Variability in estimating shunt from single pulse oximetry measurements

The fundamental principle of pulse oximetry is a light-tissue interaction. The absorption of light by tissue depends on the wavelength, the tissue composition and the distance (optical path) traveled by the light which changes with each heart beat as blood volume in vessels changes. SaO₂ describes the percentage of oxygenated hemoglobin over the total functional hemoglobin and can be estimated through measuring the ratio (R) of pulsatile and nonpulsatile components of the nonabsorbed light at two distinct wavelengths (typically 660 nm and 910 nm). The relationship between R and SaO₂ can be theoretically explained with the Beer–Lambert law. However this description ignores scattering of light (Mannheimer 2007) and various other design aspects of oximeters, such as optical shunt or probe type. To account for these effects, the relationship between R and SpO₂ is empirically determined by pulse oximeter manufacturers for each device. Even with empirical calibration, SpO₂ remains only an estimate of SaO₂.

The design of pulse oximeters is currently regulated by the International Standards Organization (ISO) specification ISO 80601-2-61 (ISO 2011). This standard describes the requirements for the design of pulse oximeters such as packaging, electro-magnetic safety, alarms, and display. Further, the standards give recommendations on the performance of calibration experiments using desaturation studies with human subjects, including testing and reporting accuracy. Manufacturers are required to specify the accuracy for each device and sensor combination since each configuration requires a different transfer function relating R to SaO₂. This accuracy is generally stated as root mean square error (A_RMS) between measured and reference values obtained during desaturation studies such as

$$\begin{eqnarray}&&{A_{{\rm{RMS}}}} = \sqrt {\frac{{\sum\limits_{i = 1}^n {{{({\rm{SpO}}_2^i - {\rm{SaO}}_2^i)}^2}} }}{n}}\end{eqnarray} \tag{ 1 }$$

where n is the number of observations and SaO₂ is the arterial saturation obtained from a reference carboxy–hemoglobin oximeter (CO-oximeter or hemoximeter) with an accuracy of ⩽1%. It is important to note that hemoximeter devices are also subject to a measurement bias between models (Gehring et al 2007). In pulse oximeter data sheets and manufacturer documentation, which are generally available to clinicians, the A_RMS value is often incorrectly referred to as 'standard deviation' (SD) (ISO 2011). According to the ISO standard, the bias B_S and the precision P_S of SpO₂ can also be used to describe device errors such as

$$\begin{eqnarray}&&{{A}_{\text{RMS}}}\approx \sqrt{P_{s}^{2}+B_{s}^{2}}.\end{eqnarray} \tag{ 2 }$$

If B_S is close to 0, A_RMS = P_S (ISO 2011). In this case the mean SpO₂ ≈ SaO₂ and A_RMS ≈ SD. Consequently, A_RMS indirectly describes the confidence interval of finding a SaO₂ for a given SpO₂ ± A_RMS, where approximately 32% of measurements will be expected to be outside of this range. It can therefore be seen as a tolerance that describes an acceptable deviation from the target value. The A_RMS error can be represented as a Gaussian distribution with a center value corresponding to the observed SpO₂ (figure 1).

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A deviation of <4% has been considered acceptable for pulse oximeters used for monitoring (ISO 2011). Commercial pulse oximeters used in current clinical practice (e.g. monitors from brands like Nellcor, Nonin, Masimo and their OEM partners) have A_RMS between 1% and 3%, depending on the testing condition (SaO₂ range, probe type and presence of artifacts). Under non-artifact conditions and normal SaO₂, an A_RMS of 2% is common. The SaO₂ estimation of a pulse oximeter with this specification will have a 95% tolerance interval between 90% and 98% when displaying an SpO₂ reading of 94%, if we hypothesize that the bias of the estimation is low (B_s ≈ 0) (figure 1). This relatively large range of 8% around this common decision threshold of 94% might be clinically acceptable for continuous monitoring applications where many measurement points are available and trend changes of vital signs are of more interest than absolute values. However, in diagnosis and treatment applications where individual measurements are performed (so called spot-checks), this wide distribution of possible SaO₂ should be taken into account when making clinical decisions. The clinical decision could be supported with a clinically relevant physiological mathematical model that accounts for the inherent inaccuracies of the measuring device.

Mathematical descriptions of the physiology of the human respiratory system and the propagation of O₂ from inspiration at the lung to consumption at the cell level are well described (West 2012). Formulae to calculate iso-shunt lines for estimation of optimal oxygen administration (Benatar et al 1973, Petros et al 1994), or for predicting arterial oxygen partial pressures (PaO₂) from administered oxygen concentrations (Kretschmer et al 2013) have been investigated and implemented in clinical practice. Pulse oximetry data have however frequently been used interchangeably with SaO₂ in these implementations (Jones and Jones 2000). In Kjaergaard et al (2003), no significant difference in shunt estimation was observed when using either SaO₂ or SpO₂. However, in addition to SaO₂ the model required measurement of expired O₂ and did not take into account the precision of pulse oximeters.

In the next sections we describe the development of a mathematical model to describe the propagation of specified tolerances of pulse oximeter readings for the estimation of the virtual shunt (VS), an indicator of the severity of the gas exchange impairment in maximizing oxygen content. We then illustrate the impact of the theoretical concepts with controlled experimental data. In the discussion, we provide a critical analysis of advantages and limitations of the suggested model and the performed experiments.

The Alveolar Gas Equation (West 2012) with a fixed respiratory quotient (RQ), water vapor pressure (PH₂O), and partial Alveolar CO₂ pressure (PACO₂) was used to estimate PAO₂ from the fraction of inspired O₂ (FiO₂). PiO₂ was defined by the ambient gas pressure (P_atm) and the fraction of inspired O₂ (FiO₂):

$$\begin{eqnarray}\begin{array}{*{35}{l}} \text{PA}{{\text{O}}_{2}}= & \text{Fi}{{\text{O}}_{2}}\cdot \left(\text{P}_{\text{atm}}^{0}-\text{P}{{\text{H}}_{2}}O\right)-\frac{\text{PAC}{{\text{O}}_{2}}}{\text{RQ}} \\ {} & + \text{PAC}{{\text{O}}_{2}}\cdot \text{Fi}{{\text{O}}_{2}}\cdot \frac{1-\text{RQ}}{\text{RQ}}, \\ \end{array}\end{eqnarray} \tag{ 3 }$$

where typically PACO₂ = 5.33 kPa (40 mmHg) and PH₂O = 6.27 kPa (47 mmHg) at 37 °C and RQ = 0.8. To simplify calculations, we assumed that these parameters are fixed for the purpose of this study. We assumed standard ambient temperature T⁰ and pressure $P_{\text{atm}}^{0}$ at sea level. FiO₂ was selected as a variable. FiO₂ above the ambient value of 21% corresponded to administration of oxygen. Decreasing $P_{\text{atm}}^{0}$ or reducing FiO₂ values have a similar effect on PAO₂, leading to a standardized oxygen concentration. Therefore FiO₂ values below 21% were equivalent to the effect of decreasing $P_{\text{atm}}^{0}$, as would be the case for subjects living at higher altitude (equivalent altitude). The lowest used FiO₂ = 11% would correspond to an altitude of approximately 5000 m above sea level.

The PxO₂ decreases between alveoli (PAO₂), end-capillaries (Pc'O₂) and systemic arteries (PaO₂). This difference in partial pressure P[A-a]O₂ is largely due to shunted blood and a mismatch of ventilation and perfusion ${{\dot{V}}_{A}}/\dot{Q}$, where ${{\dot{V}}_{A}}$ is the alveolar ventilation and $\dot{Q}$ is the blood flow. Under normal conditions, incomplete capillary diffusion is 0 and PAO₂ = Pc'O₂ (West 2012). The inequality in oxygen content between end-capillary, venous and arterial blood is commonly referred to as VS. VS can be estimated from the shunted blood flow ${{\dot{Q}}_{s}}$ and total blood flow ${{\dot{Q}}_{t}}$ (West 2012):

$$\begin{eqnarray}\begin{array}{*{35}{l}} \text{VS} & =\frac{{{{\dot{Q}}}_{s}}}{{{{\dot{Q}}}_{t}}}=\frac{\text{C}c\prime {{\text{O}}_{2}}-\text{Ca}{{\text{O}}_{2}}}{\text{C}c\prime {{\text{O}}_{2}}-\text{C}\bar{v}{{\text{O}}_{2}}} \\ {} & =\frac{\text{C}c\prime {{\text{O}}_{2}}-\text{Ca}{{\text{O}}_{2}}}{\text{C}c\prime {{\text{O}}_{2}}-\text{Ca}{{\text{O}}_{2}}+\left(\text{Ca}{{\text{O}}_{2}}-\text{C}\bar{v}{{\text{O}}_{2}}\right)}, \\ \end{array}\end{eqnarray} \tag{ 4 }$$

where CxO₂ is the oxygen content of blood, Cc'O₂ the end-capillary content, and typically (CaO₂–C$\bar{v}$ O₂) = 5 ml/100 ml of blood. CxO₂ can be obtained using

$$\begin{eqnarray}&&\text{Cx}{{\text{O}}_{2}}=\frac{1.34\cdot \text{Hb}\cdot \text{Sx}{{\text{O}}_{2}}}{100}+0.003\cdot \text{Px}{{\text{O}}_{2}},\end{eqnarray} \tag{ 5 }$$

where typically the oxygen-combining capacity of hemoglobin in blood is 1.34 ml O₂/(g Hb), Hb = 12 g/100 ml is the hemoglobin concentration in blood, SxO₂ is the oxygen saturation, 0.003 is the solubility of O₂ in Hb in $\frac{\text{ml}{{\text{O}}_{2}}}{100\text{ml}\cdot 1\text{mmHg}}$.

VS is important to the clinician as its magnitude describes the efficiency of gas exchange in the patient which would represent the severity of the underlying pathological disease process. Therefore, we use VS as an output variable that is estimated by the mathematical model. Since equations (4) and (5) can only be solved numerically, this may be inefficient to apply to a pulse oximeter. An alternative is to use P[A-a]O₂ as an estimate of VS with the inherent limitations of using partial pressure rather than oxygen content. This measure describes the gradient of PO₂ due to the underlying pathology and can be easily displayed in a oxygen cascade (figure 2). When calculating VS, PaO₂ is not estimated from FiO₂ but from SaO₂.

Hill's equation has been modified by Severinghaus to calculate SxO₂ from PxO₂ (Severinghaus 1979):

$$\begin{eqnarray}&&\text{Sx}{{\text{O}}_{2}}=\frac{1}{\frac{23400}{\text{Px}{{\text{O}}_{2}}^{3}+150\cdot \text{Px}{{\text{O}}_{2}}}+1}.\end{eqnarray} \tag{ 6 }$$

The Severinghaus–Ellis equations (Severinghaus 1979, Ellis 1989) allow for the conversion of SxO₂ to PxO₂:

$$\begin{eqnarray}&&\text{Px}{{\text{O}}_{2}}={{(B+A)}^{1/3}}-{{(B-A)}^{1/3}}\end{eqnarray} \tag{ 7 }$$

where

$$\begin{eqnarray}&&A=\frac{11700}{\frac{1}{\text{Sx}{{\text{O}}_{2}}-1}},\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&B=\sqrt{{{50}^{3}}+{{A}^{2}}}.\end{eqnarray} \tag{ 9 }$$

Equations (6) and (7) describe the nonlinear S-shaped Hb–O₂ dissociation curve in both directions (figure 3). As the term A of equation (8) is not defined for SxO₂ = 100%, we replaced 100% with 99.9% in our model calculations.

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In clinical practice SaO₂ is either obtained by analyzing an arterial blood sample in a blood gas analyzer or when not available, estimated using SpO₂ from the pulse oximeter reading. The A_RMS error in this SaO₂ estimation (as illustrated in figure 1) is propagated in our model using the Severinghaus–Ellis equation (7). The pulse oximeter can be considered as an additional stage in the oxygen cascade and SpO₂ is used as an input variable that can be measured in the clinical setting (figure 2).

After obtaining ethics approval and written consent, 19 healthy non-smoking subjects (11 males, eight females, median age 28, range 19–50 years) with no history of cardio-respiratory or neurological disease were recruited for a controlled desaturation study. The primary aim of the study was to record data for calibration experiments of a novel mobile phone pulse oximeter using a camera (Karlen et al 2012). The subjects were invited to the physiology laboratory where they rested for 5 min and then underwent a health check by measuring baseline blood pressures, heart rate and SpO₂. Fitzpatrick Skin Phototype (Fitzpatrick et al 1988) was assessed (median III, range II–V). Following the successful health check, the study sensors were applied and recording initiated. After 20 min of recording at sea level (FiO₂ = 21%), each subject entered a normobaric hypoxia chamber with a FiO₂ set to 12%. The FiO₂ was then increased stepwise to 17% and then again reduced to 12% (figure 4). After approximately 2.5 h in the chamber, when an FiO₂ = 12% was attained and maintained for 5 min, the subjects exited the chamber and were monitored for another 20 min at FiO₂ = 21%.

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The study sensors consisted of four pulse oximeters that recorded reference data simultaneously. Both, the dominant and non-dominant hands of each subject were fitted with a Nonin Xpod 16 bit (NON, Nonin Medical Inc, Plymouth, USA) and a Masimo Low Power Module (MAS, Masimo Corp, Irvine, USA) oximeter module and finger probe, each connected to a Phone Oximeter on an iPod Touch (Apple Inc, USA) (Karlen et al 2011). The devices recorded a photoplethysmogram (16 bit; MAS 62.5 Hz, NON 75 Hz), heart rate (1 Hz), and SpO₂ (1 Hz). They were configured to calculate SpO₂ over an average of 8 beats for NON and 8 s for MAS respectively. Both brands of pulse oximeters also provided indicators of artifacts, low perfusion and sensor problems. The sensor data sheets reported an A_RMS of ± 2 digits for NON and ± 2% for MAS under conditions without artifacts. Inspired and expired O₂ and CO₂ and respiratory rate were recorded using Datex-Ohmeda S5 Collect (GE Healthcare, Fairfield, USA) hardware and software at 1 Hz. All recording devices were synchronized to a common time server and a marker was pressed at the beginning of the experiment to verify synchronization.

All data from the reference oximeters on the non-dominant hand and the gas monitor were paired for this study. The data from the dominant hand were not used as they were subject to an increased level of artifacts that would potentially contaminate the data for the purposes of this study. Data pairs labeled with an artifact, low perfusion (MAS perfusion index <1 or >18, NON 'red' perfusion) or sensor warnings of either oximeter were eliminated. The first three minutes after entering or exiting the chamber and after changing FiO₂ inside the chamber, where subjects were exposed to large changes in FiO₂ in a short period of time, were also discarded to eliminate variations in response time effects of the oximeters and individual subjects physiology.

The distribution of the SpO₂ values for a given pulse oximeter at an FiO₂ of 21% ± 0.3 and 17% ± 0.3 were plotted as histograms for each device. The 17% threshold was chosen as a second level because it provided the second largest count of data pairs and contained the data with least adaptation effects (changes in respiratory rate and end tidal CO2). Also, the expected SpO₂ for this FiO₂ is in the range of 93–94%, which coincides with the commonly used threshold for clinical decision making. It was therefore particularly interesting to analyze pulse oximeter tolerance for that range. The standard deviation (SD) for all subjects and the mean of the intra-subject standard deviation (iSD) were calculated. iSD is a more conservative measure as it disregards inter-subject variations that might be present due to sensor placement, perfusion, physiology or other reasons that would lead to measurement variations between observations in different subjects. The obtained distributions were compared with the probability density functions obtained by the model for an 'ideal' pulse oximeter with A_RMS = 2% . The probability density functions were plotted for subjects with VSs of 0%, 2% and 5%, all considered normal levels.

The oxygen cascade predicts the maximal PaO₂ for a healthy subject at FiO₂ = 21% (figure 6). The maximal PaO₂ with a VS of 0% (PAO₂ = PaO₂) and FiO₂ = 21% was 98 mmHg (13 kPa) at sea level and consequently, when the Severinghaus–Ellis equations were applied, the SaO₂ was 97.8% . The oxygen cascade demonstrates that PAO₂ increases linearly with an increase in FiO₂. PaO₂ estimated from SpO₂ showed a non-linear relationship characterized by the Severinghaus–Ellis equations and PaO₂ estimates. Due to the precision of pulse oximeters, depicted in green (68% percentile) error bars, the estimated PaO₂ can be larger than the maximal possible value (PAO₂) and consequently VS is negative. The graph in figure 5 allows for an estimate of the likelihood of a shunt for a given SpO₂ observation. For example, a measured SpO₂ of 94% (black filled dot) has an 84% confidence to have a VS between 7.5% and 19%.

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A total of 32 672 sample points were obtained for FiO₂ = 21% and 6085 for FiO₂ = 17% (table 1). The SpO₂ distributions were different for the two pulse oximeter brands at FiO₂ = 21% (figure 7) and FiO₂ = 17% (figure 8). Both sensors were within range when comparing the reported SD and iSD values to the specified accuracy in the respective data sheet. SD was larger than iSD in all situations (table 1), indicating variability between individual subjects. This variability can be also observed in figure 8 where a second peak in the NON distribution can clearly be identified. This variability was also present for the FiO₂ = 21%, but was not sufficiently large to produce a secondary peak. There was a clear bias of 1.2% between the oximeters at an FiO₂ = 21%, which increased to 2.1% at an FiO₂ = 17% (table 1).

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In this work, a number of co-variates in the model such as Hb and PxCO₂ were kept invariant. While our physiological models would allow these parameters to be included as variables in the theoretical calculations, exact values are not available in clinical practice without invasive sampling and were also not available during our experiments. Our model consists of a single compartment to estimate VS. More precise models, for example with two (Karbing et al 2011, Kretschmer et al 2013), three (Jones and Jones 2000, Karbing et al 2011) and five (Wang et al 2010) compartments, have been suggested. However, considering the shown tolerance effects of pulse oximeters on the models overall accuracy, the increased complexity of higher compartment order is unlikely improving VS estimations in practice.

A low PaO₂ triggers a physiological response that compensates for the reduced O₂ intake during breathing of hypoxic gas mixes. A short term adaptation is the increase of respiratory drive and consequent reduction of PACO₂. This is a typical adaptation to higher altitudes and is also observed in our data. PACO₂ and all other CO₂ partial pressures will decrease immediately at exposure causing a left shift of the Hb–O₂ dissociation curve and a relative increase in PAO₂. This could not be avoided during data collection, but was minimized by using data for analysis that was collected after the typical short term adaptation period of 3 min. The longer term (multiple hours/days) adaptation process of increasing the concentration of 2,3-diphosphoglycerate leading to a right shift of the Hb–O₂ dissociation curve and lowering the O₂ affinity (West 2012) was considered negligible due to the short duration of our experiments.

For this study, arterial blood gas samples were not available to confirm the estimated SaO₂. We can therefore only speculate about the real PaO₂ and VS that the subjects experienced during the experiments. This corresponds to many clinical situations where arterial sampling of blood gases is not immediately available or desirable.