Statistical Description of SaO 2 –SpO 2 Relationship for Model of Oxygenation in Premature Infants

: A pulse oximeter model linking arterial (SaO 2 ) and peripheral (SpO 2 ) oxygen saturation is the terminal part of a mathematical model of neonatal oxygen transport. Previous studies have conﬁrmed the overestimation of oxygen saturation measured by pulse oximetry in neonates compared to arterial oxygen saturation and the large variability of measured values over time caused by measurement inaccuracies. This work aimed to determine the SpO 2 measurement noise that affects the biased SpO 2 value at each time point and integrate the noise description with the systematic bias between SaO 2 and SpO 2 . The SaO 2 –SpO 2 bias was based on previously published clinical data from pathological patients younger than 60 days requiring ventilatory support. The statistical properties of the random SpO 2 measurement noise were estimated from the SpO 2 continuous recordings of 21 pathological and 21 physiological neonates. The result of the work is a comprehensive characterization of the properties of a pulse oximeter model describing the transfer of the input SaO 2 value to the output SpO 2 value, including the bias and noise typical for the bedside monitoring of neonates. These results will help to improve a computer model of neonatal oxygen transport.


Introduction
The advantages of closed-loop control of oxygenation in neonates compared to manual control have been documented in recently published clinical trials. During automatic control, arterial blood oxygen saturation remains within the desired safe range for significantly longer periods [1][2][3][4][5][6]. In the last ten years, many articles have introduced closed-loop control algorithms of oxygenation in neonates, but a complex clinical study to compare the effectiveness of those various algorithms is still missing [6]. Clinical tests of the new oxygenation control algorithms of neonates bring safety and ethical risks, but a mathematical model of oxygenation in neonates can allow for the in silico simulation of oxygenation and preliminary comparison of the control algorithms [7][8][9][10].
A general scheme of a complex mathematical model of automatic oxygenation of a neonate is shown in Figure 1. The terminal part of the model is the pulse oximeter module. In the absence of arterial blood gas measurement, both the automatic and manual control of oxygenation usually depend on pulse oximetry [1,7,11]. Many studies showed inaccuracies in the pulse oximetry measurement in children and premature infants. Peripheral oxygen saturation measured by pulse oximetry (SpO 2 ) typically overestimates arterial oxygen saturation (SaO 2 ), especially at the lower values of SaO 2 that are common in critically ill premature infants and children. Bohnhorst et al. noted the presence of the SaO 2 -SpO 2 bias in Reference [12], where the authors determined the sensitivity and specificity of different pulse oximeters in the detection of hyperoxemia in 56 infants. Gerstmann et al. [13] found that SpO 2 generally overestimated SaO 2 at low levels and that the spread in the data increased with lower SaO 2 . The results were confirmed a few years later by Rosychuk et al. [14] in neonates with an average weight of 1 kg. The authors used a new generation of pulse oximeters with lower susceptibility to motion artifacts. Harris et al. [15] evaluated two sensors with improved accuracy in children with SaO 2 less than 85%. This study revealed statistically significant increasing bias and variability of SpO 2 for decreasing levels of SaO 2 and the authors concluded that pulse oximetry alone should not be relied upon for clinical decision-making when saturation is below 85%. In another study, Harris et al. [16] confirmed their previous findings and set the average bias at 4.0% and 7.4%, respectively, depending on the type of pulse oximeter. A similar average bias of 5.4% across all saturation levels was found by Murphy et al. [17] in a study of 89 patients with critical congenital heart disease. Ross et al. [11] described a significant variation in the SpO 2 accuracy. The authors computed the SaO 2 -SpO 2 bias (0.2-6.6%) and precision (3.4-6.6%), expressed as the standard deviation, for seven intervals in the range of 65-97% SpO 2 in 225 mechanically ventilated children with cyanotic congenital heart disease or acute hypoxemic respiratory failure. Very similar results were provided in the study of Bachman et al. [18] by evaluating 25 032 SaO 2 -SpO 2 measurements from 1007 critically ill neonates, or in the study of Griksaitis et al. [19] in 25 children with cyanotic congenital heart disease.
Electronics 2022, 11, x FOR PEER REVIEW 2 of 14 bias in Reference [12], where the authors determined the sensitivity and specificity of different pulse oximeters in the detection of hyperoxemia in 56 infants. Gerstmann et al. [13] found that SpO2 generally overestimated SaO2 at low levels and that the spread in the data increased with lower SaO2. The results were confirmed a few years later by Rosychuk et al. [14] in neonates with an average weight of 1 kg. The authors used a new generation of pulse oximeters with lower susceptibility to motion artifacts. Harris et al. [15] evaluated two sensors with improved accuracy in children with SaO2 less than 85%. This study revealed statistically significant increasing bias and variability of SpO2 for decreasing levels of SaO2 and the authors concluded that pulse oximetry alone should not be relied upon for clinical decision-making when saturation is below 85%. In another study, Harris et al. [16] confirmed their previous findings and set the average bias at 4.0% and 7.4%, respectively, depending on the type of pulse oximeter. A similar average bias of 5.4% across all saturation levels was found by Murphy et al. [17] in a study of 89 patients with critical congenital heart disease. Ross et al. [11] described a significant variation in the SpO2 accuracy. The authors computed the SaO2-SpO2 bias (0.2-6.6%) and precision (3.4-6.6%), expressed as the standard deviation, for seven intervals in the range of 65-97% SpO2 in 225 mechanically ventilated children with cyanotic congenital heart disease or acute hypoxemic respiratory failure. Very similar results were provided in the study of Bachman et al. [18] by evaluating 25 032 SaO2-SpO2 measurements from 1007 critically ill neonates, or in the study of Griksaitis et al. [19] in 25 children with cyanotic congenital heart disease. Several studies [11,[13][14][15]18] pointed out not only the bias between SaO2 and SpO2 (SaO2-SpO2 bias) but also quite a large variability in the bias. This variability can be caused by many factors that affect the accuracy of the pulse oximeter measurement. In the multicenter study [11], the variability in the bias could have been caused by institutional variation, for example, by using pulse oximeters from different manufacturers. A location of probe placement can influence measured SpO2 values [11,20,21]; however, Harris et al. [15] did not show a significant difference between standard and nonstandard probe location. The studies [11,18] involved premature infants who could have had increased levels of fetal hemoglobin (HbF) that may alter the SaO2-SpO2 bias [11,14,20,22]. In addition to HbF, other hemoglobin derivates, such as carboxyhemoglobin or methemoglobin, may also affect the accuracy of the measurement [21,22].
Besides the interpersonal variability, experimental data also contain intrapersonal variability in time. A low pulsatile signal (low perfusion), high noise (bright light, electromagnetic interference, or motion), or a combination of these factors can cause a low signalto-noise ratio, leading to inaccurate pulse oximeter readings [21,23,24]. Even when SaO2 remains practically unchanged, the SpO2 values presented by the pulse oximeter change in time, and in the case of abrupt motion of a neonate, they may even be falsely interpreted as rapid desaturations [23].
Recent studies dealing with noise and its filtering in relation to pulse oximetry focused on the photoplethysmography curve used by pulse oximeters. Fine et al. [25] summarized several ways of detecting noise or motion artifacts, which included using the low- Several studies [11,[13][14][15]18] pointed out not only the bias between SaO 2 and SpO 2 (SaO 2 -SpO 2 bias) but also quite a large variability in the bias. This variability can be caused by many factors that affect the accuracy of the pulse oximeter measurement. In the multicenter study [11], the variability in the bias could have been caused by institutional variation, for example, by using pulse oximeters from different manufacturers. A location of probe placement can influence measured SpO 2 values [11,20,21]; however, Harris et al. [15] did not show a significant difference between standard and nonstandard probe location. The studies [11,18] involved premature infants who could have had increased levels of fetal hemoglobin (HbF) that may alter the SaO 2 -SpO 2 bias [11,14,20,22]. In addition to HbF, other hemoglobin derivates, such as carboxyhemoglobin or methemoglobin, may also affect the accuracy of the measurement [21,22].
Besides the interpersonal variability, experimental data also contain intrapersonal variability in time. A low pulsatile signal (low perfusion), high noise (bright light, electromagnetic interference, or motion), or a combination of these factors can cause a low signal-to-noise ratio, leading to inaccurate pulse oximeter readings [21,23,24]. Even when SaO 2 remains practically unchanged, the SpO 2 values presented by the pulse oximeter change in time, and in the case of abrupt motion of a neonate, they may even be falsely interpreted as rapid desaturations [23].
Recent studies dealing with noise and its filtering in relation to pulse oximetry focused on the photoplethysmography curve used by pulse oximeters. Fine et al. [25] summarized several ways of detecting noise or motion artifacts, which included using the low-signalquality index, filters with cross-correlation, analyzing the morphology of the signal, or higher-order statistics in both the frequency and time domain. Lee et al. [26] proposed a motion artifact reduction algorithm, using independent component analysis. Other studies [27,28] assessed the quality of SaO 2 estimation from photoplethysmography and detected poor-quality segments by methods of machine learning. However, the considered global model of neonatal oxygenation [8] does not consider the pulsatility in the cardiovascular system and, thus, the photoplethysmography curve. Instead, the pulse oximeter module is treated as a black box that converts a continuous SaO 2 signal to a stream of trend SpO 2 values, reported every 2 s, that is used by automatic closed-loop control algorithms to adjust the fraction of inspired oxygen in the model input. A pulse oximeter model transforming SaO 2 to the observed SpO 2 value was used by Morozoff et al. [29]. The model adds two types of noise to the SaO 2 signal: the sensor noise modeled as the white noise and motion artifacts produced by a pulse generator. However, this approach is not based on real data from clinical practice.
The aim of this work was to statistically describe the SpO 2 measurement noise characteristic of continuous time recording of SpO 2 based on the evaluation of available clinical data and to combine the noise description with the systematic bias between SaO 2 and SpO 2 into a plausible mathematical model of the pulse oximeter output signal. The results of the work were intended for integration into an overall computer model of premature infant oxygenation.

Materials and Methods
The input of the pulse oximeter module of the overall oxygenation model is the continuous SaO 2 signal. The output signal of the pulse oximeter module consists of two principal components: the SaO 2 -SpO 2 bias and the SpO 2 measurement noise. The SaO 2 -SpO 2 bias describes a typical deviation of the SpO 2 measurement as a function of SaO 2 value. The SpO 2 measurement noise is a random process that changes the biased SpO 2 value at each time point. Data for both the components of the pulse oximeter model were processed in Matlab R2021a (MathWorks, Natick, MA, USA).

SaO 2 -SpO 2 Bias
The SaO 2 -SpO 2 bias function was determined in our previous study [30] based on clinical data acquired by Ross et al. [11]. We used the part of the data that included mechanically ventilated hypoxemic premature and term infants aged between the 37th week of gestation and the 60th day after delivery. We evaluated 1423 SaO 2 -SpO 2 data pairs. The SaO 2 values were measured by CO-oximetry, and the SpO 2 values were measured by Masimo or Nellcor pulse oximeters at the same time the arterial blood sample was taken. We calculated the bias in three neighboring intervals. For SaO 2 below 70%, the SpO 2 bias was kept constant and equal to the 7.66%, which was the SpO 2 bias at SaO 2 = 70%. For SaO 2 in the range of 70-96%, the median of measured SpO 2 values was calculated for each unit value of SaO 2 and a third-order polynomial was fitted through the medians. For SaO 2 above 96%, the bias was set as zero, that is, SpO 2 = SaO 2 . The SaO 2 -SpO 2 bias function is displayed in Figure 2 and mathematically expressed by the following equations:

SpO2 Measurement Noise
We based the model of SpO2 measurement noise on continuous neonatal SpO2 recordings from the General University Hospital in Prague. The data were collected during routine clinical care, based on a standard informed consent to hospitalization and to collection of anonymous observational data for research and educational purposes that was

SpO 2 Measurement Noise
We based the model of SpO 2 measurement noise on continuous neonatal SpO 2 recordings from the General University Hospital in Prague. The data were collected during routine clinical care, based on a standard informed consent to hospitalization and to collection of anonymous observational data for research and educational purposes that was signed by a legal representative of a neonate. Anonymized observational data were provided to the authors of this study.
In total, we evaluated SpO 2 recordings from 42 patients divided into two categories: physiological neonates and pathological neonates. Twenty-one healthy physiological patients were term infants without any known pathology who were measured during the first hours after delivery. The recording time for each physiological patient was 3-21 h. Twenty-one pathological patients were premature (born before 28th week of gestation), with various diagnosed pathologies requiring oxygen support, most commonly bronchopulmonary dysplasia. The recording time for each pathological patient was 10-95 h. SpO 2 values were measured by Masimo Rad-97 pulse oximeter (Masimo Corporation, Irvine, CA, USA), with the sampling time set to 2 s and the averaging time set to 8 s.

Data Processing
The SpO 2 measurement noise was considered as a random process that affects the biased SpO 2 values at each time point. A noise-free SpO 2 value (SpO clear 2 ) was estimated for each measured SpO 2 value (SpO meas 2 ), and the difference of these parameters was considered as the noise component. The procedure of the noise estimation is shown in the flowchart in Figure 3 and is described in detail below.
Step 1: The unstable parts of the measured SpO 2 signal (SpO meas 2 ), which were defined based on the study by Wellington et al. [31], were excluded from further processing. All SpO 2 values that met at least one of the following two criteria were excluded: (1) The SpO 2 value was measured at the time when the low-signal-quality alarm was triggered. The lowsignal-quality alarm is a pulse oximeter indicator of potentially erroneous data; however, it does not guarantee the perfect quality of all other parts of the SpO 2 signal. (2) The SpO 2 value was the middle sample of a 30 s moving window in which SpO 2 changed by more than 10%. An example of the original raw SpO meas 2 signal with identified stable and unstable parts is shown in  Outliers larger than ±6% SpO 2 were excluded. We determined the SpO 2 measurement noise for two patient categories, physiological neonates and pathological neonates. Cumulative distribution functions were constructed for all SaO 2 values, for SaO 2 ≤ 96%, and for SaO 2 ≥ 97%. Step 1: The unstable parts of the measured SpO2 signal (SpO ), which were defined based on the study by Wellington et al. [31], were excluded from further processing. All SpO2 values that met at least one of the following two criteria were excluded: (1) The SpO2 value was measured at the time when the low-signal-quality alarm was triggered. The low-signal-quality alarm is a pulse oximeter indicator of potentially erroneous data; however, it does not guarantee the perfect quality of all other parts of the SpO2 signal. (2) The SpO2 value was the middle sample of a 30 s moving window in which SpO2 changed by more than 10%. An example of the original raw SpO signal with identified stable and unstable parts is shown in Figure 4. Step 2: All null or unavailable data points in the SpO signal were replaced by the nearest preceding valid SpO2 value. The replaced values were not included in calculations of the SpO2 measurement noise. The aim of replacing the null and unavailable values was to avoid abrupt transitions to zero values in SpO , while filtering the signal in the next step.
Step 3: A median filter was applied to the preprocessed SpO2 signal. This operation resulted in SpO values that reflect SaO2 values without the presence of any measurement noise.
Step 4: Each SpO datapoint was converted to an SaO2 value, using an inverse function to the SaO2-SpO2 bias function. Each calculated SaO2 value was then paired with the respective SpO value of the raw data waveform. Steps 3 and 4 were repeated in case of the pathological patient data to find the optimal parameter values of the median filter, as described in the next section.
Step 5: The differences, SpO − SpO , of all valid datapoints (i.e., all values that were not excluded in Step 1 or 2) pooled together from all patients were used for the statistical model of the SpO2 measurement noise in the form of a cumulative distribution function. Outliers larger than ±6% SpO2 were excluded. We determined the SpO2 measurement  . An example of the original raw SpO2 data with stable parts (blue) and unstable parts (magenta and cyan). The stable parts were further processed to determine the noise, and the unstable parts were excluded due to the presence of the low-signal-quality alarm (red) or due to an abrupt change in SpO2 (cyan) within a 30 s moving window.

Median Filter Window Size
At Step 3 of the SpO2 measurement noise data processing, we applied the median filter to the SpO2 signal to obtain noise-free SpO values. The optimal median filter window size was determined from the comparison of the distribution of the SpO2 recordings (for each SaO2 unit) from 21 pathological patients and the distribution of the data Figure 4. An example of the original raw SpO 2 data with stable parts (blue) and unstable parts (magenta and cyan). The stable parts were further processed to determine the noise, and the unstable parts were excluded due to the presence of the low-signal-quality alarm (red) or due to an abrupt change in SpO 2 (cyan) within a 30 s moving window.

Median Filter Window Size
At Step 3 of the SpO 2 measurement noise data processing, we applied the median filter to the SpO 2 signal to obtain noise-free SpO clear 2 values. The optimal median filter window size was determined from the comparison of the distribution of the SpO 2 recordings (for each SaO 2 unit) from 21 pathological patients and the distribution of the data acquired by Ross et al. [11]. An assumption for comparing the two datasets was the similarity of their noise characteristics, where both datasets were of pathological patients requiring ventilatory support who were less than 60 days old. Figure 5 compares the resulting SaO 2 -SpO 2 data (generated based on the outcome of Step 4) with the data of Ross et al. [11]. For each SaO 2 unit, a histogram of the SpO 2 distribution of the calculated data was compared with the respective histogram of the data of Ross et al. [11]. The filter parameters that resulted in the best overlaps of these histograms were used to construct the cumulative distribution functions in Step 5. . An example of the original raw SpO2 data with stable parts (blue) and unstable parts (magenta and cyan). The stable parts were further processed to determine the noise, and the unstable parts were excluded due to the presence of the low-signal-quality alarm (red) or due to an abrupt change in SpO2 (cyan) within a 30 s moving window.

Median Filter Window Size
At Step 3 of the SpO2 measurement noise data processing, we applied the median filter to the SpO2 signal to obtain noise-free SpO values. The optimal median filter window size was determined from the comparison of the distribution of the SpO2 recordings (for each SaO2 unit) from 21 pathological patients and the distribution of the data acquired by Ross et al. [11]. An assumption for comparing the two datasets was the similarity of their noise characteristics, where both datasets were of pathological patients requiring ventilatory support who were less than 60 days old. Figure 5 compares the resulting SaO2-SpO2 data (generated based on the outcome of Step 4) with the data of Ross et al. [11]. For each SaO2 unit, a histogram of the SpO2 distribution of the calculated data was compared with the respective histogram of the data of Ross et al. [11]. The filter parameters that resulted in the best overlaps of these histograms were used to construct the cumulative distribution functions in Step 5. Figure 5. Comparison of SaO2-SpO2 scatterplots from the data for the noise model estimate and the data acquired by Ross et al. [11]. The diameter of a marker for each SaO2-SpO2 pair is proportional to the frequency of the value occurrence. The optimal median filter window size was selected from the range 1-1000, as illustrated in Figure 6, so that it minimizes the cost function: where h i is the value of an i-th bin of a normalized histogram of the SpO 2 distribution, SpO Ross 2 refers to the data of Ross et al. [11], and m is the number of SaO 2 units for which both the SpO meas 2 data and SpO Ross 2 data are available for a particular filter window size. The J function evaluates the extent to which normalized histograms of the SpO 2 distributions overlap at each SaO 2 unit and was based on the histogram intersection measure [32]. If two histograms overlap perfectly, the value of the histogram intersection measure at the particular SaO 2 unit is 0. On the other hand, if the histograms do not overlap at all, the value is 1. An example of partially overlapping histograms for a single window size of the median filter is presented in Figure 7. The optimal median filter window size (WDW) was set to 227 samples, which corresponds to approximately 7.5 min of pulse oximeter output samples with the sampling time of 2 s. If two histograms overlap perfectly, the value of the histogram intersection measure at the particular SaO2 unit is 0. On the other hand, if the histograms do not overlap at all, the value is 1. An example of partially overlapping histograms for a single window size of the median filter is presented in Figure 7. The optimal median filter window size (WDW) was set to 227 samples, which corresponds to approximately 7.5 min of pulse oximeter output samples with the sampling time of 2 s.  where ℎ is the value of an i-th bin of a normalized histogram of the SpO2 distribution, SpO refers to the data of Ross et al. [11], and m is the number of SaO2 units for which both the SpO data and SpO data are available for a particular filter window size. The J function evaluates the extent to which normalized histograms of the SpO2 distributions overlap at each SaO2 unit and was based on the histogram intersection measure [32]. If two histograms overlap perfectly, the value of the histogram intersection measure at the particular SaO2 unit is 0. On the other hand, if the histograms do not overlap at all, the value is 1. An example of partially overlapping histograms for a single window size of the median filter is presented in Figure 7. The optimal median filter window size (WDW) was set to 227 samples, which corresponds to approximately 7.5 min of pulse oximeter output samples with the sampling time of 2 s.   [11]. The magenta color represents the histogram intersection.

Integration of SaO 2 -SpO 2 Bias with SpO 2 Measurement Noise
The pulse oximeter model, the terminal part of a complex mathematical model of neonatal oxygenation, integrates the SaO 2 -SpO 2 bias with the SpO 2 measurement noise. The input of the pulse oximeter model is the SaO 2 value. The input, sampled every 2 s, is converted to a noise-free SpO 2 value, using the SaO 2 -SpO 2 bias function. The output of the pulse oximeter model is then obtained by adding the SpO 2 measurement noise generated randomly, following the probabilities specified by the cumulative distribution function to the noise-free SpO 2 value.

Results
The model of the output of the pulse oximeter consists of two parts: the SaO 2 -SpO 2 bias and the SpO 2 measurement noise. The SaO 2 -SpO 2 bias function was determined in our previous study [30]. The SpO 2 measurement noise was estimated for two different groups of patients, physiological neonates and pathological neonates. Figure 8 depicts the resulting normalized histograms of the SpO 2 measurement noise for both the categories for all SaO 2 values and also separately for SaO 2 ≤ 96% and for SaO 2 ≥ 97%. The statistical properties of the SpO 2 measurement noise are expressed by the cumulative distribution  Table A1.
The model of the output of the pulse oximeter consists of two parts: the SaO2-S bias and the SpO2 measurement noise. The SaO2-SpO2 bias function was determined our previous study [30]. The SpO2 measurement noise was estimated for two diffe groups of patients, physiological neonates and pathological neonates. Figure 8 depicts resulting normalized histograms of the SpO2 measurement noise for both the catego for all SaO2 values and also separately for SaO2 ≤ 96% and for SaO2 ≥ 97%. The statist properties of the SpO2 measurement noise are expressed by the cumulative distribu function presented in Figure 9. The exact numerical values of the cumulative distribu function are provided in Appendix A in Table A1.  In the pulse oximeter module, the random noise generated according to the char teristics presented in Figures 8 and 9 is added to the SpO2 value calculated by using SaO2-SpO2 bias function. Figure 10 shows the resulting SaO2-SpO2 scatterplot of the o put of the pulse oximeter module. The figure displays the distribution of SpO2 values g erated for each SaO2 value, including the frequency of occurrence of each SaO2-SpO2 p In the pulse oximeter module, the random noise generated according to the characteristics presented in Figures 8 and 9 is added to the SpO 2 value calculated by using the SaO 2 -SpO 2 bias function. Figure 10 shows the resulting SaO 2 -SpO 2 scatterplot of the output of the pulse oximeter module. The figure displays the distribution of SpO 2 values generated for each SaO 2 value, including the frequency of occurrence of each SaO 2 -SpO 2 pair. and (b) pathological patients.
In the pulse oximeter module, the random noise generated according to the characteristics presented in Figures 8 and 9 is added to the SpO2 value calculated by using the SaO2-SpO2 bias function. Figure 10 shows the resulting SaO2-SpO2 scatterplot of the output of the pulse oximeter module. The figure displays the distribution of SpO2 values generated for each SaO2 value, including the frequency of occurrence of each SaO2-SpO2 pair.

Discussion
In this work, we quantified the measurement noise that is characteristic for continuous SpO2 time recording and completed the model of the output of the pulse oximeter typical for premature infants.

Discussion
In this work, we quantified the measurement noise that is characteristic for continuous SpO 2 time recording and completed the model of the output of the pulse oximeter typical for premature infants.
The model of the output of the pulse oximeter consists of two parts: the SaO 2 -SpO 2 bias and the SpO 2 measurement noise. The SaO 2 -SpO 2 bias was determined from the Ross' clinical data [11] in the SaO 2 range of 70-100% in our previous study [30]. The bias values slightly differ between previously published clinical studies [11,14,18]. This may be due to different methods of measuring SaO 2 (co-oximetry vs. blood gas analysis), measuring SpO 2 , or monitoring of the patients with different age and diagnoses. However, the published differences are too small to affect the credibility of the simulated output of the pulse oximeter model and its applicability to the neonatal oxygen transport model. For SaO 2 values less than 70%, the bias was held constant, equal to the value of the bias for 70% SaO 2 (7.66%), due to the lack of clinical data. The constant bias for SaO 2 of less than 70% would be a sufficient approximation for the model of a neonate on oxygen support, because these values are associated with severe hypoxemia and are beyond the target range in which the saturation of ventilated premature infants should be maintained. In addition, pulse oximeter manufacturers do not guarantee the accuracy of pulse oximeter measurements of such low saturation values [33].
In addition to the interpersonal variability of SaO 2 -SpO 2 bias occurring in the studies [11,[13][14][15]18] mentioned above, intrapersonal variability due to low perfusion, motion artifacts, or another noise, such as bright light or electromagnetic interference [21,23,24], also appears in the SpO 2 time recordings of each patient during the bed-side monitoring. In our work, we characterized the intrapersonal variability of SpO 2 time recording as the SpO 2 measurement noise. The noise was determined for two groups, physiological neonates and pathological neonates, based on the clinical data of 21 patients in each group as the difference between the measured SpO 2 values and the estimated SpO 2 values without noise. The noise model was estimated by using a numerical procedure in which the window size of the median filter was varied. The median filter was chosen over the moving average filter because of the frequent sudden but short drops in the SpO 2 signal (perhaps due to moving artifacts) that we wished to remove. The advantage of the median filter is its simplicity and ease of implementation. However, other, more sophisticated denoising methods are available, based, for example, on wavelets [34] or compressed sensing [35]. The specific filter setting determined what would be considered a noise-free SpO 2 signal, thus indirectly generating variations of the noise model. We considered as the most plausible the variant of the noise model that produced our SpO 2 data distribution the most similar to that published by Ross in his study (as expressed by the minimum of the J function). The SpO 2 measurement noise was described by histograms and cumulative distribution functions not only for two categories of neonates but also for two different SaO 2 intervals, SaO 2 ≤ 96% and SaO 2 ≥ 97%. The boundary between the intervals was chosen to be identical to the intervals of the SaO 2 -SpO 2 bias.
At the beginning of noise-signal processing, we excluded the unstable parts of the measured SpO 2 signal. The exclusion of all SpO 2 values measured during the episodes of triggered low-signal-quality alarm corresponds to automatic oxygenation control algorithms that do not consider such SpO 2 values reliable for automatic adjustment of the fraction of inspired oxygen [36]. In addition, all SpO 2 values that were the middle sample of a 30 s moving window in which SpO 2 changed by more than 10% were excluded. This second criterion was introduced because the distinct drops in the SpO 2 signal can be caused not only by artifacts but also by real desaturation, which cannot be distinguished without having the simultaneous data of the patient's movement available. Previous studies have considered motion artifacts as a major factor in inaccurate pulse oximeter readings [21,23], and this is even more influential in the group of preterm and term infants [37,38]. In comparison with adults, more motion, longer periods of motion, and more intensive motion were observed in infants [37]. Fletcher et al. [38] concluded that motion artifact can affect, overall, up to 50% of SpO 2 recorded time, and actually the motion artifact was present 91% of the monitored time during infant wakefulness. During the apnea, preterm infants can desaturate with a rate of 3-8% per second [39], and the study published by Poets and Southall [40] even reported the rate of up to 12.6% per second. Therefore, motion artifact and desaturation may have a similar SpO 2 recording, and motion artifact could be interpreted as the true desaturation and vice versa, or motion artifact can obscure the true desaturation with noise [23]. Abrupt changes in the SpO 2 signal due to desaturations are reflected in the pulse oximeter module, as they are generated by the overall model of neonatal oxygenation [8]. However, motion artifacts leading to significant drops in the SpO 2 signal (drops greater than 10% within 30 s) are not included in our model and should be modeled separately in a future study with simultaneous recording of SpO 2 signal and motion capture.
Our approach to the design of the pulse oximeter model can be compared with the model used by Morozoff et al. [29] in their physiological models. The main novelty of our model is that it is based on real clinical data measured on patients who are the target group for automatic control of oxygenation. The model incorporates the bias between SaO 2 and SpO 2 , the presence of which is documented by many studies [11][12][13][14][15][16][17][18][19]. Furthermore, it proposes different noise levels according to the stability of the patients (physiological or pathological) and according to the input SaO 2 value, since, as the histograms in Figure 8 show, the noise distribution for low and high SaO 2 values is different.
The main limitations of the pulse oximeter model are the datasets used to determine the SaO 2 -SpO 2 bias and SpO 2 measurement noise. One general SaO 2 -SpO 2 bias function was based on the multicenter study on a relatively large number of patients from different PICUs, but the biases among children may systematically vary depending on its diagnosis [11,20], amount of fetal hemoglobin [11,14,20,22], or skin pigmentation [21,23]. The bias may also vary between different SpO 2 monitors or sensors [24], or between different sensor placements [15,20,21]. The noise model was determined based on comprehensive data from both physiological and pathological neonates; moreover, the amount of measured data allowed for the determination of noise characteristics for different SaO 2 intervals. The noise was estimated from the data measured at one setting of the pulse oximeter averaging time. The averaging time is usually adjustable in a range between 2 and 16 s and can significantly affect the stability of pulse oximeter readings [33] and resulting noise characteristics. We determined the noise characteristic only for 8 s averaging (with an output update every 2 s), but it might be interesting to compare the noise characteristic of different averaging time settings. Moreover, the median filter we used for the noise estimation, as discussed above, may be disputed. Finally, due to the exclusion of the unstable parts of SpO 2 signal, our model of the SaO 2 -SpO 2 relationship does not describe the effect of some motion artifacts that trigger the low-signal-quality alarm, as discussed in the previous paragraph.
The results of the work will improve the pulse oximeter model, which is the terminal part of the neonatal oxygen transport model, so that the simulated output of the model, the peripheral oxygen saturation, will more realistically represent the real SpO 2 signals observed in the clinical environment. This work and further improvements of the complex mathematical model will enhance the in silico testing and comparison of existing and future control algorithms under real clinical conditions. There is a need for improved modeling that reflects the dynamics of the neonatal oxygen transport system to achieve optimal control across the full spectrum of oxygenation disturbances, including the possibility of individualization of algorithm performance [10].

Conclusions
This work proposed methods for determining the measurement noise characteristics in peripheral oxygen saturation signal in combination with the bias between SaO 2 and SpO 2 . The terminal part of the neonatal oxygen transport model, the pulse oximeter module, was improved in two ways: we determined the characteristics of the noise presented in SpO 2 time-recordings during the premature infant bedside monitoring, and we combined the SpO 2 measurement noise with SaO 2 -SpO 2 bias. These results will improve the output of the neonatal oxygenation model and make simulations provided by the computer model of oxygenation of a neonate closer to the real situations observed in the clinical practice.