturation of retinal vessels on the optic disc.This work made three important observations.First, the ratio of optical densities measured at oxygen-sensitive and insensitive wavelengths was observed to be linearly related to the systemic oxygen saturation.This is the basis of dual-wavelength retinal oximetry, although systemic saturation references are not always used (Hammer et al., 2008).Second, using measurements in glass capillary tubes on a reflecting background, fixed alterations in saturation caused larger changes in the optical density ratio (ODR) as the capillary diameter became smaller.This was an early indication that measurements from vessels on the retina could be sensitive to vessel diameter.Last, light returning from the capillary tube increased when the background at the side of the tube was made more reflective.This indicated that in vivo vessel light measurements could be affected by light entering the vessel from the surrounding background.These early observations offer insight for evaluating the accuracy of retinal oximetry measurements using dual-wavelength oximetry.
Previously, dual-wavelength oximetry of small areas of the retina reported an inverse relationship between arterial saturation and diameter during pure oxygen breathing (Beach et al., 1999).Pure oxygen brings saturation to 100% in feeder vessels and arterioles supplying capillary networks and raises venous saturation; little or no variation in saturation would be expected at different arterial diameters, thus oxygen breathing is used for calibration of retinal oximeters.With improvements in technique, digital oximetry has captured larger areas of the retina, allowing the different orders of the vasculature to be compared within a single recording.Once colour mapping of vessel saturation was introduced, it became obvious that the measured saturation in retinal vessels is influenced by the vessel diameter.
In the artery, we expect saturation to decrease slightly as oxygen is extracted along the direction of flow, from larger to smaller diameter vessels.In measurements from a population of subjects, Hammer et al. (2008) found the opposite trend, where the artery saturation increased slightly as vessel diameter decreased.Veins also showed this trend, although increases in saturation were greater than in arteries.If measured saturations in veins exhibit strong diameter sensitivity, we would expect to see changes in saturation where the vessel diameter also changes (e.g., at a bifurcation).As an example, Figure 1a shows a venous bifurcation where vessel diameter increases along the direction of blood flow at the branch point.Saturations are nearly the same in the daughter vessels (55% and 57%).Thus, regardless of the flow in each daughter vessel, we would expect the saturation of mixed venous blood downstream of that bifurcation (i.e., closer to the optic disc) to remain nearly the same; however, we measure a lower saturation value (49%) in the larger vessel.Such a diameter-dependent decrease in measured saturation at a bifurcation may not be evident if the blood saturation in each daughter branch is different.In that case, the saturation downstream of the bifurcation would depend on the relative flow and saturation levels in each daughter; diameter effects on the measurement would remain but be less obvious.
A condition seen in many oximetry saturation maps is the apparent desaturation (yellow to green colour shift) at the junction between small side-branch venules and a large vein (Figure 1b).Large veins carry blood returning over substantial regions of the retina, including the periphery where venular saturation may be reduced (Bek & Jeppesen, 2021) due to lower blood oxygen content in terminal arterioles reaching more distal capillaries F I G U R E 1 Venous bifurcation (a) and large central veins with side branches (b, upper and lower panels).In a, saturation and vessel diameter of the main and daughter vessels are shown next to the locations of measured segments (white lines).In b, saturations are measured in large central veins upstream (US) and downstream (DS) of three intervening side branches, and the saturations in each side branch, are given next to measurement locations.In the upper panel, the fourth side branch is not considered since our estimate of its flow, determined as described in Appendix S1, was small compared to those of the other branches.(Duling & Berne, 1970;Liu et al., 2009).Blood returned by side branches in the central part of the retina may exhibit a relatively higher saturation due to greater blood flow (Feke et al., 1989;Hickam & Frayser, 1966) and reduced oxygen extraction in this region.The abrupt change in measured saturation at the side-branch junction thus could be an indication of a diameter artefact or could be the result of dilution of more highly saturated blood into a larger volumetric flow with lower saturation.However, if the abrupt saturation change is not due to a diameter artefact, one would expect saturation in large veins to increase in the direction of flow after time for oxygen uptake in red cells (Coin & Olson, 1979).Figure 1b shows this situation in two large temporal veins with branching venules from one retina where saturations are measured upstream of the most distal branch (US), downstream near the optic disc (DS) and in each branching vessel.The side-branch saturations exceed that of the large vein (upstream location) by 35% ± 9.6SD in the upper panel and by 35% ± 6.4 in the lower panel.In the first case, we observe a substantial increase in saturation in the direction of flow after inflow from side branches, suggesting oxygen is added from a relatively higher venular saturation.In the second case, saturations upstream and downstream of side branches are not appreciably different, suggesting that the actual side-branch saturations are comparable to that of the large vein and abrupt changes at side-branch junctions may result from a diameter artefact.It is possible that a diameter artefact causes measured saturations in central venous side branches to be higher than the actual value, which in turn could be higher than that in the large vein.Although mechanisms underlying these different outcomes are beyond the scope of this paper, the apparent contradictions could perhaps be resolved if blood flow measurements were combined with oximetry (see Section 4; Appendix S1).
Here we present a systematic study of arterial and venular saturation measured at different vessel diameters throughout the retinal vasculature, showing trends related to diameter in individual subjects.As a follow-up to observations in blood-filled glass capillaries (Hickam et al., 1963) and an analysis of multiple light paths in oximetry (Smith et al., 2000), we examine effects of single and double-pass light paths in vessels which interact with blood and the retinal layers differently and which may produce a diameter-dependent measurement.Other factors, such as light absorption by retinal pigment and stray light entering the recording by reflection from anterior structures, could also produce errors in saturation measurements.We note that in retinal vessels, rheological effects produce a sensitivity of measured saturation to blood flow velocity (Jeppesen & Bek, 2019), which in vivo is closely related to diameter.This finding is discussed below in relation to the structural influences on diameter sensitivity we describe from multiple light paths.We also present new insight into non-physiological (greater than 100%) arterial saturations that come from negative values of the vessel optical density obtained from oximetry images.We note that this optical effect can also cause errors in venous measurements.We conclude with suggestions for improving the accuracy of dual-wavelength oximetry.

| Selection of imagery
Images used in this work were obtained from studies approved by bioethics committees at Landspitali University Hospital in Reykjavik, Iceland and the Indiana University School of Medicine, with all subjects providing informed consent.Images were selected from 10 healthy subjects including eight Caucasian and two of African descent without signs of retinal disease.

| Oximetry image analysis
Images were acquired using mydriasis and flash imaging with a Topcon DX fundus camera equipped with the Oxymap T1 retinal oximeter.Selected images were processed offline with Oxymap Analyzer (OA) software version 2.5.2 using the version 1 vessel detection method and preinstalled calibration data (Oxymap ehf).Although previous oximetry studies have employed built-in corrections for diameter, we turn off diameter correction in this work so that we can characterize measurement error in terms of actual recorded light measurements.
Portions of vessel segments are selected manually from the Oxymap images with a point-and-drag method, avoiding overlap with bifurcations and side-branch junctions.Segments are selected from all retinal quadrants and are a minimum of 50 pixels in length with diameters ranging from 20 to 200 μm.With these constraints, opportunities for vessel selection depended on the position of vessel networks in images, resulting in as few as 16 and as many as 105 good segments in each subject.Figure 2 shows a typical selection of artery segments (bright highlights on vessels) from an oximetry image obtained from where λ represents wavelengths 570 and 600 nm, I outside is light intensity from the retina adjacent to the vessel and I inside is light intensity inside the vessel near the centre.Figure 2 (inset, lower left) shows measurement locations (white arrow) within a vessel selection.Yellow and green points give the locations of I inside measurements along the interior path at 600 and 570 nm, respectively; red points show where measures of I outside occur adjacent to the vessel.Constants α and β are experimentally determined calibration constants (Beach et al., 1999;Hardarson et al., 2006) which are supplied by the Oxymap Analyzer software (α = −0.953,β = 1.16, in version 2.5.2).ODs are normally positive-valued but can fall below zero, as discussed below.In those cases, the negative OD can cause Equation 1c to give non-physiologically high (>100%) saturation measurements.Saturation can also be determined using predetermined values of the arterial ODR obtained during pure oxygen breathing (Beach et al., 1999).This approach, which is not evaluated here, is used in a different commercial version of retinal oximetry (Hammer et al., 2008).

| Diameter dependence
The dependence of measured vessel saturation on vessel diameter is characterized by linear regression of mean segmental saturations against the mean segment diameter.This procedure is applied to measurements from individual subjects and to results pooled from all subjects.The goodness of line fits is given by the standard error (SE) of regression.Significance of non-zero trends (slopes) and significant differences between vessel slopes of paired subjects are determined from p-values (see Table 2; Appendix S2).Artery and vein dependencies are determined separately.

| Conceptual light path model
We develop a conceptual model to calculate intensities of measured light making one-way (single pass) and round-trip (double pass) paths through vessels, which exhibit absorption and scattering properties of blood (Cheong et al., 1990;Friebel et al., 2006;Prahl, 1998).As light is absorbed and scattered, intensity decreases with the distance travelled in the vessel.We use the optical penetration depth (δ, Equation 2), which gives the distance at which intensity falls to 1/e (37%) of the initial value (Jacques, 1999(Jacques, , 2018)), to define a length constant for exponential decay of the intensity: Here, μ a is the absorption coefficient and μ � s is the reduced scattering coefficient for blood.We calculate the absorption coefficient for arterial and venous blood using Equation 3: where λ is the wavelength of light (570 nm reference and 600 nm measurement) and S is vessel saturation, which is assumed to be 95% (S = 0.95) in arterial blood and 65% (S = 0.65) in venous blood.The molar concentration of haemoglobin, C Hb , is 2.33 mM (0.15 g/dL).Light extinc- tion coefficients HbO2 (oxyhemoglobin) and Hb (deoxyhemoglobin) are obtained from Prahl (1998).We assume μ � s = 22 cm −1 , which is determined for flowing red blood cells at physiological haematocrit (Friebel et al., 2006).This value is appropriate for both wavelengths since saturation has been shown to have little influence on reduced scattering at oximetry wavelengths (Friebel et al., 2009).Values of these assumed optical properties are given in Table 1.
In the conceptual model, light is transmitted over a path length, L, representing the diameter of a vessel.Equation 4 gives the light intensity (I) after transmission through a distance X, where I o is the initial intensity and δ is the length constant obtained from Equation 2: A diagram of single-pass and double-pass light paths through a vessel and the surrounding retinal layers is shown in Figure 3.Light filtered over a broad range covering oximetry wavelengths (Beach, 2014) is incident on vessels and the surrounding retina (I 0 , solid lines).Path (1a) OD meas (λ) = log 10 I outside (λ) 600) OD( 570) , 1 represents light returning from retinal and choroidal layers without passing through a vessel.Both oximetry wavelengths (570 and 600 nm) return from the retinal layer (dot-dashed line) after multiple scattering by tissue elements and backscatter from melanin granules in the retinal pigment epithelium (RPE).From the choroid (dashed line), larger blood volume results in relatively weaker light return, particularly at 570 nm where haemoglobin light absorption is higher.Path 2 (single-pass reflection) represents light return by single-pass transmission through the vessel.The composition of wavelengths returning from retinal (dot-dashed line) and choroidal (dashed line) layers is the same as for path 1; however, path 2 requires lateral spread behind the vessel (Smith et al., 2000) which results in smaller light return.The extent of lateral spread depends on vessel diameter.Path 3 (double-pass reflection) represents the light that has entered, passed through and returned through the vessel after reflection from below.The path of light under the vessel is the same as that for retinal reflection, with additional light absorption by the vessel wall.We evaluate how diameter dependence may arise from the combination of single and double-pass transmission in vessels and assess the influence of retinal reflectances associated with each path.Oximetry measurements are obtained from light near the vessel centre; due to the lateral spread in the retinal layers, these measurements depend on the vessel diameter.To model light spread in the retina and the choroid, we use point-spread functions (PSFs) with standard deviations (σ) calculated for the nasal fundus (Hodgkinson et al., 1994).These have relatively narrow (68 μm) and wide (168 μm) standard deviations at 570 and 600 nm, respectively (Smith et al., 2000).For simplicity, we use a Gaussian form of the PSF to determine the intensity of light returning at the centre of the vessel, located a distance of half the diameter from the vessel edge.A diameter-dependent reflection coefficient for the singlepass path at each wavelength, R sp (λ, L 2 ), is therefore a Gaussian function of path length, L/2, representing half the vessel diameter: where R sp (λ, 0) is single-pass reflectance at the ves- sel edge.The single-pass light intensity returning from the vessel centre is attenuated by absorption and scattering during passage through the vessel according to Equations 2 and 4 and is given by: where I o represents incident light intensity at the surface of the vessel or retina.During the development of the Oxymap retinal oximeter, the incident light intensity measured by the instrument from a spectralon target at both oximetry wavelengths was nearly equal, thus we assume a single value of I o = 1 in the model.In double-pass transmission, the retinal light path uses diffuse reflection directed backwards without a lateral component, with a reflection coefficient defined by R dp .A second exponential decay represents loss of light during the forward pass through the vessel to the retina.The model for double-pass light return is expressed in Equation 6: Light intensity returned directly from the retina is given by: where R retina (λ) is retinal reflectance at each wavelength, with values obtained from human subjects (van Norren & Tiemeijer, 1986, see Table 1).The total light return (I vessel ) at each wavelength for a pathlength of L is given by the sum of the single-pass and double-pass light returns: Results using the model can be compared with oximetry measurements most directly using the optical density ratio (ODR), which is linearly related to saturation (Equation 1c).The model ODR is calculated using the relationships from Equation 1: , ODR model = OD(600) OD( 570) where I retina replaces I outside and I vessel replaces I inside in Equation 1a.
Because the model ignores vessel circularity and does not account for recording geometry and instrument response, it does not yield actual values obtained in recordings.The model is instead used to predict the trend of measured ODRs across diameter.To make comparisons between the model and measurement values, we transform the measured and modelled ODR values to a common scale by normalizing each by their value at a pathlength (diameter) of 200 μm, the largest of our sampled venous diameters and where we find that the diameter trend is most level (Figure 6).The normalization value for veins is determined from the average ODR of the four largest diameter measurements; the normalization value for arteries is obtained from the best line fit evaluated at a diameter of 200 μm.Model parameters R sp and R dp at both wavelengths are then determined separately for the artery and vein by a least squares optimization of the normalized model inputs, fitting model-predicted ODR values to measured ODRs subject to the following constraints.We assume that similar paths through retinal layers occur for light entering at the side and under the vessel and assume a small degree of light loss at the vessel wall, so that R dp is equal to or slightly less than R retina .Since R sp involves lateral transmission in the retina, we assume its value is smaller than the direct retinal reflection R retina .Specifically, double-pass reflectance (both wavelengths) and single-pass reflectance at the vessel edge (600 nm) are each allowed to vary between 80% and 100% of R retina .Since retinal light absorption is greater at 570 nm, we constrain single-pass reflectance at 570 nm to be no more than 80% of the singlepass reflectance at 600 nm.Constraints on R sp apply only at the vessel edge, which sets the effective reflectance at vessel centre according to the Gaussian profiles.We also obtain a best possible fit where all of the model parameters (reflectances R dp , R sp and R retina ) are allowed to change without any constraint for comparison with our best-fitting model.Goodness of fits are indicated by the residual error (y − ŷ) versus diameter, where (y − ŷ) is the difference between the measured value and the model prediction at the same diameter, and by the root mean squared error ( where N is the number of observations.

| Diameter dependence of vessel saturation
Figure 4 shows saturation levels obtained from 10 subjects in segments of arteries (A) and veins (B) plotted against vessel diameter.Slopes for trend lines in Figure 4 are given in Table 2; Appendix S2.In Figure 4a, the artery saturation levels are compiled from all subjects.The red symbols indicate saturation levels obtained when any occurrence of a negative vessel optical density (OD) is excluded; black symbols give saturation values obtained when negative ODs are present in the measurement.Saturations that include negative ODs occur mostly in smaller vessels, although they extend over the entire range of vessel diameters and can cause a marked increase in saturation in the affected vessel segments.The trend-line calculations exclude negative ODs, and the pooled-data analysis accounts for withinsubject correlations by using a regression model that includes subject-specific random slopes and intercepts.The trend associated with all subjects (Figure 4a) has a non-significant slope of −0.007%/μm (standard error, SE = 0.021).Panels (i-x) show segmental saturations across diameter from individual subjects.These individual arterial saturations show both positive and negative trends that are significantly different from zero in five subjects.Overall, there is a significant difference in arterial slopes between subjects (p < 0.0001, one-way ANOVA).Posthoc analysis for significance differences between pairs showed that 16% of pairings between subjects produced a significant difference in the slope (Appendix S2).
Venous saturations pooled from all subjects (Figure 4b) show a much more pronounced diameter trend than those of the artery.Green symbols show saturations from segments that contained no negative ODs. Black symbols represent vein segments where negative ODs were found and included in the calculation of saturation.Far fewer negative ODs were observed in venous segments, and all were in segments of diameter <50 microns.In contrast to arteries, venous saturations pooled from all subjects were observed to increase at a greater rate as the diameter decreases.Over the full range of diameters, a single line fit for all subjects gives a slope of −0.180%/μm (SE = 0.022).Panels (i-x) show venous saturation trends for individual subjects.Line fits give negative slopes in the range from −0.066%/μm to −0.282%/μm, which is a four-fold difference in the diameter sensitivity across subjects.We find no effect from negative ODs on venous trend lines.Over all subjects, veins showed a significant difference in slopes, as did arteries and 18% of pairings between subjects had significantly different slopes (Appendix S2).All venous saturations obtained from our subjects were above 45%; however, lower values have been previously reported (see Section 3.3).
Effects of vessel diameter and negative OD are summarized in Table 3.Both vessel types show increases in measured saturation as vessel diameter decreases, though the change is substantially greater in the veins.When measurements from all vessel sizes and subjects are considered together, the inclusion of negative ODs results in a small (<1%) increase in the mean artery saturation and negligible change in the vein saturation.However, this apparently small effect can be misleading.As shown in the next section, considerably higher artery saturation will result when the measured vessel includes multiple occurrences of negative ODs (see Figure 6e).From all vessels measured, we find a considerably higher percentage of vessel segments with negative ODs in arteries (11.46%) than in veins (0.78%).

| Negative ODRs cause saturation over 100%
Retinal vessels, particularly arteries, can show strong reflections along the vessel centre line (central reflex), which raise the vessel brightness above that of the retinal background.Figure 5a shows images of retinal arteries obtained at oximetry reference and measurement wavelengths (570 and 600 nm).The images in the top and middle rows of Figure 5a show peripheral arteries with low and high central reflexes, respectively.The bottom row of Figure 5a shows a central artery exiting the optic disc, showing a diffuse interior illumination at 600 nm, which exceeds that of the retinal background along the length of the vessel.In Figure 5b, intensity profiles are provided for the yellow-lined regions marked on the images in panel (a).These profiles indicate a weak central reflex at both wavelengths (Figure 5b, top), a stronger reflex at 600 nm equal to the background intensity (red line, Figure 5b, centre) and a reflex exceeding the brightness of the background (red line, Figure 5b, bottom).Measurement locations inside the vessel that coincide with a strong reflex can yield saturations above 100%.Figure 5c was obtained from the Oxymap Analyzer software and shows measurement locations inside (green and yellow points) and outside (red points) of a vessel with a moderately strong reflex.Near the vessel branch, interior points move to either side of the reflex as expected, however, some overlap the reflex.In the central artery (Figure 5a, bottom row), the interior measurement continuously exceeds the retinal intensity for over several hundred microns of vessel length.In both cases, the vessel OD at 600 nm, and in turn the optical density ratio, approaches zero or becomes negative.This can be verified by Equation 1a: (OD meas (λ) = log 10 (I outside (λ )/I inside (λ)), since OD meas (λ) becomes negative if I inside (λ) is greater than I outside (λ). Figure 5d shows saturations in one subject from Figure 4a, panel (vi).Open symbols indicate saturation values where negative ODs are excluded from calculations, and black-filled symbols give the saturation levels obtained from the same segments where all measurement locations were included.Most of the symbols coincide below 95% saturation, whereas departures are observed primarily above 100%.Figure 5e shows segmental ODRs (blue symbols, all subjects) calculated from segments containing one or more negative OD plotted against the negative OD rate (fraction of negative ODs in the segment).Orange symbols show the segmental saturations obtained from these ODRs.The line fit through saturations (Figure 5e) shows that for each 10% increase in the negative ODR rate, measured saturation increases by 3.5% (e.g. a saturation of 100% becomes 103.5%).4 for root mean squared errors for constrained and unconstrained model fits.

| predictions
saturations were 48% within a confidence interval of and Geirsdottir et al. (2012), who showed that the lowest venous saturations fall outside a normal distribution.

| Diameter correction
This study reveals the very complex relationship between saturation and vessel diameter obtained from oximetry measurements.We find little change in the saturation across diameter in arteries but note significant increases in venous saturation as vessel diameter decreases (see Table 3).Across all individual subjects, we find a significant difference in the slope of measured saturations versus diameter (p < 0.0001, see ANOVA analysis in Appendix S2).This observation suggests that current compensation methods for the vessel diameter may be applicable to averaged data from pooled measurements but could be improved for measurements from individual subjects since there is no way to know if the correction is removing or adding to the effect caused by diameter.Since a priori, we cannot describe a fixed trend for all subjects, a discovery of the trend in an individual subject may be needed to improve the correction for vessel diameter.One approach would use an automated pre-analysis of the oximetry image to ascertain ODRs of large (>150 μm) and small (<50 μm) vessels to correct subsequent measurements of saturation over the range of vessel size.The applicable correction may require compensation for nonlinear trends predicted in our model for multiple light paths.This procedure may be appropriate for compensating diameter sensitivity in arteries since these vessels comprise blood flow from a single source through vessels of successively smaller diameter.It is not clear if the method can also be applied in veins since flow converges from peripheral and more central parts of the retina and these regions could differ in oxygen utilization, presenting different saturations in vessels of the same diameter.Thus, while our multiple light path model can explain diameter sensitivity in the vein, it is not the only possible explanation.If measurement over a range of vessel diameters is not required, but instead a comparison of different interventions on saturation is sought, oximetry could be improved by automatically including only those vessels within an appropriate average diameter range while excluding those vessels with larger and smaller diameters from analysis.There is no conclusive 'best diameter' to report; however, over modest ranges of diameter, the main artery or vein and most of the secondary vessels should be amenable to this approach.These corrections could be applied objectively within selected regions of interest.

| High saturation in small veins
While our model predicts a small diameter-dependent trend for artery saturation, more significant nonphysiological arterial saturation can result from occurrences of negative ODs.In veins, our measurements show a progressive increase in saturation as vessel size decreases, rising to saturation levels above 80% in vessels under 50 μm in diameter.A similar trend is reported over a smaller range of diameters using Oxymap (Paul et al., 2013).However, oxygen transport models indicate no trend in venous saturation with diameter (Arciero et al., 2013, Causin et al., 2016).Comparing mean arterial and venous saturations across all subjects (Table 3), we obtained an arteriovenous (A-V) difference of 30.6%.
We also obtain a saturation difference between small (<60 μm) and large (>100 μm) veins of 17.2%, which is greater than half of the A-V difference.If readings from the venules are correct, oxygen extraction in postcapillary blood would need to be as high as, or higher than, that in capillary blood.This seems unlikely; however, as we described in the Introduction, oxygen saturation in central retinal venules could be higher than saturation in the larger veins carrying peripheral blood and contribute to our large saturation differences.

| Interpretation of model results
The variation in diameter trends between subjects seen in Figure 4 cannot be explained by a single or small range of light absorption difference between artery and vein.However, we find that small differences in double-pass reflectance values yield relatively large changes in the model ODR curves with diameter.Since ODR and saturation are inversely related, the form of the model ODR curve will also be present, but inverted, in the relationship between saturation and diameter.In arteries, as double-pass reflectance is varied, upward and downward curvatures of the model ODR as diameter correspond to the positive and negative slopes found for saturations obtained from arteries.Similarly, the presence of only downward curvature in the model ODR of the vein may correspond to exclusively negative slopes found for saturations measured from veins.From this effect of the doublepass reflectance, we could speculate a possible role for subject-specific differences in the diameter artefact that result from differences in light absorption by the vessel wall and the nerve fibre layer under vessels, since light encounters these structures twice as often and in both directions during double passage.In addtion, some of the variations in venous diameter sensitivity could be from subject-specific differences in oxygen utilization of central and peripheral retina described above.A smaller effect on the diameter trend is found for single-pass reflection (Figure 6a,c) which we might expect due to lower reflectance predicted for this light path (Table 4).
However, since we assumed only a single literature value of retinal light reflectance, we cannot exclude the effects of subject-specific retinal light reflection as a possible contributor to the variance.Results from our conceptual model approach should be compared with optical simulations of light return by vessels over a range of diameter (Rodmell et al., 2014), where circular cross sections are assumed and where the optical density ratio is incorporated into the simulation.

| Alternate hypotheses for diameter sensitivity
Since our light path model closely follows measured ODRs across the range of vessel sizes (Figure 6), we put forward that the apparent dependence of saturation on diameter is in large part an artefact resulting from contributions of single and double-pass light transmission.Still, the possibility of a rheological basis for this trend cannot be excluded.We note that the sensitivity of measured saturations to the velocity of red cells in retinal vessels (Jeppesen & Bek, 2019) may provide an alternate hypothesis for diameter sensitivity.The mechanism is unknown; however, it is believed that blood velocity influences the orientation and profile of red cells across vessels, thus affecting light propagation (Jeppesen & Bek, 2019).By controlling flow, diameter and red cell velocity in vitro, Jeppesen et al. found using Doppler FD-OCT that saturation was most dependent on velocity.However, it is also true that in vivo red cell velocities bear a linear relationship to the vessel diameter (Riva et al., 1985(Riva et al., , 2010)).Jeppesen et al. noted that a rheological explanation by itself would also require an adjustment of the oximetry algorithm.Thus, this paradigm could be a sole or partial cause for a diameter artefact.We have shown that multiple light paths can explain diameter sensitivity, where the mechanism is instead a static interaction between light and retinal vessels, presumed to be a systematic change in OD as single and double passages of light penetrate retinal layers to cross different vessel diameters.We also find that current analytical methods need to be updated.Thus, a more complete understanding of diameter-dependency in saturation measurements may need to account for multiple factors, including variation in delivery and usage of oxygen across the retina, red cell motion within vessels and the retinal vessel architecture effecting light passage through vessels of differing diameter.To evaluate alternative mechanisms for diameter sensitivity and to gain a better understanding of oxygen transport in the retina (see Section 4.2), we envision a need to combine velocimetry and flow measurements (Werkmeister et al., 2012) with oximetry, including improved metrics for diameter and flow distance in vessels, preferably in a single instrument (see Appendix S1).

| Central vessel reflex leads to inaccurate saturation measures
We find that high arterial saturation (outside the physiological range) is measured when the vessel brightness is close to or above the brightness of the surrounding retina, causing the vessel OD and ODR to become negative.This occurs mostly in smaller vessels in 600 nm (measurement) oximetry images but can occur at either wavelength whenever the measurement encounters a strong central reflex.We note that the central reflex is more recognizable in arteries (Brinchmann-Hansen & Sandvik, 1986;Miri et al., 2017) and more prominent in red light (Narasimha-Iyer et al., 2008).The return of single-pass light in vessels from intense scatter into retinal layers from the optic nerve head can also result in negative ODs near the rim of the disc.As defined in Equation 1, where S(% ) = 100(α ODR + β), and where saturation and ODR are inversely related by a negative constant α, saturation will exceed 100% if α ODR + β is greater than one.Assuming our default software values for calibration constants (α = −0.953and β = 1.16), this occurs if ODR is less than 0.168.We note that if ODR drops below zero, the product α ODR becomes positive, leading to additional non-physiological values for saturation.In veins, a strong central reflex affects the ODR in the same way and can artificially raise measured saturations; however, the ODR of veins rarely becomes negative.
In both vessel types, accuracy depends on factors that affect the measured ODR, which include (among other effects) uneven levels of retinal reflection by choroidal capillary networks, spatial variation in pigment density and imperfect pupil alignment allowing stray light to enter the recording from iris reflections.In veins, factors affecting the ODR can produce inaccuracies that are not as obvious as those in arteries.If measurement in a vessel segment is free from undesirable effects of negative ODs, a saturation exceeding 100% represents a statistical outcome or measurement uncertainty, and therefore, a mean value obtained from many measures needs to include these high values in order to give unbiased results.The oximetry equation assumes that light return outside vessels provides a reference for determining oxygen-sensitive light absorption in the vessel.If the measurement produces a negative absorption, we lose the ability to read the saturation.When the central reflex is the cause, the measurement contains a significant amount of that does not pass through the vessel.These cases do not reflect measurement instead should be considered invalid since they do not meet the requirements assumed in the oximetry equation.We find that only a small percentage of measurements made in arteries include negative ODs (Table 2); over many measurements, these negative ODs increase mean saturation by less than 1%.However, our sampling of vessels was panretinal; we have not determined whether specific regions may display a stronger central reflex, nor do we know whether recording conditions, such as eye gaze direction, flash intensity or pupil diameter, might lead to different occurrences of negative ODs. Substantial increases in measured saturations are found where multiple negative ODs occur within a vessel segment (as could occur randomly at measurement points inside vessels, Figure 2 inset; and at reflexes, Figure 5c).If oximetry data were preprocessed to determine occurrences of negative ODs, the issue of invalid data in the oximetry equation could easily be resolved, leaving only a statistical probability for saturations above 100%.Such a process could be applied to previously scanned and archived oximetry data that include OD values.A more recent version of the Oxymap oximetry software, which excludes before pixel averaging measurements of vessel light intensity greater than the background retinal value (Karlsson et al., 2021), may be able to reduce occurrences of non-physiological artery saturations in future retinal oximetry.It will be important for any such improvement in the oximetry technology to be quantified and openly shared.

| Conclusion
To date, retinal oximetry has uncovered numerous and often unexpected relationships between oxygen utilization and different retinal disorders (Stefánsson et al., 2019).
Early work demonstrated the possibility of assessing risk for progression from background retinopathy, and possibly earlier, from venous saturation responses following glucose challenge (Ibrahim et al., 2015;Tiedeman et al., 1998).Non-invasive assessments of retinal tissue metabolism have held great promise for identifying the mechanistic drivers of retinal ganglion cell death and glaucomatous disease prior to detectable visual field loss (Carichino et al., 2016;Harris et al., 2020).The reduction in oxygen consumption detected by oximetry is a biomarker for retinitis pigmentosa and glaucoma (Stefánsson et al., 2017).Patients with higher levels of pigmentation, including those of African descent, may be at higher risk for oximetry defects and measurement errors, and although a correction for pigment is available (Hammer et al., 2008;Hirsch et al., 2022)

F
Oximetry image showing the typical arrangement of vessel segments in one arterial arcade chosen for study.Selected segments (highlighted portions of vessels) are along central and branching arteries.Similar patterns are used for all vessel arcades in an image.Saturation colour calibration bar at right.Inset: automated measurement locations given by Oxymap Analyzer inside and outside vessel near a bifurcation (white arrow).Yellow points are 600 nm locations, green points are 570 nm locations and red points are the outside locations.one of the subjects.Oxymap Analyzer automatically determines oximetry values on a pixel-by-pixel basis along a measured segment.These values, which include vessel light intensity (I), optical density (OD), the OD ratio (ODR), per cent saturation (S), vessel diameter (D) and (X, Y) coordinates of the measurement, are exported to determine segmental mean values for D, ODR and S. Oxymap calculations of OD, ODR and S are given in Equations 1a-1c:

F
Light paths in dual-wavelength oximetry.Visible light that includes oximetry wavelengths is sent from the fundus camera flash.Light passing through the pupil arrives at the retina and vessels (incident light, solid lines).Outside the vessel, light penetrates into the retinal layers and returns on path 1 as a retinal reflection.Some of the light arriving close to the vessel spreads in the retinal/choroid layers to reach behind the vessel and return through the vessel (path 2) as a single-pass reflection.In path 3, light first traverses the vessel before penetrating the retina, returning again through the vessel as a double-pass reflection.In all paths, dot-dashed and dashed lines represent light paths through the retinal and choroid layers, respectively.F I G U R E 4 Oxygen saturations plotted against vessel diameter.(a) Pooled artery saturations obtained from vessel segments of 10 subjects.Red symbols represent measurements where negative ODs are excluded.Black symbols include the negative OD.The solid line is the best-fitting linear trend when negative ODs are excluded.(i-x) saturations from individual subjects plotted as in (a).(b) Vein saturations pooled from the same 10 subjects with the best line fit.(i-x) saturations from individual subjects with linear trends depicted.See also Table 2. Grey-shaded regions on each plot indicate 95% confidence intervals.

F I G U R E 5
Effects of negative vessel OD.(a) Oximetry images showing central reflex on the retinal artery.(b) Intensity profiles along bands (yellow) across vessels in (a).(c) Automated vessel measurement locations by Oxymap Analyzer near a reflex, showing locations inside (yellow and green) and outside (red) the vessel (see text).(d) Arterial saturations from one subject (Figure 4a, panel vi) plotted against vessel diameter.Open symbols indicate negative ODs were removed before calculations.Dark symbols include negative OD in the calculation.Dashed lines indicate where the inclusion of the negative OD resulted in a substantial increase in saturation.(e) The effect of the negative OD rate (percentage of negative ODs in a measured vessel segment) on the calculated arterial ODR (blue) and saturation (orange).

Figure 6
Figure6shows normalized ODRs (black points) obtained from all subjects in arteries (panel A) and veins (panel B) together with model ODR curves plotted against vessel diameter.In both vessel types, negative ODs were excluded.The optimized values for reflectance parameters, taking into account the assumed constraints on retinal, single and double-pass reflectance (Section 2.3), are given in Table4.Our model finds similar values of double-pass reflectance for the artery and vein, slightly below that of extravascular retinal reflectance.Single-pass reflectance (light coming from the side and returned at vessel centre) differs between the vessel types and is markedly less than retinal and doublepass reflectance.In both vessel types, the removal of model fitting constraints does not appreciably improve goodness of fits.The model simulations for these values give ODR curves that change exponentially with Optical properties of blood.

range Mean percent oxygen Saturation a (weighted mean ± SD) Artery Vein
Oxygen saturation in retinal arteries and veins of varying diameter.
T A B L E 3a Obtained in all retinal quadrants.bFor small (<60 μm) and large (>100 μm) veins; see section 4. *p < 0.0001 with respect to saturation in the largest diameter range.
Light path model-best reflectance fits.
T A B L E 4 , its full assimilation and testing across race would significantly improve the reliability of retinal oximetry for clinical applications.Such studies are representative of many latent possibilities for clinical diagnostics; however, this goal remains unattained in part due to uncertainty concerning accuracy and calibration.Here, we have described systematic errors of current retinal vessel oximetry that are sensitive to vessel diameter and errors leading to non-physiologically high saturation caused in part by the central vessel reflex.These issues are related to tissue-optic mechanisms and would likely be present in different oximetry implementations that are based on reflected light.Correction for oximetry artefacts prior to calculation of saturation is important to ensure rigour of retinal oxygen extraction data and future clinical utility for conclusions about retinal metabolism and health.Preliminary aspects of this work were presented at the 6th International Virtual Workshop on Ocular Oximetry, January 2022.JA gratefully acknowledges NSF DMS-1654019 and NSF DMS-1852146.JA, AH and BS gratefully acknowledge NIH R01EY030851.AH also acknowledges NSF DMS (1853222/2021192), NYEE Foundation grants and a Challenge Grant award from Research to Prevent Blindness, NY.JB acknowledges NIH R01EY012606 and R43EY014776 and private research support from James E. Garrette.