2.1.

Theoretical Frameworks: Self-Calibrated Method for ${\mathrm{SpO}}_2$ Calculation

In this section, we outline the formulation of our self-calibrated algorithm designed for ${\mathrm{SpO}}_2$ calculation. The central premise is that $⟨L⟩$ is a function of ${\mu }_a$ and ${\mu }_s^{\prime }$, and given that ${\mu }_a$ depends on the saturation level, $⟨L⟩$ also varies accordingly.

In the following discussion, we use the term “$⟨L⟩$ ratio” to denote the ratio ${⟨L⟩}^{{\lambda }_n}/{⟨L⟩}^{{\lambda }_1}$, where ${\lambda }_n$ represents NIRS measurements at the $n$’th $(n=1,2,3,\dots )$ wavelength. Thus when multiple wavelengths (at least two) are used, the $⟨L⟩$ ratio describes a spectral shape across these wavelengths with the value at the first wavelength being 1. In contrast to common notation in pulse oximetry, where red and infrared wavelengths are typically used and $⟨L⟩$ ratio is usually represented by ${⟨L⟩}^{\text{infrared}}/{⟨L⟩}^{\text{red}}$, which is a singular value.

Building on this foundation, if we obtain a measured $⟨L⟩$ ratio from experiment that is ${\mathrm{SpO}}_2$ dependent, and we also have its analytical representation that depends on ${\mathrm{SpO}}_2$, it becomes feasible to align or fit the analytical $⟨L⟩$ ratio with the measured version to deduce ${\mathrm{SpO}}_2$. Our self-calibrated algorithm thus consists of three main components: (1) constructing a measured $⟨L⟩$ ratio from the continuous-wave near-infrared spectroscopy (CW-NIRS) measurement using mBLL, (2) constructing an analytical $⟨L⟩$ ratio derived from the analytical equation of $⟨L⟩$, and (3) determining the ${\mathrm{SpO}}_2$ value that best matches the measured and analytical $⟨L⟩$ ratios.

2.1.3.

${\mathrm{SpO}}_2$ determination by aligning the measured and analytical $⟨L⟩$ ratios

With ${⟨\hat{L}⟩}_{\text{measured}}^{\lambda }$ and ${⟨\hat{L}⟩}_{\text{analytical}}^{\lambda }$ defined in Eqs. (3) and (5), respectively, our objective is to find the ${\mathrm{SpO}}_2$ value that will best align the spectral shapes of these two ratios. To achieve this, we quantify the similarity between ${⟨\hat{L}⟩}_{\text{measured}}^{\lambda }$ and ${⟨\hat{L}⟩}_{\text{analytical}}^{\lambda }$ using the residual sum of squares (RSS) error ($\mathrm{RSS}={\mathrm{\Sigma }}_{\lambda }{[{⟨\hat{L}⟩}_{\text{measured}}^{\lambda }-{⟨\hat{L}⟩}_{\text{analytical}}^{\lambda }]}^2$) for the full range of ${\mathrm{SpO}}_2$ (0% to 100%). The ${\mathrm{SpO}}_2$ value yielding the minimum RSS provides our best estimate for true arterial oxygen saturation (${\mathrm{SaO}}_2$). Here we chose an iterative approach rather than a direct fitting approach for demonstration purpose, but computational efficiency can be improved if needed.

To illustrate the implementation of the self-calibrated algorithm, we provide a flowchart in Fig. 1. In this example, we assume the ground truth ${\mathrm{SaO}}_2$ to be 60%. As indicated in the rightmost box, an accurate estimation of ${\mathrm{SaO}}_2$ (with ${\mathrm{SpO}}_2=60\%$) results in spectral shapes of the measured and analytical $⟨L⟩$ ratios that closely resemble each other. It is important to note that this is only a visual representation—the actual values of $⟨L⟩$ ratios and their shapes across wavelengths will vary based on the specific experiment.

Fig. 1

Flowchart of the self-calibrated pulse oximetry algorithm.

2.2.

Algorithm Validation with Monte-Carlo Simulation

To validate the self-calibrated algorithm, we used Monte-Carlo photon simulation (MCX Lab)¹⁷ to generate synthetic data. To simulate cardiac pulsations, simulations at systolic and diastolic ${\mu }_a$ were performed. We modeled a semi-infinite homogenous media with ${\mathrm{SaO}}_2$ at 20%, 40%, 60%, 80%, and 100% at eight wavelengths (740, 760, 780, 800, 820, 840, 860 and 880 nm). A 2% change from baseline total hemoglobin concentration ([HbT], assumed to be $50\mu \mathrm{M}$) due to pulsation was assumed. Reduced scattering ${\mu }_s^{\prime }$ was assumed to be constant across ${\mathrm{SpO}}_2$ levels and pulsation. In ${\mathrm{cm}}^{-1}$, ${\mu }_s^{\prime }$ was calculated using ${\mu }_s^{\prime }=a·{\lambda }^b$, where $\lambda$ is in nm, and $a=260.7$ and $b=-0.4668$ were obtained from the literature for human subdermal tissue.^12,18 For example, at 800 nm and ${\mathrm{SaO}}_2=100\%$, ${\mu }_s^{\prime }=11.51{\mathrm{cm}}^{-1}$ and ${\mu }_a=0.0930$ and $0.0949{\mathrm{cm}}^{-1}$ at diastolic and systolic states, respectively. The dimensions of the modeled media were set to be 350, 350, and 200 mm for its width, length, and height, respectively. A simulated isotropic source with a photon count of $5\times {10}^8$ was placed at the center of the modeled tissue surface, and intensities emitted from the surface (as a reflectance measurement) at a source–detector distance of 3 cm were recorded. The simulated intensities at the systolic ($I_d$) and diastolic ($I_s$) states were then converted to $\mathrm{\Delta }\mathrm{OD}$ by $\ln (I_d/I_s)$.

Using the simulated $\mathrm{\Delta }\mathrm{OD}$, we first calculated ${\mathrm{SpO}}_2$ using the conventional method:¹¹

2.3.

${\mathrm{SpO}}_2$ Extraction Using Two Versus Eight Wavelengths in the Presence of Noise in $\mathrm{\Delta }\mathrm{OD}$

As described in Sec. 2.2, we simulated $\mathrm{\Delta }\mathrm{OD}$ over eight wavelengths, all of which were incorporated into the self-calibrated algorithm. In the human freediving experiment (detailed in the next section), however, data were only available for two wavelengths—760 and 840 nm. Recognizing that real experiments introduce noise into the $\mathrm{\Delta }\mathrm{OD}$, we aimed to assess the impact of noise on ${\mathrm{SpO}}_2$ extraction when using two versus eight wavelengths with the self-calibrated algorithm.

To examine this, we began with the noise-free simulated $\mathrm{\Delta }\mathrm{OD}$ at $r=3\mathrm{cm}$ and introduced Gaussian white noise [with standard deviation (STD) of 0.002] to $\mathrm{\Delta }\mathrm{OD}$ values at each wavelength (740, 760, 780, 800, 840, 860, and 880 nm). We then input the noisy $\mathrm{\Delta }\mathrm{OD}$ either at all eight wavelengths or just the two (760 and 840 nm) to the self-calibrated algorithm to derive the ${\mathrm{SpO}}_2$. This entire procedure was repeated 100 times, each with a unique Gaussian white noise introduced at every wavelength. Subsequently, we calculated the mean and STD of the extracted ${\mathrm{SpO}}_2$ values for comparison.

2.4.

Data Collection from Human Freediving Experiment

To evaluate the algorithm on human data, we analyzed measurements from breath-hold divers. The study’s experimental protocol received approval from the University of Padua’s Department of Biomedical Science Human Ethical Committee and followed the guidelines established by the declaration of Helsinki. Prior to their participation in the study, all subjects provided their written informed consent.

The freediving experiment was conducted in the 42 m deep indoor thermal pool “Y-40 THE DEEP JOY” in, Padua, Montegrotto, Italy. For each experimental trial, the participant diver completed a 15 m deep breath-hold dive and then recovered at the surface before completing a 42 m deep breath-hold dive. The descent and ascent of all dives were completed with the assistance of a weighted sled, meaning the divers experienced minimal physical exertion. The start, bottom, and end time of each dive was recorded. In this study, nine subjects performed sled-assisted dives. NIRS measurement and blood draws were carried out on these subjects during seven dives of 15 m and nine dives of 42 m.

An arterial cannula was inserted in the radial artery of each diver’s nondominant arm as described by Bosco et al.¹⁵ and Paganini et al.²⁰ for underwater blood sample collection at different stages of the dive as shown in Fig. 2: A (15 m start), B (15 m bottom), C (15 m end), D (42 m bottom), and E (42 m end). There was no blood sample collection at the start of the 42 m dives to limit the amount of blood taken from the divers. All the blood samples used for this study were taken while the divers’ heads were submerged under water (i.e., the predescent blood sample for the 15 m dive (point A) was taken right after the divers’ submerged under water and began their breath-hold, and the end-ascent blood samples (points C and E) were taken when the divers’ surfaced and before they took their first breath). The approximate timings for all the blood draws were recorded. A blood gas analyzer that uses CO-Oximetry (Stat Profile Prime Plus, Nova Biomedical Italia S.r.l., Lainate, Italy) was used for obtaining the ${\mathrm{SaO}}_2$ on site.

Fig. 2

Illustration for 15 and 42 m dives and arterial blood extraction time points.

In addition, CW-NIRS measurements were taken during the dives using a waterproof device, PortaDiver, which was developed from the PortaLite mini (Artinis, Medical System BV, Netherlands).¹⁴ The sensor contains three light-emitting diodes (3, 3.5, and 4 cm source–detector distance) with two wavelengths each (760 and 840 nm) and one photodiode with ambient light protection. It operates at a sampling rate of 10 Hz and was placed on the diver’s forehead and secured by a swimming cap.

2.5.

Application of Self-Calibrated Algorithm and Conventional Method on Freediving Data

First, we calculated ${\mathrm{SpO}}_2$ using the conventional method with a constant $⟨L⟩$ ratio of 0.87 derived from the literature (see Sec. 2.2).¹⁹ Specifically, we applied the mBLL [Eq. (1)] to the measured OD data at 760 and 840 nm from the CW-NIRS device to obtain time traces of $\mathrm{\Delta }[\mathrm{HbO}]$ and $\mathrm{\Delta }[\mathrm{Hb}]$. In MATLAB (Mathworks, Inc., Natick, Massachusetts, United States), a third-order zero-phase Butterworth high-pass filter at 1 Hz was applied to the $\mathrm{\Delta }[\mathrm{HbO}]$ and $\mathrm{\Delta }[\mathrm{Hb}]$ time traces. Spectrograms of $\mathrm{\Delta }[\mathrm{HbO}]$ and $\mathrm{\Delta }[\mathrm{Hb}]$ were then generated with a segment length of 10 s and an overlapping percentage of 99%. From these spectrograms, we identified the HR time trace using MATLAB’s “tfridge” function. Finally, ${\mathrm{SpO}}_2$ time traces were calculated by ${\mathrm{SpO}}_2=\mathrm{\Delta }[\mathrm{HbO}](\mathrm{HR})/(\mathrm{\Delta }[\mathrm{HbO}](\mathrm{HR})+\mathrm{\Delta }[\mathrm{Hb}](\mathrm{HR}))$, and a 5-s moving average was applied to smooth the time trace.

For the self-calibrated algorithm, we first extracted the ΔOD time trace at the HR with the spectrogram and “tfridge” function. We then input $\mathrm{\Delta }\mathrm{OD}$ at each time point for two wavelengths (760 and 840 nm) into Eq. (3) to calculate the measured $⟨L⟩$ ratio. The same ${\mu }_a$ (as a function of ${\mathrm{SpO}}_2$) and ${\mu }_s^{\prime }$ used in Sec. 2.2 were employed to calculate the analytical $⟨L⟩$ ratio. Finally, ${\mathrm{SpO}}_2$ at each time point was derived by minimizing the RSS $({\mathrm{\Sigma }}_{\lambda }{[{⟨\hat{L}⟩}_{\text{measured}}^{\lambda }-{⟨\hat{L}⟩}_{\text{analytical}}^{\lambda }]}^2)$ between the measured and analytical $⟨L⟩$ ratio as described in Sec. 2.1.

After obtaining the ${\mathrm{SpO}}_2$ time traces from both the self-calibrated algorithm and the conventional method, we compared them to the ${\mathrm{SaO}}_2$ measurements, which were regarded as the ground truth, from the blood gas analyzer. To account for the uncertainties in the recorded blood draw timings, we averaged the ${\mathrm{SpO}}_2$ around $\pm 2\mathrm{s}$ of the recorded timepoints and then used these values to compare to the ${\mathrm{SaO}}_2$. In our analysis, we first plotted ${\mathrm{SpO}}_2$ against ${\mathrm{SaO}}_2$ for both methods. This was complemented with Bland–Altman analysis to demonstrate the accuracy of ${\mathrm{SpO}}_2$ extraction for each method. Furthermore, we divided our data based on ${\mathrm{SaO}}_2$ levels—high ($>90\%$) and low ($<90\%$). For each category, we assessed the ${\mathrm{SpO}}_2$ extraction performance by calculating the absolute difference between ${\mathrm{SpO}}_2$ and ${\mathrm{SaO}}_2$.

As shown in Fig. 2, ${\mathrm{SaO}}_2$ was extracted at time points A, B, and C at 15 m dives and at E and F for 42 m dives. All 9 subjects completed the 42 m dive with ideally $9\times 2=18$ data points expected from two blood gas measurements per dive. However, two subjects had unreliable data due to very noisy signals, likely from probe movement under the swimming cap. This resulted in a reduction to 14 data points. Additionally, two data points had missing information in the measurement log, further reducing the count to 12 data points for the 42 m dives. 7 out of the 9 subjects completed the 15 m dive, with ideally $7\times 3=21$ data points expected from three blood gas measurements per dive. Noisy signals in three measurements reduced this to 18 data points. One missing entry in the measurement log further reduced the final count to 11 data points for the 15 m dives. In summary, we have 12 data points from the 42 m dives and 11 from the 15 m dives, totaling 23 data points for the ${\mathrm{SaO}}_2$ and ${\mathrm{SpO}}_2$ comparison. When categorized by ${\mathrm{SaO}}_2$ levels, we had 10 data points for ${\mathrm{SaO}}_2<90\%$ and 13 for ${\mathrm{SaO}}_2>90\%$.

3.1.

Validation of the Self-Calibrated Algorithm with Simulated Data

In Monte-Carlo simulation with MCX Lab, we simulated changes in ${\mu }_a$ corresponding to the systolic and diastolic phases of cardiac pulsation. The simulated $\mathrm{\Delta }\mathrm{OD}$ ranging from ${\mathrm{SaO}}_2=20\%$ to 100% is shown in Fig. 3. As expected, the spectral shape of $\mathrm{\Delta }\mathrm{OD}$ changes with ${\mathrm{SaO}}_2$ levels.

Fig. 3

Simulated $\mathrm{\Delta }\mathrm{OD}$ versus wavelength at ${\mathrm{SaO}}_2=20\%$ to 100%.

Taking the simulated $\mathrm{\Delta }\mathrm{OD}$ at ${\mathrm{SaO}}_2=60\%$ as an example, we applied both our algorithm (optimizing for $⟨L⟩$ ratio changes) and the conventional method (with constant $⟨L⟩$ ratio) as described previously. For our algorithm, Figs. 4(a)–4(c) represent input ${\mathrm{SpO}}_2$ at 20%, 60%, and 100%, respectively. The spectral shapes of analytical and measured $⟨L⟩$ ratio closely overlap when ${\mathrm{SpO}}_2$ reaches ${\mathrm{SaO}}_2=60\%$. By iterating ${\mathrm{SpO}}_2$ $\mathrm{values}$ from 0% to 100% in the algorithm, we plotted the RSS (${\mathrm{\Sigma }}_{\lambda }{[{⟨\hat{L}⟩}_{\text{measured}}(\lambda )-{⟨\hat{L}⟩}_{\text{analytical}}(\lambda )]}^2$) between the analytical and measured $⟨L⟩$ ratio. The RSS versus ${\mathrm{SpO}}_2$ in Fig. 4(d) shows a U-shape with its minimum error at the expected ${\mathrm{SpO}}_2=60\%$. In Fig. 4(e), we show all the extracted ${\mathrm{SpO}}_2$ values from our algorithm (red squares) and those from the conventional method—Eq. (6) with constant $⟨L⟩$ ratios (${⟨L⟩}^{840\mathrm{nm}}/{⟨L⟩}^{760\mathrm{nm}}$) of 0.61 (yellow triangles) and 0.87 (blue circles).

Fig. 4

(a)–(c) Comparison between analytical (solid purple) and measured (dotted green) $⟨L⟩$ ratio at ${\mathrm{SpO}}_2=20\%$, 60%, and 100%, respectively. (d) RSS between analytical and measured $⟨L⟩$ ratio at ${\mathrm{SpO}}_2=0\%$ to 100%. (e) Comparison between ${\mathrm{SpO}}_2$ calculated by the self-calibrated algorithm (red square) and the conventional method with constant $⟨L⟩$ ratio = 0.61 (yellow triangle) and 0.87 (blue circle).

In Fig. 4(e), we observe that the ${\mathrm{SpO}}_2$ values derived from our algorithm accurately follow the unity line. ${\mathrm{SpO}}_2$ values calculated by the conventional method, however, depends on the manually set constant $⟨L⟩$ ratios. When a ratio of 0.61 (yellow triangle) was used, the calculated ${\mathrm{SpO}}_2$ values are relatively close to the unity line at higher ${\mathrm{SaO}}_2$ levels (80% to 100%). However, the accuracy diminishes for lower saturation levels. In contrast, when using a ratio of 0.87 (blue circles) taken from the literature, the extracted ${\mathrm{SpO}}_2$ values align more closely to the unity line at lower ${\mathrm{SaO}}_2$ levels, particularly around 20%.

It is curial to clarify that the ratio of 0.87, taken from the literature, does not generally ensure enhanced ${\mathrm{SpO}}_2$ extraction accuracy at lower saturation levels. This specific value was derived from measurements on the frontal human head, which consists of various tissue layers that combine artery, venous, and capillary components.¹⁹ In contrast, our simulations utilized a semi-infinite rectangular slab volume, a geometry distinctly different from the human head’s structure. As we will demonstrate in the next section, the ratio of 0.87 is quite adequate for measurement taken from the human forehead, resulting in good ${\mathrm{SpO}}_2$ extraction accuracy at higher saturation levels. These findings thus highlight the importance of carefully tailoring the $⟨L⟩$ ratio within the mBLL to the specific context and application.

3.2.

${\mathrm{SpO}}_2$ Extraction Using Two Versus Eight Wavelengths in the Presence of Noise in $\mathrm{\Delta }\mathrm{OD}$

In Sec. 2.3, we described the process of adding Gaussian white noise to simulated noise-free $\mathrm{\Delta }\mathrm{OD}$ across eight wavelengths. We then processed these noisy $\mathrm{\Delta }\mathrm{OD}$ values using either two (760 and 840 nm) or all eight wavelengths with the self-calibrated algorithm to compute ${\mathrm{SpO}}_2$. Figure 5(a) shows an example comparison between the clean $\mathrm{\Delta }\mathrm{OD}$ at ${\mathrm{SaO}}_2=100\%$ (yellow squares) and three noisy $\mathrm{\Delta }\mathrm{OD}$ examples (green circles).

Fig. 5

(a) Clean $\mathrm{\Delta }\mathrm{OD}$ at ${\mathrm{SaO}}_2=100\%$ (yellow square) and three examples of noisy $\mathrm{\Delta }\mathrm{OD}$ (green circle). (b) ${\mathrm{SpO}}_2$ extraction results from the self-calibrated algorithm using two (blue circle) versus eight (red triangle) wavelengths.

For simulated $\mathrm{\Delta }\mathrm{OD}$ at each ${\mathrm{SaO}}_2$ value (20%, 40%, 60%, 80%, and 100%), we processed 100 distinct noisy $\mathrm{\Delta }\mathrm{OD}$ sets through the self-calibrated algorithm, leading to 100 ${\mathrm{SpO}}_2$ estimations per ${\mathrm{SaO}}_2$ level. Figure 5(b) shows the mean and STD of these ${\mathrm{SpO}}_2$ values. To enhance visual clarity, we slightly offset the datasets representing results from two and eight wavelengths on the $x$ axis. We can see that the ${\mathrm{SpO}}_2$ results, both in terms of mean and STD, from two (blue circles) and eight (red triangles) wavelengths are notably similar and align closely with the unity line. This similarity indicates that with the self-calibrated algorithm, accurate ${\mathrm{SaO}}_2$ estimation is feasible using just two wavelengths, as will be demonstrated in the subsequent human freediving experiment.

3.3.

Application of the Self-Calibrated Algorithm and Conventional Method to Freediver Data

Following the conventional approach to calculate ${\mathrm{SpO}}_2$, we took the measured data at $r=3\mathrm{cm}$ and calculated the $\mathrm{\Delta }[\mathrm{HbO}]$ and $\mathrm{\Delta }[\mathrm{Hb}]$ time traces using mBLL as described in Sec. 2.4, with the assumption of a constant $⟨L⟩$ ratio (${⟨L⟩}^{840\mathrm{nm}}/{⟨L⟩}^{760\mathrm{nm}}=0.87$). An example result from a 42 m dive is presented in Figs. 6(a) and 6(b), which show example spectrograms for $\mathrm{\Delta }[\mathrm{HbO}]$ and $\mathrm{\Delta }[\mathrm{Hb}]$ with their identified HR (red solid line). ${\mathrm{SpO}}_2$ time trace—${\mathrm{SpO}}_2=\mathrm{\Delta }[\mathrm{HbO}](\mathrm{HR})/(\mathrm{\Delta }[\mathrm{HbO}](\mathrm{HR})+\mathrm{\Delta }[\mathrm{Hb}](\mathrm{HR}))$—is depicted in Fig. 7. Comparing the ${\mathrm{SpO}}_2$ time trace (green dotted line) to the blood gas measurement (red circles), we observed that the calculated ${\mathrm{SpO}}_2$ of 99.6% at 42 m (“bottom”) is consistent with the measured ${\mathrm{SaO}}_2$ of 99%. However, at the end of the 42 m dive, near the surface, the calculated ${\mathrm{SpO}}_2$ of 87.8% largely overestimates the ground truth ${\mathrm{SaO}}_2$ of 61% (a 27.8% difference).

Fig. 6

(a), (b) Spectrogram for $\mathrm{\Delta }[\mathrm{HbO}]$ and $\mathrm{\Delta }[\mathrm{Hb}]$ with HR time trace (red line).

For the self-calibrated algorithm, we extracted $\mathrm{\Delta }\mathrm{OD}$ at the HR from the spectrograms for each wavelength (760 and 840 nm) and input them to the algorithm using the same data. The purple solid line in Fig. 7 shows the ${\mathrm{SpO}}_2$ time traces calculated by the algorithm. The ${\mathrm{SpO}}_2$ calculations are consistent with the ground-truth ${\mathrm{SaO}}_2$ measured from the blood gas (red circles) not only at 42 m bottom, where ${\mathrm{SpO}}_2$ is 100% and ${\mathrm{SaO}}_2$ is 99%, but also at the surface, where ${\mathrm{SpO}}_2$ is 63.2% and ${\mathrm{SaO}}_2$ is 61% (a 2.2% difference).

Fig. 7

Comparison between ${\mathrm{SpO}}_2$ extracted from self-calibrated algorithm (solid purple line) and the conventional method (dotted green line) to ground truth ${\mathrm{SaO}}_2$ from blood gas (red circles).

In Figs. 8(a) and 8(b), we plotted ${\mathrm{SpO}}_2$ against ${\mathrm{SaO}}_2$ for all the data. For the self-calibrated algorithm, the ${\mathrm{SpO}}_2$ values predominantly fall in line with the unity line (represented by the black dashed line). In contrast, the conventional method often overestimates ${\mathrm{SaO}}_2$, particularly at lower saturation levels.

Fig. 8

${\mathrm{SpO}}_2$ versus ${\mathrm{SaO}}_2$ for all data points: (a) conventional method and (b) self-calibrated algorithm. Data from 15 and 42 m dives are marked by circles and squares, respectively. (c), (d) Bland–Altman plots for (a) and (b), respectively.

This difference becomes more evident in the Bland–Altman plots of Figs. 8(c) and 8(d). The self-calibrated algorithm exhibits a modest average bias of 3.29% between ${\mathrm{SpO}}_2$ and ${\mathrm{SaO}}_2$, with most of the data points clustering near this central bias line. Additionally, the limits of agreement, capturing 95% of the differences between ${\mathrm{SpO}}_2$ and ${\mathrm{SaO}}_2$, are narrow ($-8.14\%$ to 14.72%). Importantly, the residuals show no systematic biases, confirming the reliability of this method.

In contrast, the conventional method presents a substantially higher average bias of 11.16% between ${\mathrm{SpO}}_2$ and ${\mathrm{SaO}}_2$. The limits of agreement for this method are also broader ($-15.92\%$ to 38.24%), signifying greater variability. Moreover, a noticeable trend in the residuals reveals an increased bias at lower saturation levels.

Taken together, these findings demonstrate the precision and consistency of our self-calibrated algorithm, indicating its superior performance in estimating ${\mathrm{SpO}}_2$ across various saturation levels compared to the conventional method.

To further evaluate the robustness of the two methods under different saturation levels, we divided our dataset based on ${\mathrm{SaO}}_2$ levels into two categories: high ($>90\%$) and low ($<90\%$). The metric used to quantify performance was the absolute difference between ${\mathrm{SpO}}_2$ and ${\mathrm{SaO}}_2$ ($|{\mathrm{SpO}}_2-{\mathrm{SaO}}_2|$). Figure 9 illustrates the mean and standard error of these differences for each method.

Fig. 9

Absolute difference between the extracted ${\mathrm{SpO}}_2$ and ground-truth ${\mathrm{SaO}}_2$ at low ($<90\%$) and high ($>90\%$) saturation levels. *$p<0.05$.

For high ${\mathrm{SaO}}_2$ levels ($>90\%$), both methods perform similarly well, with differences of 0.91% for the conventional method and a slightly higher 1.05% for the self-calibrated algorithm. However, the distinction between the two methods becomes pronounced at lower ${\mathrm{SaO}}_2$ levels ($<90\%$): the self-calibrated algorithm shows a discrepancy of only 8.37%, whereas the conventional method has a 24.7% difference. The statistical significance of this variation was further demonstrated by the Wilcoxon rank sum test with the significance level was set to be 0.05.

These results emphasize the superior accuracy of the self-calibrated algorithm, particularly when the ${\mathrm{SaO}}_2$ is low, when compared to the conventional approach.