# Data-Driven Calibration Estimation for Robust Remote Pulse-Oximetry
## 2 Materials and Methods
### Apbv
#### 2.2.1 Opto-Physiological Modelling  
In order to be able to determine the parameters for the calibration model, the PPG amplitude spectrum can be modelled, as we did in our earlier work [8]. A first order approach to estimate the relative PPG amplitudes and their dependence on wavelength is to simply weigh the absororptivity spectra for (deoxy)hemoglobin (Hb) and oxyhemoglobin (HbO\({}_{2}\)). However, this would disregard the complex light interaction with the various skin layers. Since pulsatile arterioles are not distributed homogeneously throughout the skin, different wavelengths will experience different degrees of PPG intensity modulation not only due to the wavelength-dependent blood absorption coefficient, but also due to their different skin penetration depths. Various skin properties such as dermal scattering, venous blood concentration and blood vessel diameter [18] impact these skin penetration depths differently for different wavelengths. Therefore, we modeled diffuse reflectance for a layered skin model using the analytic approach by [PERSON] et al. [19]. We also adopted their best guesses for the skin geometrical and optical properties for normal skin (e.g., epidermal and dermal scattering = \(\frac{29\times 10^{6}}{\lambda}\)). Three layers were modeled: epidermis, upper dermis and deep dermis, with thicknesses of 0.05, 0.5 and 4 mm, respectively, with skin blood volume fractions of 0%, 1% and 2%, respectively, both for venous and arterial blood. Saturation for venous blood was 70% while four different values for arterial blood were modeled: 70%, 80%, 90% and 100%, representing four SpO\({}_{2}\) values. At each arterial saturation level, we computed a PPG modulation as the normalized diffuse reflectance difference between a systolic and diastolic phase (slightly more/less arterial blood was modeled). One important departure from the modeling by [PERSON], however, is that we used the original absorptivities for Hb and HbO\({}_{2}\) measured by [PERSON] and [PERSON] [20]. Using their tabulated values (their Table 3) along with the plots (their Figure 2), we interpolated absorptivities for Hb and HbO\({}_{2}\) at 5 nm increments from 400-1000 nm. The modeled relative PPG spectrum is visualized in Figure 1. Finally, the SpO\({}_{2}\)-dependent parameters of the pulse signature vector \(\vec{P}_{bv}^{*}\) can be determined by:  
\[\vec{P}_{bv}^{i}(SpO_{2})=\int_{\lambda=400}^{1000}PPG(SpO_{2},\lambda)I( \lambda)Cam^{i}(\lambda)Filter^{i}(\lambda)\,\mathrm{d}\lambda\qquad\text{for i = 1,...,N}, \tag{4}\]  
where \(I(\lambda)\), \(Cam(\lambda)\) and \(Filter(\lambda)\) are the illumination spectrum, the camera sensitivity and the transmission spectrum of the filter, respectively. This model is very generic, but clearly practical values for all parameters had to be chosen. The values we used, and which determined the experimental setup we used to create our dataset (Section 2.2.3), led to the following APBV calibration model:  
\[\vec{P}_{bv}^{i}=\begin{bmatrix}0.388\\ 0.805\\ 1\\ 1.320\end{bmatrix}_{[\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}]=[675,760,8 00,905]nm},\vec{P}_{bv}^{i}=\begin{bmatrix}0.0224\\ 0.0103\\ 0\\ -0.0036\end{bmatrix}_{[\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}]=[675,76 0,800,905]nm}. \tag{5}\]  
Figure 1: The modeled relative PPG spectrum using the analytic approach by [PERSON] et al. [19] for four SpO\({}_{2}\) values.