Calculation of blood oxygen saturation

The calculation of blood oxygen saturation (StO2) is based on the modified Beer–Lambert law applied at two wavelengths (660 nm and 850 nm). Briefly, the measured optical attenuation at each wavelength is related to the concentrations of oxyhemoglobin (C_Ox) and deoxyhemoglobin (C_Deox) through their wavelength-dependent extinction coefficients.

1.Known parameters

• Wavelengths: λ1 = 660 nm and λ2 = 850 nm.

• Effective path length: d_eff (accounts for geometric separation and the differential pathlength factor, if used).

• Reference intensities: I0,660 and I0,850 (incident/reference intensities).

• Extinction coefficients (α):

  • α_Ox,660 and α_Deox,660 at 660 nm

  • α_Ox,850 and α_Deox,850 at 850 nm

     

2. Measured optical signals and attenuation

The probe measures detected light intensities I_660 and I_850 in the blood sample. We compute the optical attenuation at each wavelength as:

A_660 = -(1/d_eff) * log10(I_660 / I0,660)
A_850 = -(1/d_eff) * log10(I_850 / I0,850)

3. Two-wavelength modified Beer–Lambert equations

Under the modified Beer–Lambert approximation, the attenuations satisfy:

α_Ox,660 * C_Ox + α_Deox,660 * C_Deox = A_660
α_Ox,850 * C_Ox + α_Deox,850 * C_Deox = A_850

Solving these two linear equations yields C_Ox and C_Deox, and StO2 is defined as:

StO2 = C_Ox / (C_Ox + C_Deox)

4. StO2 expressed directly in terms of the absorbance ratio

Defining the absorbance ratio R = A_850 / A_660, StO2 can be written as:

StO2 = (α_Deox,850 * R - α_Deox,660) / ((α_Deox,850 - α_Ox,850) * R + (α_Ox,660 - α_Deox,660))

This form enables direct computation of StO2 from the measured ratio R and known extinction coefficients.

5. In vitro calibration to ensure accuracy in the reservoir geometry

Although the above relationship is physically motivated, the benchtop cylindrical reservoir does not perfectly satisfy the underlying assumptions of the modified Beer–Lambert model (e.g., homogeneous semi-infinite medium, well-defined effective pathlength, and wavelength-independent optical coupling). In practice, boundary reflections (air–blood and blood–reservoir interfaces), wavelength-dependent scattering/coupling, and uncertainty in d_eff can introduce systematic scale/offset errors in A_660 and A_850, and therefore bias the ratio R.

To account for these non-idealities, we apply an empirical affine calibration to the measured ratio:

R_cal = k * R + b,

where k and b are fitting parameters. We then compute StO2 from R_cal using the above equation. The parameters (k, b) are obtained by least-squares fitting using paired measurements from the wireless probe and the reference catheter oximeter (Swan Ganz 777F8, Edwards Life Science Inc.) across the full sodium dithionite titration range (35–73% StO2). This approach compensates for systematic optical artifacts introduced by the in vitro geometry while preserving the physically motivated functional form relating StO2 to the measured optical ratio.