**Purpose**:
Two-wavelength algorithms aimed at the extrapolation of retinal vasculature optical properties are being used in the clinical setting. Although robust, this approach has some clear mathematical limitations. We have conducted an in-depth study of this methodology and report on the limits and benefit of this approach.

**Methods**:
We used a well-tested, voxel-based Monte Carlo model of light transfer into biological tissue combined with a seven-layer model of the human fundus to create reflectance maps of retina vessels at different oxygenation levels.

**Results**:
This study shows that the two-wavelength approach works remarkably well in the optimal scenario of known calibration arteries and veins. Errors as a result of choroidal pigmentation and discrepancies in vessel size can be minimized with numerical approaches. When the calibration process deviates largely from physiological values, the technique fails with large errors.

**Conclusions**:
The two-wavelength approach is convenient, easy to implement, and suitable in studies where relative rather than absolute knowledge of retinal oximetry is necessary. A robust calibration step is paramount when using this approach.

_{2}) can provide key insight into the pathophysiological status of the eye. Unusual saturation levels have been linked to different ocular diseases, including diabetic retinopathy, retinopathy of prematurity, photoreceptor degeneration, and visual loss.

^{1–3}The course of treatment is often contingent on the stage of the disease.

^{4,5}

^{6–10}and used to investigate diabetic retinopathy,

^{11,12}glaucoma,

^{13–17}ocular hypertension,

^{13}Eisenmenger syndrome,

^{18}macular degeneration,

^{19}retinitis pigmentosa,

^{20}vein occlusions,

^{21–23}and artery occlusions.

^{24}Several authors are reporting high reproducibility in their oximetry measurements,

^{10,25,26}yet most researchers agree that measurement values obtained with retinal oximetry must be considered as relative values and not absolute. As this technology enters the clinical realm it is paramount to accurately measure its strength and limitation.

_{2}. An excellent review of these techniques is available here.

^{27}To summarize briefly, Hickam et al.

^{28}used a two-wavelength apparatus to measure retinal vein oxygen saturation in 1963. Two-wavelength systems exploit hemoglobin's absorbance spectrum at one isosbestic point and at a wavelength different from the isosbestic point.

^{29,30}Three-wavelength retinal oximeters were also proposed,

^{31,32}taking into account the scattering of red blood cells in blood.

^{33}Multispectral techniques were also utilized by several groups both with point measurements and imaging.

^{34–36}

^{37–39}These systems are generally developed utilizing a fundus camera combined with a scientific grade digital camera. The light illuminating the retina is either filtered before impinging on the eye or at the detector. Typical wavelengths combinations are 586/605 nm,

^{40}570/600 nm,

^{7}and 548/610 nm.

^{39}

^{39,41–43}changes in total or heterogeneous distribution of pigmentation in the eye,

^{39,42}and crosstalk with hemoglobin in the choroidal vasculature.

^{42,44}Algorithms compensating for these factors have been proposed.

^{29,39}

^{45,46}model of light transport to analyze the error associated with two-wavelength algorithms under different scenarios. We investigate the accuracy of these algorithms for incorrect assumption of SO

_{2}in the calibration step, for variations of vessel diameter, for variations in choroidal pigmentation, and for choroidal vessel crosstalk.

^{29}and later by Hardarson et al.

^{40}is based on the calculation of the optical density (OD) of the retina at two different wavelengths. By measuring the reflectance on the vessel (

*I*) and next to the vessel (

*I*

_{0}), the OD at each wavelength is calculated using Equation 1. The OD ratio (ODR) is defined as the ratio of the ODs at two different wavelengths (Equation 2). OD is the OD at one wavelength (

*λ*

_{1}) and is the OD at a different wavelength. ODR is assumed to have a linear relationship with SO

_{2}, hence Equation 3 is used to calculate the oxygen saturation within a vessel, where

*a*and

*k*are constants obtained through a calibration procedure.

_{2}values for arteries and veins reported by several investigators,

^{28,32,47,48}we estimate typical values to be 96% and 54%, respectively. The optical density ODR of first-degree arteries and veins are also calculated with the retinal oximeter on healthy individuals. For example, Hardarson reported ODR

_{a}= 0.209 for arteries and ODR

_{v}= 0.502 on veins based on a study of 18 healthy individuals.

_{2},

*a*and

*k*can be calculated by solving the equation pair. Finally, once

*a*and

*k*are known, Equation 3 can be used to extrapolate any vessel oxygen saturation level.

_{2}Algorithm

^{49}can be applied to quantify the transmission of light through a sample. Second, we assume that no other absorber is present in the optical path aside from hemoglobin at various levels of oxygenation. Using tabulated data of human oxygenated and deoxygenated hemoglobin,

^{49}we can then calculate transmittance values through samples of a predetermined size.

*a*and

*k*are calculated using Equation 4. This step mimics a perfect calibration scenario where all of the elements of the experiment are known.

_{2}levels in the test vessel are allowed to change from 0% to 98% with a 1% step. The vessel SO

_{2}is obtained using the previous values of

*a*and

*k*and Equation 2. The results of the calculated values are compared with the expected value of SO

_{2}, Figure 1. Even in this oversimplified approach an error is produced, especially toward lower values of SO

_{2}. At physiological relevant levels of SO

_{2}(40% to 100%), the error is a very low < 0.1 percentage points.

_{2}. For calibration, a representative artery and vein are necessary (Equation 4). The representative arterial value from the literature (96%) was used. In this test, the SO

_{2}of the calibration artery was varied from 90% to 100%. Figure 2b shows how a faulty calibration assumption impacts the calculated results of SO

_{2}for the test vessel. The further the calibration true value is from the assumption of 96%, the higher is the error. The same process is followed for a faulty assumption of calibration vein SO

_{2}, with the test vessel in this case being a vein at 58% SO

_{2}. The assumed value for calibration is 54% (Equation 2), whereas the true value of the calibration vessel is varied between 40% and 60%.

^{50,51}A voxel-based Monte Carlo model was utilized in this work.

^{45}This model allows for the analysis of three-dimensional geometries with heterogeneous structures. Photons are launched and traced independently, propagating within voxels experiencing scattering and/or absorption events, depending on the optical properties of the tissues being modeled. Each voxel may represent a different tissue or structure based on the optical properties assigned to it. Voxel sizes were 4 μm in length, width, and height. Two hundred million photons were launched in each simulation. Photons reflected out of the geometry were recorded to obtain intensity images of the geometry under investigation. Prior to conducting any test, our model was further validated against a Monte Carlo model of photon transport in multilayered tissue

^{50}and resulted in errors well below statistical significance.

^{51}Utilizing the advantages of the voxel-based analysis, we embedded a vessel within the retinal layer. Our tissue geometry of investigation spanned a volume of 800 μm by 800 μm laterally with a tissue thickness of 1160 μm, with retinal thickness of 20 0μm, RPE thickness of 10 μm, choroid thickness of 250 μm, and sclera thickness of 700 μm.

^{52}A vessel was placed 10 μm below the surface of the neural retina. Unless otherwise stated, the calibration vessels were 100 μm in diameter, and the detection vessel diameter was varied in different tests. The optical properties of the layers at the wavelengths of interest can be seen in the Table.

^{52,55,56}Melanin is known to be a strong absorber and has been extensively studied because of its prevalence in the skin and retina. Glickman et al.

^{53}reported melanin to have an absorption coefficient of 2237 cm

^{−1}. Feeney-Burns et al.

^{54}showed that the RPE can be estimated to be composed of approximately 12% melanin.

^{3}as reported by Koblova et al.

^{57}The absorption and scattering coefficients of blood were calculated from data published by Bosschaart et al.

^{49}The assignment of scattering angles is achieved in the Monte Carlo through a Henyey Greenstein phase function,

^{58}and this has been shown to provide an effective phase function for blood with physiological haematocrit in the visible range.

^{59}The index of refraction for all layers was kept at 1.37.

*I*was calculated by averaging reflection intensities within a neighboring area of the vessel. To be certain that

_{0}*I*was not affected by the vessel (Fig. 4b),

_{0}*I*was calculated > 316 μm away from

_{0}*I*(Fig. 4a).

*I*was calculated by averaging reflection intensities within 25% of the corresponding vessel diameter around the center of the vessel.

_{2}= 96%) and a perfect vein (SO

_{2}= 54%). ODR for each vessel was then computed, and their actual SO

_{2}was used within the algorithm. Under this assumption, constants

*a*and

*k*were calculated by solving a simple system of equations.

_{2}measurement accuracy.

^{60}using 520 and 630 nm as the primary wavelengths. This strategy is necessary because of the particular experimental layout (single camera) as well as the necessity of simultaneously measuring oxygenation and flow. Here we show the impact of the choice of different wavelength pairs in the accuracy of the two-wavelength algorithm. In this scenario, calibration was performed using vessels of 100 μm in diameter and assumed SO

_{2}of 54% and 96% for the calibration vein and artery, respectively. Figures 6a and 6b show the ideal case where the calibration assumptions were correct. Results were similar to those produced using 570 and 600 nm wavelengths, although a higher error is shown at very low SO

_{2}levels (SO

_{2}< 45%).

_{2}measurement accuracy. A 570 and 600 nm wavelength pair was used with 5-nm band-pass filters centered at these wavelengths. The choice of wavelength and filter bandwidth is guided by current methodologies.

^{7}Implementing a 5-nm band-pass filter was achieved by assuming a Gaussian distribution centered at each wavelength with a full width half max of 5 nm to calculate the amount of photons launched per wavelength, as shown in Figure 7.

_{2}ranging from 44% to 100%. The equations were calibrated based on correct assumptions of SO

_{2}. The results are shown in Figures 8a and 8b.

_{2}values to an artery and vein relating as a way to establish a linear relationship between ODR and SO

_{2}. The assumed values are based on previous work by different investigators. In the ideal case we have shown how this approach introduces an inherent error whose magnitude depends on the degree of separation between assumed and true calibration values. Utilizing this model, we tested cases where the assumed SO

_{2}values used for the calibration vein and artery (54% and 96%) were incorrect. SO

_{2}calculations were performed on fundus vasculature with varying oxygen saturations: 44% to 100% SO

_{2}in steps of 2%.

_{2}assumption can produce widely varying levels of error. When the calibration artery is assumed to be 96% SO

_{2}but its true value is 84% SO

_{2}, a vessel with 100% SO

_{2}will be calculated as having values of up to 20 percentage points greater than its true level. Similar results are presented in Figures 4c and 4d, where the calibration vein SO

_{2}was assumed incorrectly. The actual SO

_{2}of calibration veins ranged from 44% to 60%. Errors in SO

_{2}were up to 10 percentage points. Figures 9e and 9f show the case where SO

_{2}was assumed incorrectly for both the calibration artery and vein. In these cases, errors in estimating SO

_{2}approached 15 percentage points.

_{.}Monte Carlo simulations were performed on vessels with varying diameters. In this scenario, calibration vessel SO

_{2}values were assumed correctly. SO

_{2}was measured for blood vessels with SO

_{2}equal to 92% and diameters ranging from 50 to 190 μm in steps of 10 μm, utilizing a reference vessel of diameter equal to 100 μm. Figure 10a shows the calculated SO

_{2}for each vessel; these values exhibit a strong monotonic decrease with vessel size. Errors resulting from very large vessels (190 μm in diameter) were as large as 11.9 percentage points below true vessel SO

_{2}(92%).

_{2}introduced by vessel diameter, Geirsdottir et al.

^{7}developed a vessel size correction. The correction factor assumes that the SO

_{2}of a vessel just before a bifurcation is the same as those in both branches after the bifurcation. Equation 5 was used to calculate

*k*, a correction factor, where , , and are the measured values without correction for the first branch, second branch, and the primary vessel, respectively, and , , and

*d*

_{pri}are the diameters of the first branch, second branch, and primary vessel, respectively. The constant

*k*was then used in Equation 6, where

*d*is the vessel diameter,

*d#x0304;*is the mean vessel diameter, and SO

_{2(cor)}is the corrected saturation value. SO

_{2}values before correction were gathered from the previous section, testing a 92% SO

_{2}vessel with varying diameter. A theoretical bifurcation based on Murray's principle

^{61}was assumed, where a primary 120-μm diameter vessel bifurcated into 90-μm and 100-μm diameter branches to calculate the correction factor,

*k*.

_{2}values for a vessel with a true SO

_{2}of 92%. The correction works very well when the mean diameter is very close to the diameter of the vessels used to calibrate the algorithms. As shown in Figure 12, if the calibration vessels are larger than the mean diameter, then the correction will shift calculated values of SO

_{2}up; conversely, if the calibration vessel is smaller than the mean diameter, then the correction will shift the calculated SO

_{2}values downward. This result greatly reduces the benefit of the compensation.

^{55}and Liu et al.

^{51}Nevertheless, choroidal vessels can often be resolved in long wavelength imagery and disrupt the otherwise homogeneous background of fundus imaging. This section will investigate this phenomenon. For this purpose, we have embedded a small vessel with blood oxygenation of 95% in our standard choroid. The vessel is placed 10 μm below the top of the choroid layer, underneath the test vessel in the retina and parallel to the surface. SO

_{2}calculations were made on the vessel in the retinal layer. Calculations were performed utilizing the well-calibrated equation, where the SO

_{2}of both calibration vessels were assumed correctly. Simulations were performed on 96% SO

_{2}, 100-μm test retinal vessels. Choroidal vessel diameters ranged from 20 to 100 μm, with a step size of 10 μm and an SO

_{2}of 95%. Figure 13b shows the error caused by the introduction of a discrete choroidal vessels in the uniform choroid directly below the vessel in the retina.

^{39,51}Variations in melanin concentration exist among different people, but may also exist in different regions in the eye because the choroid is not a homogenous structure. For these calculations, calibration vessel SO

_{2}values were assumed correctly. Simulations were performed on 100-μm retinal vessels with 96% SO

_{2}. The concentration of melanin in the choroid was varied from 18 to 40 mg/ml, with a step size of 2 mg/ml.

^{39}in their experimental work.

_{2}values very close to the true value. The most significant errors were found when there were incorrect calibration assumptions, resulting in differences from true values as high as 20 percentage points in SO

_{2.}Recent versions of commercial retinal oximeter do not rely on individual calibration but a set of tabulated data for ODR values of arteries and veins, hence the possibility of large errors in this domain has been reduced.

^{7}worked rather well under ideal circumstances. Some error arises in cases where calibration vessels and test vessels have very different diameters, resulting at times in errors larger than uncorrected SO

_{2}measurements.

_{2}under very ideal circumstances, but greater than 20 percentage points when a very poor measurement technique is applied.

^{33,62}some pertaining to technical choices for the apparatus. For example, dark current, readout noise, and shot noise camera noise level must be carefully estimated and corrected when using retinal oximeter as they may limit the detector dynamic range and become additive noise in the oximetry measurements. Subtraction of a dark noise images has proven to be essential when acquiring oximetry measurements. Other errors may occur as a result of poor or inconsistent illumination, which would reflect in the calculation of ODR. Here we have limited our study to an ideal scenario of a perfect instrument, but further work needs to be done to characterize the two-wavelength technique in less ideal situations.

*. 1995; 44: 603–607.*

*Diabetes**. 2003; 121: 547–557.*

*Arch Ophthalmol**. 2000; 41: 4275–4280.*

*Invest Ophthalmol Vis Sci**. 2000; 107: 19–24.*

*Ophthalmology**. 2010; 117: 1147–1154.*

*Ophthalmology**. 2009; 50: 2308–2311.*

*Invest Ophthalmol Vis Sci**. 2012; 53: 5433–5442.*

*Invest Ophthalmol Vis Sci**. 2015; 10: e0128780.*

*PLoS One**. 2011; 249: 1311–1317.*

*Graefes Arch Clin Exp Ophthalmol**. 2012; 53: 1729–1733.*

*Invest Ophthalmol Vis Sci**. 2012; 96: 560–563.*

*Br J Ophthalmol**. 2014; 92: 34–39.*

*Acta Ophthalmol**. 2009; 93: 1064–1067.*

*Br J Ophthalmol**. 2009; 50: 5247–5250.*

*Invest Ophthalmol Vis Sci**. 2011; 52: 6409–6413.*

*Invest Ophthalmol Vis Sci**. 2014; 92: 105–110.*

*Acta Ophthalmol**. 2014; 98: 329–333.*

*Br J Ophthalmol**. 2011; 52: 5064–5067.*

*Invest Ophthalmol Vis Sci**. 2014; 92: 27–33.*

*Acta Ophthalmol**. 2014; 92: 449–453.*

*Acta Ophthalmol**. 2010; 150: 871–875.*

*Am J Ophthalmol**. 2012; 90: 466–470.*

*Acta Ophthalmol**. 2015; 253: 1653–1661.*

*Graefes Arch Clin Exp Ophthalmol**. 2013; 91: 189–190.*

*Acta Ophthalmol**. 2013; 91: e590–e594.*

*Acta Ophthalmol**. 2015; 93: e439–e445.*

*Acta Ophthalmol**. 2014; 3: 2.*

*Trans Vis Sci Tech**. 1963; 27: 375–385.*

*Circulation**. 1999; 86: 748–758.*

*J Appl Physiol**. 2013; 91: 489–490.*

*Acta Ophthalmol**. 1975; 38: 315–320.*

*J Appl Physiol**. 1988; 27: 1113–1125.*

*Appl Opt**. 1999; 38: 258–267.*

*Appl Opt**. 2011; 6: e24482.*

*PLoS One**. 2013; 113: 143–147.*

*Exp Eye Res**. 2014; 28: 1190–1200.*

*Eye (Lond)**. 2012; 319: 75–83.*

*Bull Soc Belge Ophtalmol**. 2009; 247: 1025–1030.*

*Graefes Arch Clin Exp Ophthalmol**. 2008; 13: 054015.*

*J Biomed Opt**. 2006; 47: 5011–5016.*

*Invest Ophthalmol Vis Sci**. 1993; 45: 134–148.*

*Microvasc Res**. 2003; 34: 152–164.*

*Ophthalmic Surg Lasers Imaging**. 2015; 15: 184.*

*BMC Ophthalmol**. 2000; 39: 1183–1193.*

*Appl Opt**. 1996; 2: 934–942.*

*IEEE J Sel Top Quant**. 2013; 18: 10504.*

*J Biomed Opt**. 1999; 46: 1454–1465.*

*IEEE Trans Biomed Eng**. 1993; 40: 817–823.*

*IEEE Trans Biomed Eng**. 2014; 29: 453–479.*

*Lasers Med Sci**. 1995; 47: 131–146.*

*Comput Methods Programs Biomed**. 2013; 18: 066003.*

*J Biomed Opt**. 1995; 40: 963–978.*

*Phys Med Biol**. 2001; 4257: 134–141.*

*Proc SPIE Int Soc Opt Eng**. 1984; 25: 195–200.*

*Invest Ophthalmol Vis Sci**. 2002; 47: 2863–2877.*

*Phys Med Biol**. 2001; 23 (4–6): 279–289.*

*Int Ophthalmol**. 2005; 5688: 302–311.*

*Proc SPIE Int Soc Opt Eng**. 1941; 93: 70–83.*

*Astrophys J**. 2001; 46: N65–N69.*

*Phys Med Biol**. 2015; 5: 86–96.*

*Quant Imaging Med Surg**. 1926; 9: 835–841.*

*J Gen Physiol**. 2011; 52: 2851–2859.*

*Invest Ophthalmol Vis Sci*