Melanin- and linewidth-corrected pulse-oximetry

Abstract

Current pulse oximeters tend to overestimate arterial saturation in darkly pigmented patients, increasing occult hypoxemia rates. One proposed mechanism is spectral coloring: melanin contributes to reshaping the detected spectrum of a broad-linewidth source, making the wavelength-dependent parameters behind calibration skin-tone-dependent. Diffuse optics theory allows for a melanin- and linewidth-corrected calibration equation, which to our knowledge has never been tested in vivo. In this work, we fit the empirical linear equation, the diffuse optics theoretical equation, and its corrected form to the public OpenOximetry Repository—considering 98 patients with paired arterial blood-draws, raw photoplethysmograms, and skin reflectance spectra. Accuracy is practically indistinguishable between the three approaches, yet the saturation error’s dependence on melanin falls from +2.8 to +1.2 to +0.1 percentage points per unit melanin volume fraction—consistent with the spectral-coloring theory. None of these dependencies on melanin are statistically resolved here—a cohort roughly ten times larger is needed—so the correction is backed by theory and its expected direction. All three calibration approaches carry two degrees of freedom: the theoretical equation drops into existing calibration procedures, and its melanin-aware form needs a per-patient skin-tone measurement. This work demonstrates how the theoretical pulse oximetry model can be applied in calibration.

Introduction

Pulse oximetry is a non-invasive optical technique that aims to measure a patient’s arterial Oxygen Saturation (SaO₂) using red and InfraRed (IR) light that has traversed tissue. The index recovered by a pulse oximeter is typically called the pulsatile Oxygen Saturation (SpO₂) (n.b., the “p” in SpO₂ can denote “peripheral” as well as “pulsatile”); however, in this work we will use SaO₂ since it represents the target quantity. Pulse oximeters focus on measuring the ratio of relative pulsatile amplitudes—from cardiac pulsation—at red versus IR, called the Ratio-of-Ratios (RoR)^1,2. Connecting the measured RoR to SaO₂ is typically achieved using an empirical calibration study on a relatively small cohort, on the order of 10 patients³. However, in recent years it has become clear that current pulse oximetry devices exhibit a skin-tone-correlated bias causing erroneously high SaO₂ readings for Black versus White patients^4–9. A consequence of this bias is higher rates of occult hypoxemia in patients with darker skin pigmentation^10–12 (n.b., occult hypoxemia occurs when true SaO₂ is low enough to warrant intervention, but intervention is not offered since the oximeter incorrectly indicates a high SaO₂). The current United States Food and Drug Administration (US FDA) calibration protocol from 2013 recommends at least two darkly pigmented subjects, or 15 % of the calibration cohort, whichever is larger³. This protocol itself is under active reconsideration, with a 2024 US FDA advisory panel reviewing the 2013 clinical study design and weighing pigmentation scales—including the Individual Typology Angle (ITA) considered here—for performance evaluation¹³. This asymmetric cohort requirement may explain the skin-tone bias, but raises the question of why the calibration would depend on skin tone and how calibration can be modified to account for it.

One proposed mechanism for the skin-tone bias is spectral coloring of broad-linewidth light sources^14–18. The argument is that if broad-linewidth sources are used (e.g., Light Emitting Diodes (LEDs)) as opposed to monochromatic sources (e.g., Laser Diodes (LDs)), the detected light spectrum and its nominal wavelength depend on the tissue optical properties, including a strong dependence on epidermal Melanin volume fraction (M). Ultimately, this means that the physical constants that depend on wavelength (e.g., the hemoglobin extinction coefficients) need to be adjusted for different skin tones, necessitating M-dependent calibration equations. This mechanism has recently received direct experimental support: substituting narrow-linewidth LDs for the LEDs of a conventional oximeter removed the pigmentation-related bias in both a benchtop test and a clinical study of 18 participants, while the two LED-based devices tested exhibited a bias¹⁹. Alternative mechanisms causing the skin-tone bias have also been proposed, such as a changing dermal sensitivity with skin tone²⁰ or non-linear effects involving changes in the background optical properties during the cardiac cycle interacting with M²¹. It is unclear whether one particular mechanism is the dominant source of the skin-tone bias, and it is possible that the bias is in fact the result of a combination of these mechanisms.

The empirical nature of pulse oximeter calibration itself has also been called into question, since the calibration is somewhat blind to what may confound it and how its recovered coefficients relate to physical quantities^{22, 23}. Recently, we derived the linear pulse oximeter calibration equation from the modified Beer-Lambert Law (mBLL), allowing one to relate the linear coefficients to physical quantities²⁴. This work also allowed us to arrive at a non-linearized theoretical expression for the relationship between SaO₂ and RoR based on hemoglobin molar extinction coefficients and optical path-lengths. We further tested the spectral coloring mechanism for skin-tone bias on the basis of this derivation, entirely in simulation, and initially proposed a spectral-coloring correction to pulse oximeter calibration¹⁸. This correction may now be tested on in vivo pulse oximeter calibration data. Such a test requires raw red and IR PhotoPlethysmoGram (PPG) traces paired with Arterial Blood Gas (ABG) SaO₂ in a skin-tone-diverse population, which the public OpenOximetry Repository provides²⁵. An important aspect of the proposed correction is a quantitative measure of skin tone directly related to M, which may also be derived from the OpenOximetry Repository’s raw skin reflectance spectra.

The core of this work considers three calibration cases: first, the standard linear empirical calibration; second, the mBLL-derived expression with an extra additive term; and third, the same mBLL-based expression including a correction for spectral coloring based on M. Importantly, in our investigation we consider only calibration methods with two Degrees of Freedom (DoF), so that the three cases are compared fairly—for this reason we do not include the sometimes-used empirical quadratic calibration, which would require three DoF. To our knowledge, this work therefore constitutes the first investigation of the mBLL-derived pulse oximeter calibration equation²⁴ and its spectral-coloring, skin-tone-dependent correction¹⁸ on an experimental controlled-desaturation dataset where M may also be quantitatively determined (i.e., the OpenOximetry Repository²⁵).

Results

OpenOximetry analyzable cohort

Oximetry parameters

The two oximetry parameters required for calibration are SaO₂ and RoR. Of the total 233 patients in the dataset, 106 patients satisfied this requirement (Figure 1 – orange set)—of which no M was recovered for 8 patients, leaving 98 patients in this work’s calibration set (Figure 1 – red intersection). The ABG SaO₂ requirement is the least restrictive since almost every patient in the dataset included these data with 218 patients contributing (Figure 1 – blue set). The majority of the 135 patients excluded from calibration were excluded due to no PPG record because the OpenOximetry dataset collected these data on only 139 patients (Figure 1 – green set). Therefore, the signal processing methods herein excluded 33/139 or 24 % of the PPG records when recovering RoR.

Figure 1.

UpSet plot of the OpenOximetry cohort divided into sets relevant for this work. All values refer to numbers of patients and the sum of all intersection bars represents the total number in the dataset (233 patients). (Blue) 218 patients have arterial Oxygen Saturation (SaO₂) values from Arterial Blood Gas (ABG). (Green) 139 patients have raw PhotoPlethysmoGram (PPG) records. (Orange) Ratio-of-Ratios (RoR) values were recovered for 106 patients. (Purple) 221 patients have reflectance spectra. (Brown) Finger epidermal Melanin volume fraction (M) values were recovered for 196 patients. (Red) 98 patients satisfy the intersection of the three sets needed for calibration (ABG SaO₂, Recovered RoR, and Recovered M).

The statistics of the SaO₂ and RoR data in the calibration set are reported in Table 1. The calibration set’s SaO₂ range is narrower than the 58.1 % to 100.0 % range of accepted SaO₂ values (n.b., the full dataset includes values >100 %, which are not accepted). This is because very low values of SaO₂ were often associated with no recovered RoR. Despite this, the calibration set’s range extends below the US FDA lower calibration bound of 70 %³.

Table 1.

Summary of the calibration set’s oximetry parameters (98 patients and 2991 samples)

Arterial Oxygen Saturation (SaO2) range	65.2 % to 99.4 %
SaO2 inter-quartile range	76.4 % to 93.1 %
SaO2 median	85.0 %
Ratio-of-Ratios (RoR) range	0.376 to 1.740
RoR inter-quartile range	0.740 to 1.211
RoR median	0.986

Skin-tone parameters

Skin tone was parameterized by M and skin color by ITA, both of which were derived from reflectance spectra in the visible range (Figure 2(a)). These are not interchangeable—M is a spectroscopic concentration recovered through a dermis-optics model, whereas ITA is a perceptual summary of that same spectrum. Both are reported due to the popularity of ITA, but only M is treated as a physical quantity. M is plotted versus ITA in Figure 2(b) showing a non-linear relationship. This relationship between M and ITA is largely due to ITA being an indirect measure of the perceived lightness while M is the concentration of a pigment, therefore in the limit of ITA = −90° (i.e., completely black), M would blow up to infinity (i.e., completely absorbing).

Figure 2.

Skin-tone or -color parameters derived from the reflectance spectra. (a) 791 spectra representing unique patient-and-site combinations. Line color represents recovered epidermal Melanin volume fraction (M), darker being greater. (b) Recovered M versus colorimetric Individual Typology Angle (ITA) for each of the spectra. Point colors match spectra in (a). (c) Distribution of recovered Ms at each of the four fitted sites. (d) The finger-site pooled M values used in further analysis, one value for each of the 196 patients with recoverable M.

The spectral fit was executed for all 791 spectra representing unique patient and site combinations (n.b., this included non-finger sites). This estimated 795 free parameters from the 791 spectra in one least-squares minimization (i.e., a free M for each spectrum and a free Total Hemoglobin (HbT), tissue Oxygen Saturation (StO₂), Rayleigh scattering fraction, and specular reflectance factor shared across all spectra), returning shared values of 34 μmol L⁻¹ for HbT, 0.76 for StO₂, a Rayleigh scattering fraction of 0.52, and 0.087 for the specular reflectance factor, with a coefficient of determination for the entire fit of 0.97. M recovered values in the range 0.0068 to 0.34 across all spectra (Figure 2(c)). However, only values for finger dorsal and palmar sites were continued into the pulse-oximetry analysis, as those were the sites relevant to the pulse oximeter finger clip measurement (Figure 2(d) and Table 2).

Table 2.

Summary of finger skin-tone or -color parameters for the 98 patients included in and the 98 excluded from calibration, of the 196 with a recovered epidermal Melanin volume fraction (M)

	Included in calibration	Excluded from calibration
M range	0.022 to 0.193	0.027 to 0.220
M inter-quartile range	0.042 to 0.083	0.038 to 0.074
M median	0.053	0.055
Individual Typology Angle (ITA) range	−27.9° to 47.3°	−34.9° to 43.6°
ITA inter-quartile range	6.6° to 29.4°	11.5° to 32.8°
ITA median	21.7°	23.7°

The 2013 US FDA guidance imposes a darkly pigmented quota on the calibration set without defining how to determine that criterion³. Therefore, we assumed that this US FDA criterion would typically be determined by self-reported race. Across the 98 patients used for calibration the reported categories are Asian (27), Caucasian (25), African American (24), and Hispanic (7), with 12 spread over multi-ethnic combinations and 3 not reported; 26 of them fall in the two categories treated here as darkly pigmented (i.e., African American and Other/Multiethnic African American). Race is a coarse and poor stand-in for skin pigmentation, and it is used only to replicate what the US FDA guidance suggests.

Calibration accuracy and skin-tone dependence

Three calibration cases

The first calibration case considered in this work is the conventional empirical linear (Equation 9); case two considers a calibration model based on mBLL assuming monochromatic light sources²⁴ (Equation 10); and case three is that same recovery with detected-spectrum-weighted coefficients which account for spectral coloring of polychromatic sources¹⁸ (Equation 12). Cases one and two are blind to melanin—neither takes M as an input. Case three is the only melanin-aware case, since its detected spectrum, and hence its weighted extinction coefficients and weighted path-length ratio, depends on M. Case three also requires either knowledge or an assumption of the sources’ bandwidth; in this work we assumed a red bandwidth of 25 nm and an IR bandwidth twice that (i.e., 50 nm). All three cases have two DoF and differ only in form.

Fitted over the whole calibration set of 98 patients, case one gave a = −32.53 % with b = 116.73 % both in SaO₂ units (Equation 9), case two gave path-length ratio Γ = 0.987 with offset o = 10.21 % in SaO₂ (Equation 10), and case three Γ = 0.957 with o = 9.85 % in SaO₂ (Equation 12; Figure 3). Accuracy—considering the whole fit of 98 patients—is nearly identical across the three cases, at A_rms of 2.420 %, 2.416 % and 2.418 % in SaO₂ units, respectively. Similar average accuracies were found in a 200-fold cross-validation using US FDA complaint training sets³. The median held-out A_rmss for the three cases were 2.494 %, 2.482 % and 2.484 % in SaO₂, respectively (Figure 4(a)). Every held-out fold of every case in the 200-fold cross-validation fell below the 3.0 % transmittance limit set by the US FDA guidance³. Therefore, it is reasonable to conclude that these three calibration cases perform practically identically in terms of overall accuracy.

Figure 3.

Qualitative comparison of the three calibration cases. Plots show arterial Oxygen Saturation (SaO₂) from Arterial Blood Gas (ABG) analysis versus the measured Ratio-of-Ratios (RoR) for 98 patients and 2991 blood-draws. Curves in these plots are fitted to this entire dataset. (a) Empirical linear calibration (Equation 9) recovering a = −32.53 % and b = 116.73 % in SaO₂ units. (b) Modified Beer-Lambert Law (mBLL)-derived calibration with offset (Equation 10) recovering path length ratio Γ = 0.987 and offset o = 10.21 % in SaO₂. (c) Detected-spectrum-weighted mBLL-derived calibration with offset accounting for the spectral coloring of epidermal Melanin volume fraction (M) (Equation 12) recovering Γ = 0.957 and o = 9.85 % in SaO₂. This case gives a curve that depends on M. (d) Zoomed region shown as a pink box in (a)–(c) showing the slight difference among the three curves and the differing M-dependent curves for case (c).

Figure 4.

Quantitative comparison of the three calibration cases using 200-fold cross-validation across 98 patients. Training sets consisted of 10 patients that satisfied the 2013 United States Food and Drug Administration (US FDA) guidance³; the remaining 88 patients formed the testing set. Plots present testing set metrics. (a) Root-mean-square accuracy in arterial Oxygen Saturation (SaO₂) including a green dashed line at the US FDA acceptance limit. (b) Skin-tone or -color dependence of the calibration expressed as the slope of the residuals between recovered and Arterial Blood Gas (ABG) SaO₂ versus either epidermal Melanin volume fraction (M) (orange) or Individual Typology Angle (ITA) (pink). A slope of 0 (dotted black line) indicates no M or ITA dependence.

Skin-tone dependence

We measured skin-tone or -color dependence in terms of the linear slope of the SaO₂ residual—the difference between the recovered value and the gold standard ABG value—versus either M or ITA. These results are shown in Figure 4(b) for the 200 cross-validation folds. Considering M as the skin-tone variable, the median held-out slope was +2.76 percentage points of SaO₂ per unit volume fraction for case one, +1.24 for case two, and +0.10 for case three. Against ITA the same pattern appears with the opposite sign—since ITA is more negative with higher M (Figure 2(b))—at median held-out slopes of −0.0040, −0.0017, and +0.0007 percentage points of SaO₂ per degree ITA. The ordering above therefore indicates the direction in which a spectral-coloring correction moves the residual. However, we cannot conclude that a melanin-dependent bias has been measured in these data, since the fold-to-fold range of each slope contains zero. Overall, we report the correction as applied and physically motivated while making no claim that it removes a measured skin-tone bias in these data.

Discussion

This work considered three pulse oximetry calibration cases which were fit to the same 2991 blood-draws from 98 patients. The cases differ only in the function relating SaO₂ to RoR; importantly each contains two Degrees of Freedom (DoF). They are indistinguishable in accuracy, differing by less than 0.02 % in SaO₂ units considering root-mean-square error. These calibration cases differ in what each form is able to represent, and how much the residuals depend on skin tone.

The first calibration case (Equation 9 and Figure 3(a)) is entirely empirical. Its linear equation’s slope and intercept were not derived from physical principles; however, these parameters can be expressed in terms of physical quantities if the mBLL-derived second case is linearized²⁴. The mBLL-derived form shows SaO₂ as a ratio of two linear expressions with RoR (i.e., linear-fractional: y = (ax+b)/(cx+d)). However, the linear-fractional expression only contains one unknown DoF (i.e., the ratio of path lengths Γ) with other coefficients being combinations of extinction coefficients which are fixed if the optical wavelengths are known^{18, 24}. Calibration cases two and three (Equations 10&12 and Figure 3(b)&(c)) consider a combination of the theoretical mBLL expression plus an additive empirical constant (i.e., the offset o). This offset parameter was adopted for two reasons: first, to match the number of DoF in case one; second, to substantially improve fit quality as a single DoF fit was unable to adequately capture the SaO₂ versus RoR relationship in the experimental data. The mBLL-derived linear-fractional form generates a curved line—with or without the offset parameter—which a simple linear equation with the same number of DoF cannot recreate. Case three (Equation 12 and Figure 3(c)) adds no additional DoF, but considers spectral coloring of broadband sources by tissue including M¹⁸. This makes M an independent variable and turns the single curve into a family of curves dependent on skin tone. Therefore, case three would need a measurement of M and knowledge of the light source bandwidth to implement in practice. All three cases carry two DoF, but only cases two and three are physically derived and interpretable.

The curvature of cases two and three is small but not negligible over the range of SaO₂ considered. Across the calibration set’s RoR range of 0.376 to 1.740, the case-two curve departs from the best straight line through it by up to 1.36 % in SaO₂ and the case-three curve by up to 1.39 % in SaO₂. This curvature results in a fit curve offset from the linear case despite having the same number of DoF (Figure 3(d)) and may have implications for values recovered outside the calibration range—where extrapolation relies solely on the model’s form since no data was present in the fit. Despite the different curves recovered by the fit for the three cases, A_rms is almost unaffected (Figure 4(a)) because the differences in the curves are largest where there is the least experimental data—only 5.8 % of blood-draws lie above a RoR of 1.4 and 5.2 % below 0.6—so the three curves separate mainly outside the crowded middle of the distribution that dominates A_rms. Since there is no observable difference in accuracy of the three calibration cases, we cannot recommend one over another on that basis, but we do expect a difference in measurements which extend below the calibrated SaO₂ range where the shape of the extrapolated curve matters^{26, 27}. We would expect cases two and three to perform better in this scenario; however, we have no evidence of this in these data, since extremely low SaO₂ s were not reached.

The mBLL-based calibration cases’ fitted path-length ratios (i.e., Γs) recover values above the modeled monochromatic value of Γ_mono = 0.850 considering the total path-lengths in a bulk tissue slab as modeled in Ref. 18. However, agreement was not expected, since we know considering Γ in that way is, strictly speaking, not correct and only gives a rough estimate of the true value²⁴. The mBLL expression underlying Equations 10&12 contains a ratio of mean partial optical path-lengths in the pulsatile arterial volume—the tissue regions the arterioles expand into—and not a ratio of total optical path-lengths through the entire bulk tissue. A preliminary Monte Carlo study of these partial path-lengths supports that reading, indicating that the relevant quantity is the path-length in the annulus surrounding the blood vessels rather than the bulk path-length, and that this ratio itself varies with SaO₂²⁸. At 0.987 for case two and 0.957 for case three they are much closer to unity compared to the bulk value, which is consistent with the pulsatile partial paths at the two wavelengths being nearly equal.

Interpretation of the additive offset (i.e., o) is more difficult since it is not arrived at by the mBLL derivation. Cases two and three recovered 10.21 % and 9.85 % for the offset, respectively. An offset of that magnitude is unlikely due to rounding or noise—it says the mBLL expression models roughly 10 % in SaO₂ low across the whole cohort—and it is stable across the cross-validation folds, suggesting it is systematic. Candidate explanations include venous, tissue, or water contamination of the pulsatile signal^{29, 30} or an offset in the ABG values (e.g., the dataset contains SaO₂ from ABG above 100 %), though the latter is unlikely given the magnitude of the offset. To interpret this offset one would need to carefully consider which assumptions in Ref. 24 are broken in the real-world measurement. Distinguishing among these explanations is beyond the scope of this work, therefore in this context we consider o an uninterpreted empirical parameter similar to b in the linear case.

Moving to the three calibration cases’ skin-tone dependence, we can discuss the interpretation of Figure 4(b). The ordering of the calibration cases shows what each case is able to represent: the empirical linear case shows the most uncorrected skin-tone dependence, the nominal mBLL-derived recovery—which considers monochromatic wavelengths thus no source linewidth—shows less dependence, and case three, which accounts for spectral coloring of a finite linewidth given M, sits near zero skin-tone or -color dependence. An important caveat is that none of these slopes are resolved by these data (e.g., every case’s 95 % fold-to-fold spread interval contains zero dependence). We therefore report an observed change in dependence across three ways of fitting the same data, and not a measured skin-tone bias in case one nor a statistically demonstrated removal of a skin-tone bias. The expected cohort size required to resolve such a skin-tone dependence in the residuals is therefore a useful point of comparison. This calibration set holds 98 patients, whereas the clinical reports of skin-tone-dependent pulse-oximeter error utilize far larger cohorts—1565 patients in one and 8392 patients in another cohort, both from Ref. 4. Therefore, a study aiming to resolve a significant slope of SaO₂ residual versus M is expected to need on the order of ten times the patients here. So the absence of a resolved dependence in these data does not negate the change we observe between the calibration cases, but points to the need for a larger cohort study. Public cohorts of the required size are becoming available—for example, the Blood-gas and Oximetry Linked Dataset (BOLD) combines 49 099 paired measurements from three intensive-care databases³¹—though such resources record paired commercial oximeter and ABG recovered values rather than the raw PPG traces—or RoR—and skin reflectance spectra that the correction tested here requires. Nonetheless, our interpretation of these results shows promise for calibration cases two and three, especially three which may correct for the spectral-coloring portion of the skin-tone bias in pulse oximetry.

In conclusion, this work proposes and tests two mBLL-derived pulse-oximetry calibration methods (i.e., cases two and three). Although case three’s correction was suggested in Ref. 18, this is the first work with sufficient detail to implement it—see the Methods section—and test it on experimental data. While case three is the best-performing case, it is also the most complex to implement due to its requirement of a separate measurement of M for each patient. This suggests that case two is the most feasible to implement for current oximeter technology; however, we must note that case two is melanin-blind and only case three is melanin-aware. Furthermore, since we propose calibration methods, the melanin and linewidth corrections cannot be directly applied to current pulse oximeters. Future work may utilize the theory and model presented here to derive such an oximeter correction. Derivation of such a correction may be necessary to reach a cohort size with enough statistical power to show a statistically significant skin-tone bias and its correction, since datasets of such size exist for clinical commercial oximeter readings but not for controlled desaturation studies. As low-hanging fruit, in future desaturation-based pulse-oximeter calibration studies we suggest that mBLL-based calibration models be considered, especially case two which does not require a separate skin pigmentation measurement and could easily replace linear calibration since it contains the same number of DoF.

Methods

The analysis in this work can be reproduced using the code in the accompanying public GitHub repository³² and the data that are publicly available in the OpenOximetry Repository v1.1.1 from PhysioNet²⁵. These data were collected under the UCSF Hypoxia Lab controlled-desaturation protocol, aligned with ISO 80601-2-61, in which healthy adult subjects breathe titrated nitrogen, oxygen, and carbon dioxide gas mixtures to hold a sequence of stable arterial oxygen saturation plateaus spanning about 70 % to 100 %. At each plateau, arterial blood drawn from a radial-artery catheter was analyzed by ABG oximetry (Radiometer ABL90 Flex Plus) to yield a gold-standard SaO₂ measurement, which was recorded simultaneously with a custom pulse-oximeter reporting raw PPG (Analog Devices MAX86171). Skin pigmentation was quantified with a reflectance spectrophotometer designed for color measurement (Konica Minolta CM-700d). No human-subject data other than the OpenOximetry repository were collected or used in this work.

The OpenOximetry study was approved by the UCSF IRB #21-35637 and #23-40212—the authors of this work were not associated with or involved in the study. Informed consent, including consent to share de-identified data, was obtained from all participants by the original study investigators, and all data included in the repository were collected in accordance with the ethical principles of the Declaration of Helsinki²⁵. Large language models were used for proofreading and editing of the work; they were not used to generate or interpret results, and the authors take full responsibility for the content of this work.

Indexing ABG data

The first stage of the analysis indexed the OpenOximetry²⁵ repository to extract SaO₂ data from the ABG measurements. From the blood-gas table the patient identifiers, encounter identifiers, sample numbers, and SaO₂ measurements were extracted—rows missing one of these values were excluded. Rows in the blood-gas table were also excluded if SaO₂ > 100 % or if their encounter and patient identifiers did not appear together as a registered pair in the encounter table. For each sample the median of repeated ABG draws was considered. In total, the blood-gas table included 32 877 rows associated with 224 patients which were processed to result in 17 756 samples of SaO₂ from 218 patients (Figure 1 – blue set). This step of the analysis is found in the A_loadSaO2_openox.m script in the GitHub repository³².

Ratio-of-ratios extraction from raw PPG

Each encounter’s raw PPG data was read using the PhysioNet WaveForm DataBase (WFDB) Toolbox³³. These consisted of 86 Hz recordings at 660 nm and 910 nm (i.e., red and IR) optical wavelengths. Missing samples were linearly interpolated with nearest-value end filling. Then, spike or step artifacts were detected and their indexes saved (Algorithm 1). The red and IR artifact flags were combined by logical union into one mask, and only contiguous valid segments of at least 3 s were considered.

Algorithm 1.

Spike and step artifact detection

Require: Full-encounter PPG trace $I$; sampling rate $f_s$; spike window $w_{sp}\leftarrow 2\mathrm{s}$; spike threshold $k_{sp}\leftarrow 5$; step half-window $w_{st}\leftarrow 0.25\mathrm{s}$; step threshold $k_{st}\leftarrow 6$; step floor $m_{st}\leftarrow 0.0015$; min valid segment length $\mathrm{\Delta }{t}_{\mathit{min}}\leftarrow 3\mathrm{s}$

Ensure: Valid artifact-free segments $\{s\}$

1: moving baseline $B\leftarrow$ moving median of I over max(3, $\mathrm{round}\left(w_{sp}{f}_s\right)$) samples

2: relative change $X\leftarrow (I-B)/B$

3: pseudo standard deviation ${\sigma }_X\leftarrow \mathrm{median}(|X-\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{n}(X)|)/\mathrm{norminv}(0.75)$

4: spike mask $A_{sp}\leftarrow \left(|X|\ge k_{sp}{\sigma }_X\right)$

5: trailing baseline $B^{-}\leftarrow$ moving median of I over max(2, $\mathrm{round}\left(w_{st}{f}_s\right)$) samples before the sample

6: leading baseline $B^{+}\leftarrow$ moving median of I over max(2, $\mathrm{round}\left(w_{st}{f}_s\right)$) samples after the sample

7: relative edge change $Y\leftarrow \left|B^{-}-B^{+}\right|/B$

8: pseudo edge standard deviation ${\sigma }_Y\leftarrow \mathrm{median}(|Y-\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{n}(Y)|)/\mathrm{norminv}(0.75)$

9: step mask $A_{st}\leftarrow \left(Y\ge \mathrm{m}\mathrm{a}\mathrm{x}\left(m_{st},k_{st}{\sigma }_Y\right)\right)$

10: artifact trace $A\leftarrow A_{sp}\cup A_{st}$

11: valid segments $\{s\}\leftarrow$ all segments between artifacts in A longer than $\mathrm{\Delta }{t}_{\mathit{min}}$

12: return $\{s\}$

The RoR was extracted on two types of fixed-length analysis windows: 1) 20 s windows swept in 10 s steps to form a RoR trace for synchronization with ABG, and 2) 30 s windows centered on each ABG blood-draw to yield a RoR for each SaO₂ measurement. Within each analysis window, the heart rate $\left(f_{HR}\right)$ was estimated as the dominant Fourier-domain magnitude peak in the range 0.8 Hz to 3.5 Hz considering the PPG signal for the IR optical wavelength. Per-beat RoR was estimated based on the peak-to-peak amplitude of the raw PPG (Algorithm 2). A window’s RoR was the median of its per-beat values after rejecting beat-level outliers beyond 3 pseudo standard deviations (i.e., 3 × median(|X – median(X)|)/norminv(0.75)) of the median. A window was accepted only when it retained at least 20 beats, its pseudo standard deviation normalized by its median was at most 0.08, and the number of artifact samples was less than 0.3 times the window length. These criteria are deliberately not applied to the sliding windows of the synchronization trace (Algorithm 3).

Algorithm 2.

Peak-to-peak RoR estimator

Require: Windowed red/IR PPG traces $I_r,I_i$; valid segments $\{s\}$; heart rate $f_{HR}$; sampling rate $f_s$; baseline span $w_b\leftarrow 3$ periods; min trough spacing $w_t\leftarrow 0.5$ periods; min trough prominence $p\leftarrow 0.2$

Ensure: Per-beat RoR

1: baseline window $W\leftarrow \mathrm{m}\mathrm{a}\mathrm{x}\left(3,\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\left(w_b{f}_s/f_{HR}\right)\right)$ samples

2: de-trended average $I^{\prime }\leftarrow \left[\left(I_r-\mathrm{m}\mathrm{o}\mathrm{v}\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{n}\left(I_r,W\right)\right)+\left(I_i-\mathrm{m}\mathrm{o}\mathrm{v}\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{n}\left(I_i,W\right)\right)\right]/2$

3: for each segment $s$ do

4:  troughs $T\leftarrow$ minima of $I^{\prime }$ on $s$, spaced $\ge w_t$ periods, prominence $\ge p\times \left(Q_{0.95}-Q_{0.05}\right)$

5:  for each period between consecutive troughs in $T$ do

6:   $A{C}_y\leftarrow \mathrm{m}\mathrm{a}\mathrm{x}\left(I_y\right)-\mathrm{m}\mathrm{i}\mathrm{n}\left(I_y\right)$, with $y\in \{r,i\}$

7:   $D{C}_y\leftarrow \left(\mathrm{m}\mathrm{a}\mathrm{x}\left(I_y\right)+\mathrm{m}\mathrm{i}\mathrm{n}\left(I_y\right)\right)/2$, with $y\in \{r,i\}$

8:   if $A{C}_r,A{C}_i,D{C}_r,D{C}_i>0$ then

9:    append $\left(A{C}_r/D{C}_r\right)/\left(A{C}_i/D{C}_{\mathit{i}}\right)$ to RoR

10:   end if

11:  end for

12: end for

13: return RoR

Synchronization was required since the raw PPG and the monitor files containing ABG blood-draw times did not share a time axis, so the offset was determined from waveform shape using Algorithm 3. This was achieved by finding the temporal offset corresponding to the maximum cross-correlation between the RoR synchronization trace and the negative of the reported SpO₂ from a commercial oximeter. The synchronization quality was determined using the Pearson correlation between these two signals at that temporal offset, and encounters with Pearson correlation coefficient below 0.9 were excluded.

This stage of the analysis processed the 139 patients associated with an encounter carrying a raw PPG record (Figure 1 – green set). This totaled 7723 samples from candidate blood-draws, of which 3376 samples were associated with RoR values which passed quality thresholds, resulting in that number of RoR samples for 106 patients (Figure 1 – orange set bar). The step of the analysis described in this section can be found in the B_extractRoR_openox.m script in the GitHub repository³².

Skin tone and melanin fitting

Skin optics model

The analysis in this work focuses on $M$ because melanin is the optical absorber that models different skin tones directly^{34, 35}. $M$ was fit for using the measured 400 nm to 700 nm reflectance spectra. The fit modeled the epidermis as a transmissive filter with effective optical thickness of 0.2 mm on top of a semi-infinite medium representing the dermis similar to previous work^{18, 36}. This model determined the absorption coefficient (${\mu }_a$) of the epidermis solely from the M absorption, which has been reported as follows³⁷:

$${\mu }_{a,epi}(\lambda )=M\times 51.9{\mathrm{m}\mathrm{m}}^{-1}{\left(\frac{\lambda }{500\mathrm{n}\mathrm{m}}\right)}^{-3.5}$$ (1)

where $51.9{\mathrm{m}\mathrm{m}}^{-1}$ represents the ${\mu }_a$ for $M=1$ at 500 nm and the −3.5 power represents the wavelength dependence. For the dermis, model determined the ${\mu }_a$ from the HbT concentration, ${\mathrm{S}\mathrm{t}\mathrm{O}}_2$, and Water volume fraction $(W)$ as follows²⁰:

$${\mu }_{a,\mathit{der}}(\lambda )={\mathrm{S}\mathrm{t}\mathrm{O}}_2[\mathrm{H}\mathrm{b}\mathrm{T}]{\epsilon }_{{\mathrm{H}\mathrm{b}\mathrm{O}}_2}(\lambda )+\left(1-{\mathrm{S}\mathrm{t}\mathrm{O}}_2\right)[\mathrm{H}\mathrm{b}\mathrm{T}]{\epsilon }_{\mathrm{H}\mathrm{b}}(\lambda )+W{\mu }_{a,\mathit{water}}(\lambda )$$ (2)

where ε’s represent Molar extinction coefficients of either OxyHemoglobin $\left({\mathrm{H}\mathrm{b}\mathrm{O}}_2\right)$ or deoxyHemoglobin (Hb) according to the subscript. The reduced scattering coefficient $\left({\mu }_s^{\prime }\right)$ of the dermis was modeled as a combination of Rayleigh and Mie scattering as follows^{20, 37}:

$${\mu }_{s,\mathit{der}}^{\prime }(\lambda )={\mu }_{s,\mathit{der}}^{\prime }(500\mathrm{n}\mathrm{m})\left(f_{\mathit{Ray}}{\left(\frac{\lambda }{500\mathrm{n}\mathrm{m}}\right)}^{-4}+\left(1-f_{\mathit{Ray}}\right){\left(\frac{\lambda }{500\mathrm{n}\mathrm{m}}\right)}^{-b_{\mathit{Mie}}}\right)$$ (3)

Algorithm 3.

PPG RoR and ABG blood-draw time synchronization

Require: Full-encounter red/IR PPG traces $I_r,I_i$, reference ${\mathrm{S}\mathrm{p}\mathrm{O}}_2$ trace $S$ (in ABG time); ABG blood-draw times $\{D\}$; sync window $w_s\leftarrow 20\mathrm{s}$; sync step $\mathrm{\Delta }{t}_s\leftarrow 10\mathrm{s}$; resample grid $\delta \leftarrow 1\mathrm{s}$; quality threshold $r_{\text{min}}\leftarrow 0.9$; blood-draw half-window $h\leftarrow 15\mathrm{s}$

Ensure: RoR for each ABG blood-draw

1: slow RoR trace RoR′ ← median of the windowed peak-to-peak estimator (Alg. 2) on $w_s$ windows spaced by $\mathrm{\Delta }{t}_s$, requiring that the window contains at least two beats

2: monitor start $t_0\leftarrow$ min(time of $S$)

3: ${\mathrm{R}\mathrm{o}\mathrm{R}}_z^{\prime },S_z\leftarrow z$-scores of ${\mathrm{R}\mathrm{o}\mathrm{R}}_{pp}^{\prime }$ and $-S$, each resampled to $\delta$ spacing on their own time axis

4: lag $\mathrm{\Delta }\leftarrow$ lag time of max($\mathrm{x}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\left({\mathrm{R}\mathrm{o}\mathrm{R}}_{pp,z}^{\prime },S_z\right)$)

5: synced $r\leftarrow$ Pearson correlation coefficient of ${\mathrm{R}\mathrm{o}\mathrm{R}}_{pp,z}^{\prime }$ and $S_z$ after shifting $S_z$ by $\mathrm{\Delta }$

6: if $r<r_{\text{min}}$ then

7:   return exclude the encounter

8: end if

9: temporal sync offset $\tau \leftarrow t_0-\mathrm{\Delta }$, which maps monitor time onto PPG-record time

10: for blood-draw time $D$ do

11:   compute RoR by Alg. 2 on the $\pm h$ window at $D-\tau$

12: end for

13: return RoR

where $f_{\mathit{Ray}}$ is the fraction of Rayleigh scattering and $b_{\mathit{Mie}}$ is the Mie scattering power.

Spectra were acquired using a Konica Minolta CM-700d on the small-aperture setting—three repeated measurements per site—under a D65 / 2° illuminant–observer setting³⁸; in that configuration the instrument diffusely illuminates a 6 mm diameter port and views a 3mm diameter measurement area at 8° from the normal³⁹. The illuminated port and the collection area are coaxial disks of different radii, $r_{\mathit{ill}}=3\mathrm{mm}$ and $r_{\mathit{col}}=1.5\mathrm{mm}$, so the source–detector separations ($\mathit{\rho }\mathrm{s}$) that contribute to the measurement are distributed following the overlap area ($\mathrm{\Lambda }$) of those two disks with centers separated by ρ:

$$\mathrm{\Lambda }(\mathit{\rho })=\left\{\begin{array}{ll}\pi r_{col}^2& \mathit{\rho }\le r_{ill}-r_{col}\\ r_{col}^2{\mathrm{c}\mathrm{o}\mathrm{s}}^{-1}\left(\frac{{\mathit{\rho }}^2+r_{col}^2-r_{ill}^2}{2\mathit{\rho }{r}_{col}}\right)+r_{ill}^2{\mathrm{c}\mathrm{o}\mathrm{s}}^{-1}\left(\frac{{\mathit{\rho }}^2+r_{ill}^2-r_{col}^2}{2\mathit{\rho }{r}_{ill}}\right)& r_{ill}-r_{col}<\mathit{\rho }<r_{ill}+r_{col}\\ -\frac{1}{2}\sqrt{\left[{\left(r_{ill}+r_{col}\right)}^2-{\mathit{\rho }}^2\right]\left[{\mathit{\rho }}^2-{\left(r_{ill}-r_{col}\right)}^2\right]}& \\ 0& \mathit{\rho }\ge r_{ill}+r_{col}\end{array}\right.$$ (4)

Considering the dermis Green’s function, $G_{\mathit{der}}(\mathit{\rho },\lambda )$, for the continuous-wave diffuse reflectance⁴⁰ from a semi-infinite medium at separation $\mathit{\rho }$, and the epidermal transmission, $T_{epi}(\lambda )$, through a non-scattering thin slab¹⁸, the total diffuse reflectance ($R_{\mathit{skin}}(\lambda )$) leaving the skin over the measurement aperture is (n.b., in this formulation, $R_{\mathit{skin}}(\lambda )$ is unit-less as it is normalized by the source power-spectral-density entering the boundary):

$$R_{skin}(\lambda )=T_{epi}(\lambda ){\int }_0^{r_{ill}+r_{col}}{G}_{der}(\mathit{\rho },\lambda )\frac{\mathrm{\Lambda }(\mathit{\rho })}{\mathrm{\Lambda }(0)}2\pi \mathit{\rho }d\mathit{\rho }$$ (5)

where $G_{\mathit{der}}(\mathit{\rho },\lambda )$ depends on the dermal absorption (Equation 2) and scattering (Equation 3), and $T_{\mathit{epi}}(\lambda )$ depends on the epidermal absorption (Equation 1) and effective optical thickness $L_{\mathit{epi}}$:

$$T_{epi}(\lambda )=e^{-2L_{epi}{\mu }_{a_{,epi}}(\lambda )}$$ (6)

(n.b., $L_{\mathit{epi}}$ is multiplied by two to account for the round trip distance). Finally, the measured total diffuse reflectance $\left(R_{\mathit{meas}}(\lambda )\right)$ considers the front-surface Saunderson specular reflection term $\left(K_{\mathit{spec}}\right)$ in which a fraction $K_{\mathit{spec}}$ of the incident light is reflected at the air skin boundary^{41, 42}:

$$R_{\mathit{meas}}(\lambda )=K_{\mathit{spec}}+\left(1-K_{\mathit{spec}}\right)R_{\mathit{skin}}(\lambda )$$ (7)

(n.b., $R_{\mathit{meas}}(\lambda )$ is unit-less as it is normalized by the power-spectral-density of the source emission). Internal re-reflection at that boundary is accounted for by the refractive-index mismatch considered in the $G_{\mathit{der}}(\mathit{\rho },\lambda )$ term. The fitting procedure minimized the squared difference between the modeled and observed $R_{\mathit{meas}}(\lambda )$.

Spectra pre-processing and fitting

The fit was restricted to spectra collected on the finger dorsal (top) side, palmar (bottom) side, forehead, and inner upper arm measurement sites³⁸. Each spectrum sampled 400 nm to 700 nm every 10 nm resulting in 31 individual wavelengths. A spectrum was rejected if any of its 31 reflectance values was non-positive, or if its peak reflectance fell outside [0.01, 0.90). Of the 9517 individual spectra at the four fitted sites, 288 were incomplete and 1 carried a non-positive value, leaving 9228. Surviving spectra were grouped by patient and site, and a group was carried forward only if it held at least 3 replicates; 2 groups held only 2 replicates each and were excluded, removing 4 spectra and leaving 9224 in 810 groups (i.e., patient-site combos). A replicate was discarded when its level departed from the group median by more than both 5 pseudo standard deviations and 0.25 in log₁₀ units; the group itself was retained and pooled from the replicates that survived—the pseudo standard deviation again being $\mathrm{median}(|X-\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{n}(X)|$)/ norminv(0.75) over the levels in that group. This discarded 7 replicates drawn from 5 groups, and the remaining 9217 spectra were averaged wavelength by wavelength into one spectrum per patient-and-site combination (i.e., group), giving 810 averaged spectra.

Water volume fraction (0.65), dry-tissue refractive index (1.514), reduced-scattering amplitude (${\mathit{\mu }}_s^{\prime }$(500 nm) = 4.36 mm⁻¹), and Mie scattering power (b = 0.562) were fixed at dermis literature values^{20, 37}. Each remaining optical-property parameter was fit for, either with a unique value for each spectrum or a shared value across all spectra. Four population-shared parameters (i.e., shared values across all spectra) were considered, namely the three dermis properties HbT, StO₂ (Equation 2), and the Rayleigh scattering fraction (Equation 3), together with the front-surface specular reflectance factor (Equation 7). The parameter of interest, $M$ (Equation 1), found a unique value for each average spectrum. Any individual spectrum whose coefficient of determination—referenced to that spectrum’s own mean—fell below 0.50 was discarded and the fit rerun until all remaining spectra met this threshold; 19 of the 810 average spectra were discarded by this threshold.

Color quantification

Skin pigmentation was also quantified in terms of color using the ITA. ITA was computed from CIE $L^{\mathrm{*}}{a}^{\mathrm{*}}{b}^{\mathrm{*}}$ coordinates $\left(L^{\mathrm{*}}-\right.$ lightness; $a^{\mathrm{*}}$ – red/green; $b^{\mathrm{*}}$ – yellow/blue) derived from the measured reflectance spectrum—integrated against the CIE 1931 2° color-matching functions under D65 illumination⁴²—as:

$$\mathrm{I}\mathrm{T}\mathrm{A}={\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}\left(\frac{L^{\mathrm{*}}-50}{b^{\mathrm{*}}}\right)$$ (8)

with ITA expressed in degrees, and larger ITA corresponding to lighter skin. Despite this colorimetric descriptor not directly representing melanin, it was retained because it is simple and directly comparable to the repository’s own colorimetry.

Skin tone summary

In total, this stage of the analysis processed 810 spectra from 221 patients (Figure 1 – purple set) each representing a unique patient and site combination for any of the four fitted sites. Of these, 791 spectra satisfied the goodness-of-fit requirement and finger $M$ values for 196 patients were recovered (Figure 1 – brown set). This entire step of the analysis is found in the C_extractM_openox.m script in the GitHub repository³².

SaO₂ versus RoR calibration

A blood-draw data point (i.e., a SaO₂ and RoR pair) was used for calibration when it passed all quality thresholds and was matched with a $M$ estimate—this criterion was met by 2991 blood-draws from 98 patients (Figure 1 – red intersection). Three SaO₂ versus RoR models were considered for calibration. The first case utilized the conventional empirical linear calibration:

$${\mathrm{S}\mathrm{a}\mathrm{O}}_2(\mathrm{R}\mathrm{o}\mathrm{R})=a\times \mathrm{R}\mathrm{o}\mathrm{R}+b$$ (9)

where $a$ and $b$ are empirical slope and intercept free parameters, respectively. The second case considered the mBLL-derived expression^{18, 24} with an empirical additive offset ($o$) included to improve fit quality and match the number of DoF in the empirical linear case:

$${\mathrm{S}\mathrm{a}\mathrm{O}}_2(\mathrm{R}\mathrm{o}\mathrm{R})=\frac{-{\mathit{\epsilon }}_{\mathrm{H}\mathrm{b},r}+{\mathit{\epsilon }}_{\mathrm{H}\mathrm{b},i}\times \mathrm{\Gamma }\times \mathrm{R}\mathrm{o}\mathrm{R}}{\left({\mathit{\epsilon }}_{{\mathrm{H}\mathrm{b}\mathrm{O}}_2,r}-{\mathit{\epsilon }}_{\mathrm{H}\mathrm{b},r}\right)+\left({\mathit{\epsilon }}_{\mathrm{H}\mathrm{b},i}-{\mathit{\epsilon }}_{{\mathrm{H}\mathrm{b}\mathrm{O}}_2,i}\right)\times \mathrm{\Gamma }\times \mathrm{R}\mathrm{o}\mathrm{R}}+o$$ (10)

where the $\mathit{\epsilon }$’s are the molar extinction coefficients for either ${\mathrm{H}\mathrm{b}\mathrm{O}}_2$ or Hb according to the subscript, and either red or IR wavelengths—$r$ or $i$ subscripts, respectively—and $\mathrm{\Gamma }$ is the ratio of average optical path-lengths:

$$\mathrm{\Gamma }=\frac{⟨\ell {⟩}_i}{⟨\ell {⟩}_r}$$ (11)

at IR and red wavelengths. The third case utilized mBLL the same way as Equation 10, but accounted for polychromatic sources, whereas Equation 10 assumed monochromatic light sources. Polychromatic sources with Full-Width at Half-Maximum (FWHM) w were considered by using detected spectrum weighted extinction coefficients $(\bar{\mathit{\epsilon }})$ and a weighted path-length ratio $(\stackrel{\mathrm{‾}}{\mathrm{\Gamma }}$)¹⁸:

$${\mathrm{S}\mathrm{a}\mathrm{O}}_2(\mathrm{R}\mathrm{o}\mathrm{R},M,w)=\frac{-{\bar{\mathit{\epsilon }}}_{\mathrm{H}\mathrm{b},r}(M,w)+{\bar{\mathit{\epsilon }}}_{\mathrm{H}\mathrm{b},i}(M,w)\times \stackrel{\mathrm{‾}}{\mathrm{\Gamma }}(M,w)\times \mathrm{R}\mathrm{o}\mathrm{R}}{\left({\bar{\mathit{\epsilon }}}_{{\mathrm{H}\mathrm{b}\mathrm{O}}_2,r}(M,w)-{\bar{\mathit{\epsilon }}}_{\mathrm{H}\mathrm{b},r}(M,w)\right)+\left({\bar{\mathit{\epsilon }}}_{\mathrm{H}\mathrm{b},i}(M,w)-{\bar{\mathit{\epsilon }}}_{{\mathrm{H}\mathrm{b}\mathrm{O}}_2,i}(M,w)\right)\times \stackrel{\mathrm{‾}}{\mathrm{\Gamma }}(M,w)\times \mathrm{R}\mathrm{o}\mathrm{R}}+o$$ (12)

where a weighted extinction coefficient is expressed as:

$$\bar{\mathit{\epsilon }}(M,w)=\frac{\int \mathrm{ }I(\lambda ,M,w)\mathit{\epsilon }(\lambda )d\lambda }{\int \mathrm{ }I(\lambda ,M,w)d\lambda }$$ (13)

where $I(\lambda ,M)$ is the power-spectral density of the detected light which depends on $M$ due to spectral coloring if the light source is polychromatic¹⁸. The weighted path-length ratio is expressed as:

$$\stackrel{\mathrm{‾}}{\mathrm{\Gamma }}(M,w)=\mathrm{\Gamma }\times \gamma (M,w)$$ (14)

where:

$$\gamma (M,w)=\frac{⟨\stackrel{\mathcal{‾}}{\ell }{⟩}_i(M,w)/⟨\stackrel{\mathcal{‾}}{\ell }{⟩}_r(M,w)}{{\mathrm{\Gamma }}_{\mathit{mono}}}$$ (15)

and a weighted optical path-length ($⟨\stackrel{\mathcal{‾}}{\ell }⟩$) is:

$$⟨\stackrel{\mathcal{‾}}{\ell }⟩(M,w)=\frac{\int I(\lambda ,M,w)⟨\ell ⟩(\lambda )d\lambda }{\int I(\lambda ,M,w)d\lambda }$$ (16)

where ${\mathrm{\Gamma }}_{\mathit{mono}}$ and $⟨\ell ⟩$ are derived from the slab bulk tissue model presented in Ref. 18 and ${\mathrm{\Gamma }}_{\mathit{mono}}$ is the modeled value for the nominal monochromatic wavelengths. The purpose of the variable $\gamma$ is to capture the spectral coloring dependence of $\mathrm{\Gamma }$ on $M$ using an analytical model, but $\mathrm{\Gamma }$ (Equation 14) is still directly fit for in case three. This makes the dependence of $\stackrel{\mathrm{‾}}{\mathrm{\Gamma }}$ on $M$ model dependent, but not the amplitude of the fit parameter $\mathrm{\Gamma }$. The takeaway of case three (Equation 12) is that the detected spectrum depends on epidermal Melanin volume fraction $(M)$, making $M$ an independent variable in the calibration. All three of these calibration cases contain two free parameters: Equation 9 – a and b; Equations 10 & 12 – Γ and o. Importantly, the $\mathrm{\Gamma }$ parameter has a physical interpretation²⁴ while the interpretation of the empirical parameters—$a,b$, and $o$—is less clear.

In Equation 10, the nominal extinction coefficients used the values at 660 nm and 910 nm for red and IR, respectively. In Equations 12, 13, and 16, the detected power-spectral density $(I)$ was modeled as the product of the source emission power-spectral density $(P)$, the epidermal transmission $\left(T_{epi}\right)$, and the bulk-tissue transmittance Green’s function ($G_{\text{bulk}}$)¹⁸:

$$I(\lambda ,M,w)=P(\lambda ,w)\times T_{epi}(\lambda ,M)\times G_{bulk}(\lambda )\times {\mathrm{\Lambda }}_{det}$$ (17)

where ${\mathrm{\Lambda }}_{\mathit{det}}$ is the area of the detector and $G_{\mathit{bulk}}$ was modeled using a 15 mm thick slab as in Ref. 18—the same model from which $⟨\ell ⟩$ values were calculated in Equation 16 (n.b., for computation a unit detector area was considered). In this work, $P$ was modeled as a Gaussian broad-linewidth source:

$$P(\lambda ,w)=P_0\times e^{-4\mathrm{l}\mathrm{n}(2)\times {\left(\lambda -{\lambda }_0\right)}^2/w^2}$$ (18)

where $w$ is the FWHM, ${\lambda }_0$ is the nominal wavelength, and the expression is normalized so that $P\left({\lambda }_0\right)=P_0$ where $P_0$ is the peak power-spectral density (n.b., for computation a unit peak power-spectral density was considered). For the results in this work, we assumed a $w_i$ of 50 nm for the IR source (i.e., ${\lambda }_{0,i}=910\mathrm{n}\mathrm{m}$) and a FWHM of half that for the red source (i.e., $w_r$ of 25 nm with $\left.{\lambda }_{0,r}=660\mathrm{n}\mathrm{m}\right)$^{15, 18}. At zero FWHM—the monochromatic case—the weighted extinction coefficients reduce to their nominal values (i.e., same values as Equation 10) and $\gamma =1$, so case three reduces to case two exactly when $w_r=w_i=0\mathrm{n}\mathrm{m}$.

To execute the calibration, case one (Equation 9) was fit by ordinary least squares. Cases two and three (Equations 10 and 12) were fit by bounded nonlinear least squares—Γ was bounded at 0.1 to 10 and o was bounded at −100 % to 100 %. A blood-draw’s residual was defined as its recovered SaO₂ using the calibrated model minus its measured SaO₂, and calibration accuracy was summarized as the root-mean-square residual, $A_{\mathit{rms}}$ To test the calibration’s dependence on skin tone, a linear regression considered the patient-mean residual versus patient pigment (i.e., $M$ or ITA), weighting each patient by the inverse squared standard error of that patient’s mean residual. The quantity taken from this fit was its slope, in units of SaO₂ percentage per unit pigment (e.g., % per volume-fraction for $M$ and % per ° for ITA).

This calibration procedure was carried out globally for the whole dataset and across 200 cross-validation folds. For cross-validation, each fold drew a training set that satisfied the US FDA 2013 premarket-notification guidance for pulse oximeters³. Following that guidance, a training set held 10 patients of both assigned sexes, 2 of them darkly pigmented—determined by their reported race (i.e., “African American” or “Other/Multiethnic African American”) since the guidance does not define darkly pigmented—contributing at least 200 blood-draws inside the 70 % to 100 % SaO₂ range. Patients were drawn at random and redrawn until these conditions were met. The 88 held-out patients (n.b., 98 total patients were considered for calibration) comprised the testing set which was used to determine the held-out $A_{\text{rms}}$ and held-out slope of SaO₂ residual on pigment to test accuracy and skin-tone dependence, respectively (n.b., testing sets did consider blood-draws with SaO₂ < 70 % while training sets did not). This calibration step of the analysis is found in the D_calibrate_openox.m script in the GitHub repository³².

Data availability

The source dataset is the OpenOximetry Repository v1.1.1 on PhysioNet²⁵, which is subject to the repository’s data-use requirements. No OpenOximetry data are redistributed with this manuscript.

Code availability

The analysis code supporting this work is available at a public GitHub repository³².