## 2 Methodology  
In a recent study [21], we built a finger model based on the hypothesis that when blood enters a tissue the total volume of the tissue expands, increasing the amount of light traversing a path through the tissue medium.  
Moreover, the study also hypothesized that the blood constitutes oxyhemoglobin (HbO), deoxyhemoglobin (HHb), and glycated hemoglobin (HbA1c). The HbA1c component of blood is stated to be fixed at a mixture of 98% oxygenated and 2% deoxygenated HbA1c. So, the total absorption coefficient of the blood solution becomes  
\[C_{a}=\epsilon_{a}^{\text{HbA1c}}(\lambda)\times c_{\text{HbA1c}}+\epsilon_{a}^ {\text{HbO}}(\lambda)\times c_{\text{HbO}}+\epsilon_{a}^{\text{HHb}}(\lambda) \times c_{\text{HHb}} \tag{1}\]  
\[C_{a}=\mu_{a}^{\text{HbA1c}}(\lambda)+\mu_{a}^{\text{HbO}}(\lambda)+\mu_{a}^{ \text{HHb}}(\lambda) \tag{2}\]  
In (1) and (2), \(C_{a}\), \(\epsilon\), and \(c\) are the total absorption coefficient of the blood solution, molar absorption coefficient, and molar concentration of the individual component of the solution, respectively.  
Figure 1: Discrete-LED/PD-based transmission- (**left**) and reflection-mode (**right**) PPG acquisition devices.  
Figure 2: Camera-sensor white-LED MW-PPG system. The (**left**) figure is the transmission mode and the (**right**) figure is the reflection mode.  
From another study [20], we can obtain a different blood-solution hypothesis to estimate the amount of glycated hemoglobin in the blood. According to the hypothesis, the blood solution only contains glycated and non-glyacted hemoglobin components. So, the blood solution absorption coefficient expression becomes  
\[C_{a}=e_{a}^{\text{HDA1c}}c_{\text{HDA1c}}+\epsilon_{a}^{\text{NonHbA1c}}c_{ \text{NonHbA1c}} \tag{3}\]  
\[C_{a}=\mu_{a}^{\text{HbA1c}}+\mu_{a}^{\text{NonHbA1c}} \tag{4}\]  
The system from (3) and (4) greatly simplifies the actual finger structure. The nonHbA1c component of (3) and (4) mostly indicates the homogenous mixture of 98% oxyhemoglobin and 2% deoxyhemoglobin compounds. So,  
\[\epsilon_{a}^{\text{NonHbA1c}}=\epsilon_{a}^{\text{HDO}}\times 0.98+\epsilon_{a} ^{\text{HHb}}\times 0.02 \tag{5}\]  
Now, from the Beer-Lambert law  
\[A=C_{a}d=-\log\biggl{(}\frac{I}{I_{0}}\biggr{)} \tag{6}\]  
In (6), \(A\), \(d\), \(I\), and \(I_{0}\) denote the total solution absorbance, light transmission path length, received light, and the incident light, respectively. Applying (6) in (1) and (3) with different wavelengths of light can enable us to calculate the parameters of (1) and (3), respectively.  
Now, placing (1) in (6), we get  
\[A(\lambda)=\Bigl{(}\epsilon_{a}^{\text{HbA1c}}(\lambda)\times c_{\text{HbA1c}} +\epsilon_{a}^{\text{HDO}}(\lambda)\times c_{\text{HDO}}+\epsilon_{a}^{\text{ HHb}}(\lambda)\times c_{\text{HHb}}\Bigr{)}d=-\log\biggl{(}\frac{I(\lambda)}{I_{0}( \lambda)}\biggr{)} \tag{7}\]  
Similarly, placing (3) in (6), we get  
\[A(\lambda)=\Bigl{(}\epsilon_{a}^{\text{HDA1c}}c_{\text{HbA1c}}+\epsilon_{a}^{ \text{NonHbA1c}}c_{\text{NonHbA1c}}\Bigr{)}d=-\log\biggl{(}\frac{I(\lambda)}{ I_{0}(\lambda)}\biggr{)} \tag{8}\]  
From the discussion of these models, it can be deduced that the oxyhemoglobin and deoxyhemoglobin are kept fixed in the latter model, which may lead to more errors due to the change of blood oxygenation level (SpO2) in the bloodstream. The processes of error analysis are described in the following subsections.