Multiple forward scattering reduces the measured scattering coefficient of blood in visible-light optical coherence tomography

Optical properties of blood encode oxygen-dependent information. Noninvasive optical detection of these properties is increasingly desirable to extract biomarkers for tissue health. Recently, visible-light optical coherence tomography (vis-OCT) demonstrated retinal oxygen saturation (sO2) measurements using the depth-resolved spectrum of blood. Such measurements rely on differences between the absorption and scattering coefficients of oxygenated and deoxygenated blood. However, there is still broad disagreement, both theoretically and experimentally, on how vis-OCT measures blood’s scattering coefficient. Incorrect assumptions of blood’s optical properties can add additional uncertainties or biases into vis-OCT’s sO2 model. Using Monte Carlo simulation of a retinal vessel, we determined that vis-OCT almost exclusively detects multiple-scattered photons in blood. Meanwhile, photons mostly forward scatter in blood within the visible spectral range, allowing photons to maintain ballistic paths and penetrate deeply, leading to a reduction in the measured scattering coefficient. We defined a scattering scaling factor (SSF) to account for such a reduction and found that SSF varied with measurement conditions, such as numerical aperture, depth resolution, and depth selection. We further experimentally validated SSF in ex vivo blood phantoms pre-set sO2 levels and in the human retina, both of which agreed well with our simulation.


Introduction
Optical coherence tomography (OCT) enabled noninvasive three-dimensional (3D) retinal imaging at micrometer-scale volumetric resolutions [1,2]. Since its first report 30 years ago, OCT has become the clinical gold standard for diagnosing and monitoring nearly all major ocular diseases [3,4].
Researchers attributed such a reduction in μs to the blood's 'packing factor', which describes correlated optical interactions between densely packed RBCs and their hematocrit-dependence [9,29]. Our group previously used the packing factor (denoted as W) to scale μs in the vis-OCT inverse fitting model for sO2 measurement [31]. Specifically, our group found that the model's goodness of fit (R 2 ) maximized when W was between 0.2 and 0.4. This W value was consistent with the definition of the packing factor, which scales μs by ~ 1/3 at physiological hematocrit [9].
However, reported vis-OCT oximetry methods accounted for blood's μs differently, among which significant discrepancies exist [7,22,23,31,[33][34][35][36]. Since vis-OCT oximetry fits the measured spectrum to the literature-reported μa and μs, deviations between the measured and reported μs can introduce sO2 measurement error. Therefore, accurate and consistent quantification of sO2 benefits from a systemic investigation on how vis-OCT measures μs.
We systemically investigated measuring optical scattering properties of blood detected by vis-OCT. First, we performed Monte Carlo (MC) simulations of a retinal blood vessel and photon detection by vis-OCT. MC simulation is not susceptible to systemic biases present in practical OCT detection or image reconstruction [37][38][39] and, therefore, provides a fundamental understanding of light-tissue interaction. For each photon packet exiting the tissue, we monitored the number of scattering events and optical pathlength traveled in tissue to investigate the impact of multiple scattering on the vis-OCT signal. We reconstructed simulated vis-OCT A-lines to establish a direct relationship between multiple scattering and the measured μs. Then we established the scattering scaling factor (SSF), a generalized scaling coefficient for μs. Since multiple scattering influenced the measured μs, we further investigated photon detection by different numerical apertures (NA's), because NA acts as a geometric filter in detecting multiply scattered photons. Second, we compared our simulation results to experimental vis-OCT imaging of ex vivo blood phantoms. We found excellent agreement between the simulated and experimentally measured values across pre-set oxygenation levels. Finally, we validated our SSF analysis in human retina vis-OCT imaging and found a strong agreement with our simulated results. Validated by simulation, ex vivo blood phantom imaging, and in vivo human retinal imaging, we provide evidence that vis-OCT measured μs is smaller than the reported packing factor but higher than the reduced scattering coefficient ( ) [37,40]. This work sets the foundation for a unified theory of vis-OCT-measured optical properties of blood and more robust retinal oximetry.

Detecting scattered light with vis-OCT
Most OCT's use NIR (800 nm -1300 nm) illumination, where biological tissues have lower optical scattering (μs < 100 cm -1 ), lower optical absorbing (μa < 1 cm -1 ), and moderately high scattering anisotropy (0.7 < g < 0.9) [41]. Such optical properties yield mean-free-paths (MFPs) of several hundred micrometers in tissue, meaning photons can travel deep into tissues before being multiply scattered. Since OCT's axial resolution is dominated by light's coherence length and not geometrical optics, OCT can use a low NA to image deeply penetrating photons across several hundred micrometers with axial resolutions < 10 µm [42]. Another benefit of imaging weakly scattering tissues with a low NA is high sensitivity to single-scattered or ballistic photons [43]. In this work, we define ballistic photons as Class I photons [37], which satisfy and 0.7 < g < 0.9) using a low NA (e.g. < 0.2). Fig. 1A shows a global illustration of tissue with optical properties within the NIR spectral range. The OCT sample arm focuses light on a spot in the tissue (illustrated by the green oval), which creates an A-line at that location. Due to the low NA, most incident photons (green arrow) are perpendicular to the tissue surface. Since the focus spot creates a conjugate point with the sample arm detector, it also acts as a geometric projection of the detector itself [44]. Only photons collected within the spot and the solid angle defined by the illumination NA will contribute to the A-line.
where [arbitrary unit] is the A-line intensity; [mm] is the depth in the tissue; and the coefficient 2 in the exponential term indicates round trip in tissue. Eq. 3 has been thoroughly derived and experimentally validated for Class I photons [6].
Vis-OCT imaging of blood represents a special scenario that deviates from the assumptions described in Figs. 1A & 1B. The reported μs for oxygenated and deoxygenated blood states are > 3000 cm -1 and the W-scaled μs are near 1000 cm -1 [9]. Both these μs values are over an order of magnitude greater than μs values (< 100 cm -1 ) typical of most tissues within NIR spectral range. Furthermore, blood is more highly forward scattering ( ≥ 0.98) than most tissues (0.7 < < 0.9). Therefore, photons are likely to travel only a few μm or less along thedirection after each scattering event. Assuming normal incidence of light on a vessel (Fig. 1C), photons can be multiply scattered and still satisfy the Class I condition. Fig. 1D illustrates such a path following the green arrow. We hypothesize that the path shown in Fig. 1D is a common, if not dominant, detection scenario in blood imaging using vis-OCT. A Class I photon that travels deeper than its single scatter assumption is equivalent to reducing its μs in Eq. 3. Therefore, the Beer-Lambert Law may be rewritten as ∝ where SSF [dimensionless] is the scattering scaling factor, which is < 1 and scales μs to account for the increased detection of photons deeper in tissue. The SSF is a generalized scaling factor and incorporates any other reductions of μs, including W.

MC simulation parameters
We simulated vis-OCT detection and reconstruction in a retinal blood vessel using MC simulation [8,27,37,40,[47][48][49][50][51]. Fig. 2 shows the multi-layered 3D model of a blood vessel embedded in the retina. We modeled the blood vessel using an infinitely long cylinder located 55 µm below the vitreous-retina interface. The vessel has three concentric layers: blood, cell-free zone (CFZ) [35,36], and the vessel wall. The CFZ is a thin layer consisting primarily of plasma between the blood and the vessel wall. It arises from the difference in viscosity between RBCs and plasma and is described by the Fahraeus Lindqvist (FL) effect [52]. The CFZ has been previously observed in OCT images [53] and is noticeable in our vis-OCT data. Table I summarizes the optical and geometrical parameters. We extracted the properties of the retina [27,49], vessel wall [27], and CFZ [54] from the literature. We used the theoretical optical properties of blood previously derived by Faber et al. [12,27]. To increase the speed of our simulation, we used the average µa and µs between 520 nm and 600 nm (center wavelength 560 nm). Although there is still uncertainty in the exact values of µa, µs, and g [9], our values are well within the reported range [9][10][11][27][28][29][30]. To account for correlated optical interactions between densely packed RBCs, we scaled blood's µs using the packing factor 4 = 1 − 6 , where 6 is . Assuming a hematocrit of 45% [55], we have W = 0.3025

MC simulation algorithm
MC simulation of photon propagation in biological tissue has been widely reported [8,27,37,40,[47][48][49][50][51]56]. We followed the algorithm of simulation photon transport in multi-layered tissues (MCML) [40]. Briefly, we launch a photon packet towards the retina and blood vessel, as illustrated in Fig 2. The photon packet's launching position is at the vitreous-retina interface where ; = + and < [dimensionless] is a random variable following an even distribution between 0 and 1. After traveling distance (, the photon packet interacts with tissue and deposits a fraction : of its weight. At each interaction location, the photon packet is scattered by an angle = relative to its current propagation direction determined by the Henyey-Greenstein phase At the interface between two media, the photon packet either reflects or transmits with probabilities according to the Fresnel's equations [57]. Upon entering a new tissue region, ; is adjusted accordingly. The simulation continues until the photon packet exits the retina into the vitreous or the optical path distance traveled is greater than 2000 m, well beyond the depth of the blood vessel.
If the photon packet exits the retina into the vitreous, we recorded the remaining weight of the photon packet, the optical pathlength traveled within the tissue, the total number of scattering events in all tissue regions, the total number of scattering events in blood, the exiting position, and the exit angle. We simulated 10 10 photon packets for each A-line. We implemented the simulation in MATLAB 2020 using parallel computing on a PC with a 3.4-GHz Intel Core i7-6800K CPU and 64-GB RAM. The simulation of an A-line took approximately 120 hours to complete.

Photon detection
To simulate OCT detection, we geometrically filtered photons exiting the retina [51] (blackdashed lines in Fig. 2). We determined photon acceptance aperture and angle according to the NA of the light incident on the retina. We tested the NA value from 0.015 to 0.094, which follows the optical properties of normal human eyes [58]. We calculated the NA as where K [dimensionless] = 1.35 is the refractive index; [cm] is the 1/ diameter of the collimated beam incident on the cornea; V [cm] = 1.8 cm is the focal length of a normal eye [59]; and tan ? R S is the acceptance angle with respect to normal incidence. We used the NA to calculate the focal spot beam waist where % = 560 nm is the central wavelength of the vis-OCT probing light. For simplicity, we detect photons using a uniform circle with a radius of W. In the results, we use an NA = 0.05, equivalent to a 2.9º acceptance angle and a 7.0 m detection aperture diameter unless otherwise specified.

Vis-OCT A-line reconstruction
We reconstructed simulated vis-OCT A-lines using the recorded photon weights and optical path distances in the simulation. We only used photons detected under the acceptance conditions.
Adopting the methods in Kirillin et al. [51], we reconstructed the OCT A-line as is the axial resolution. We set the distance between adjacent z positions as 1 m and used an axial resolution of 9 m defined by the h E of the STFT sub-band window centered at 558 nm with an FWHM bandwidth of 11 nm [42]. We used this sub-band window size for spectroscopic A-line reconstruction in our experimental studies.

Experimental measurements of ex vivo blood samples
We used the vis-OCT system operating from 510 to 610 nm described by Beckmann et al. [60] to image ex vivo blood samples in phantom vessels. The imaging objective in the sample arm had an NA of 0.05 [61], consistent with our simulation and human imaging. Briefly, we constructed a vessel phantom by pulling a glass capillary tube to have an inner diameter of 200 m and embedded and stabilized the tube in a plastic well. To reduce the influence of specular reflections, we added immersion oil to the well until the tube was covered. We prepared whole bovine blood (Quadfive, Ryegate, MT) of hematocrit 45% of oxygen levels ranging from 45% to > 99%. To oxygenate the blood, we added a constant stream of pure oxygen and stirred the blood with a magnetic stir bar. To deoxygenate the blood, we added sodium dithionite [62] to the solution and stirred. We repeated these processes until reaching the desired oxygen level. We monitored blood's partial pressure of oxygen (pO2), partial pressure of carbon dioxide (pCO2), pH, and temperature using a blood-gas analyzer (Rapidlab 248, Siemens Healthcare Diagnostics, Malvern, PA) and estimated the corresponding sO2 [63]. Before loading the tube with blood, we

Experimental measurement of human retinal vessels
For human imaging, we used the system described by Rubinoff et al. [64]. We used Eq.

Contribution from multiple forward scattering in vis-OCT blood signal
We investigated how blood's scattering properties influence the detection of photons in vis-OCT. 3A was scattered 11 times inside the blood vessel. Notably, the photon packet travels mainly along the -axis, consistent with a high scattering anisotropy (g = 0.987). This allows the photon packet to backscatter to nearly the same x-position as it launched, similar to the illustration in Fig. 1D. Despite the multiple scattering events, we classified this photon packet to be Class I.
Based on the optical properties shown in Table 1, the calculated MFP in blood was 8.5 m; however, the photon packet travels 60 m into the vessel (Fig. 3A).
To assess whether the multiple scattering observed in Fig. 3A was a frequent occurrence, we measured the proportion of all detected singly scattered photons packets that entered the blood region (Fig. 3B)  It is commonly believed that Class I photons are singly scattered and Class II photons are multiply scattered in OCT tissue imaging. Fig. 3 suggests that photon propagation in the blood is similar to a scenario where singly scattered photons are detected from a medium with a significantly smaller . This is why we introduce the scattering scaling factor SSF for in the modified Beer-Lambert model in blood as defined in Eq. 4. with log-scale decay described by the Beer-Lambert law [45]. When measuring the optical properties of blood in vivo, it is critical to start at least from the BM rather than assuming BSD occurs at the boundary of the AW and vessel lumen. Finally, the last peak represents the posterior wall (PW) of the vessel.
We include only the oxygenated coefficients since the simulated sO2 was 100%. The term 'avg' indicates that simulation used optical properties that were the average value between 520 nm and 600 nm. The double pass Beer-Lambert law does not need to be squared in the simulated A-line since it is not the product of interference between the sample and the reference electromagnetic fields. Although there are a handful of ways to extract *+, -jk from this equation, we elected to compute a depth-average of Eq. 11, which we empirically found robust against noise [64]. The starting measurement depth is m and the depth range is ∆ . The full region of measurement is highlighted by the red dashed line. We first normalized by its amplitude at m , which shifted the coordinate system to m = 0. The average intensity becomes Dividing by ∆ , we are left only with the linear combination of *+, -jk and 001 × *+, -jk . By subtracting *+, -jk , whose value is from the literature, we are left with 001 × *+, -jk , which can be compared with the literature µs value used in the simulation.
The SSF can be calculated as

Influence of the number of scattering events on SSF
To understand how multiple scattering influences SSF, we set a graded threshold to the number of detected scattering events in blood. If a photon packet that entered the blood region is scattered more times than the set threshold, it was not included in the simulated A-line. Fig. 5A shows the simulated A-line for different scattering thresholds. The different shades of red plot different scattering threshold levels. The darkest shade has no threshold and collects photon packets from all scattering events (same as Fig. 4A). A-line amplitude decays slower with higher threshold levels, consistent with the notion that increased multiple scattering reduces SSF.
Specifically, when the threshold level is 1, the A-line is reconstructed by singly-scattered Class I photon packets and decays within 20 m in blood. As sO2 calculation fits A-lines beyond 20 m in blood, multiply scattered photons are required for accurate sO2 measurement in vivo.
Additionally, it becomes clear that multiple scattering enables deeper photon penetration necessary for visualizing the PW. The PW becomes weak when the threshold level is less than 7 and invisible when the threshold level is less than 3. This suggests that multiple forward scattering also facilitates the visualization of the PW, a landmark commonly used in vis-OCT oximetry to indirectly measure [31]. we did not include this data point due to insufficient signal-to-noise-ratio along the measured depths.

Influence of numerical aperture on SSF
As described in photon packet detection in our MC simulation, the detection NA acts as a spatial filter for photon scattering events. The detection aperture (radius) and angle limit an existing photon packet's position and propagation direction. Therefore, the detection criteria can potentially influence the measured SSF.
We examined the relationship between NA and SFF for m = 17 m and ∆ = 33 m based on the conclusion from the above section. The tested NA's are 0.015 to 0.094, which are based on physically reasonable imaging NA's in the human eye. In OCT, retinal imaging NA is almost always less than the maximum possible NA (e.g. 0.2 [58]), since researchers must consider limiting factors like reduced depth-of-focus, aberrations, and eye dilation [44].
Therefore, we varied the acceptance radius from 1.9 µm to 11.9 µm and varied the acceptance angle from 0.9 o to 5.4 o and found that SSF varied between 0.02 to 0.09 (Fig. 6A). We found that SSF reduces with increased acceptance radius, allowing more photon Class II photons (result here). We also found that SSF increased with increased acceptance angles. Photon packets traveling deeper in tissue are less likely to be filtered by the acceptance angle as most photon packets within the acceptance radius for deeply traveling photon packets are within the acceptance angle in deeper tissues.  Fig. 6A, and shows that SSF increases with increased NA.
However, the relationship between NA and SSF is nonlinear and appears to saturate at the highest NA. For all the tested NA values, SSF has always been less than 0.1, which is still less than one third of the literature value 4 = 0.3025. Fig. 6C plots the simulated A-line under different NA values. As NA increases, the amplitudes corresponding to AW and BM increase with respect to PW, which intrinsically increases the slope of the blood decay and leads to an increased SSF. Since the number of scattering events increases with imaging depth (Fig. 3E), It is important to note that this simulation presents a simplified view of the influence of OCT detection on SSF. Other variables, including the Gaussian beam profile, longitudinal chromatic aberrations, lateral chromatic aberrations, defocusing, eye geometry, oblique incidence, etc., will collectively influence the illumination and detection criteria as well [65].
However, the above simulation on the relationship between acceptance aperture and angle and SSF establishes a critical foundation for multiple scattering analysis in OCT. Such a relationship suggests no 'one-size-fits-all' SSF value exists.   7C plots a least-squares fit of the measured attenuation spectrum of fully oxygenated blood (red line) to its theoretical attenuation spectrum (black-dashed line). We performed leastsquares fitting using Eq. 4 to measure sO2 using the ratio of the oxygen-dependent coefficients [64] and estimate the SSF. The fitting (R 2 = 0.98) in Fig. 7C yields sO2 = 100%, *+, -jk = 205 cm -1 , and SSF = 0.06, which agrees with the simulated SSF for fully oxygenated blood.

Ex vivo experimental results
We measured sO2 and SSF from five sO2 levels between 40% and 100%. We bootstrapped 100 measurements for each level by shuffling each dataset and randomly selecting 50 B-scans. Fig. 7D shows the mean and SD of measured sO2 (circles) and SSF (triangles). The black dashed line shows the linear best fit relationship ({ = 1.05 − 3.30) between the bloodgas machine measurements and vis-OCT measurements, indicating excellent agreement between these two independent measurements. The average SSF is 0.060 +/-0.021, and the average spectroscopic fit R 2 is 0.99. The agreement between simulated and experimentally measured SSF values suggests that our work does not contradict the previously suggested packing factor in whole blood [9] but, instead, adds an additional correction for vis-OCT oximetry.

In vivo experimental results
We validated SSF measurement in vis-OCT imaging of human retinas.  4A) and ex vivo A-line (Fig. 7B). In the artery plotted in Fig. 8C, this delineation is less obvious, and there is a change in slope near 260 m depth where the BM is typically located, which can be caused by higher, pulsatile blood flow in arteries leading to less precise spatial averaging. We also observe a small valley near the center of the vessel in Fig. 8C, which may also be associated with more turbulent flow patterns in arteries. Nevertheless, the measured optical properties of blood also agreed with our simulated and ex vivo experimental results. Fig.   8D shows a least-squares fit of the attenuation spectrum measured in the vein. The best fit yields sO2 = 59% (R 2 = 0.99) and SSF = 0.07. Fig. 8E shows the fitting results for the artery, where sO2 = 100% (R 2 = 0.97) and SSF = 0.07.

Discussion and conclusion
Accurate We imaged ex vivo bovine blood phantoms using vis-OCT as the first validation of our simulation findings. After correcting systemic biases from the background and roll-off, we measured sO2 and SSF and compared vis-OCT measurements with blood-oxygen analyzer measurements. We found that the average SSF was 0.060 +/-0.021, almost identical to the simulated results. We further performed vis-OCT sO2 measurements in the human retina and found that SSF was 0.06 and measured sO2 values consistent with physiological ranges for arteries and veins. This work is the first comprehensive investigation and validation of blood's attenuation spectrum in vis-OCT using simulation, ex vivo phantom imaging, and human retinal imaging.
Using the packing factor to scale the scattering coefficient in our MC simulation resulted in SSF values in excellent agreement with experimental data. As suggested in the literature, the packing factor is the result of correlated scatterings among densely packed RBCs [9,29,67,68].
As RBC concentration increases, coherent interferences can affect the far-field scattered field, which is nonlinearly correlated with RBC density and is likely dependent on the orientations of individual RBCs. While MC simulation does not directly account for orientation-dependent scattering, it can replicate their statistical influence by scaling the input by 4. Furthermore, previous experimental tests of blood's scattering coefficient were performed using an integration sphere [10,11,29,47,69], which does not spatially filter detected photons as vis-OCT does. Our SSF combines the influence of scattering effects from blood hemodynamics (e.g., packing factor) and spatial filtering by the imaging modality (e.g., acceptance aperture and angle) on the effective measured by vis-OCT.
One limitation in our simulation is the assumption of blood as a homogenous medium. In reality, RBC packing density and orientations are affected by blood flow, blood velocity, vessel size, and incident angle [11,50,[70][71][72]. These factors can alter scattering cross-section and directionality, changing the optical properties assumed in this work. Although significant spatial averaging may suppress these variations, researchers should carefully monitor the variability of SSF to ensure a suitable oximetry model.
We investigated a suite of parameters influencing vis-OCT oximetry. We found that the combination of a low NA and high forward scattering breaks the common belief that OCT is mainly sensitive to blood's scattering coefficient because ophthalmic vis-OCT almost exclusively detects the unique Class I photons in the blood. Furthermore, considering the contribution of SSF, the detected scattering coefficient is significantly lower than the expected scattering coefficient, allowing absorption-dominated measurements in vis-OCT oximetry.