## 3 Results and Discussion  
### System Identification  
#### 3.1.1 Data-Based Mechanistic Modelling of CIVC  
The measured blood flow, conducted during the first set of experiments (Exp.1), showed a decreasing pattern (average drop of \(400\pm 53\) PU) coupled with decreased finger skin temperatures in response to the applied localized cooling (for more information see [20]). In agreement with previous study [20], the SRIV algorithm combined with _YIC_ and \(R_{2}^{T}\) suggested that a second-order (with number of poles \(n=2\) and zeros \(m=2\)) discrete-time TF described the dynamic responses of finger skin temperature to step-decreases in the input (Peltier element's temperature) most accurately (i.e., \(R_{2}^{T}=0.91\pm 0.04\) and _YIC_\(=-11.23\pm 2.33\)) for all 33 test subjects. More specifically, the SRIV algorithm identified the following general discrete-time DTF model structure,  
\[y(k)=\ \frac{b_{1}z^{-1}+b_{2}z^{-2}}{1+a_{1}z^{-1}+a_{2}z^{-2}}u(k-\delta)+ \xi(k) \tag{3}\]  
where the pure time delay (\(\delta\)) was different from test subject to another (interpersonal variation) with an average (\(\bar{\delta}=2.3\) min) and standard deviation (\(SD=1\)). The variations in the time delay of the model reflects the different individual biological responses to colder environment. Figure 4 shows a simulation example of the resulting model (3), showing the simulated dynamic responses of the finger skin temperature to step decrease (cooling) in the Peltier element temperature. Figure 4: A simulation example of the second-order DTF (3) that represents the finger’s temperature response (CIVC reaction) to the step decrease in Peltier element temperature (cooling). To understand the underlying mechanism of the resulting second-order system (3), describing the CIVC process, it was decomposed into two simpler first-order transfer functions (TF). These two first-order TFs can be interconnected in three well-known configurations, namely, series, parallel and feedback configurations. The results showed that the parallel configuration (Figure 5) was mathematically most suitable (average \(R_{2}^{T}=0.91\pm 0.01\)) for describing the dynamic response of the CIVC process in all test subjects. As depicted in Figure 5, the second-order system (3), representing the CIVC process, was decomposed into two first-order TFs, defined as follows:  
\[TF_{1}=\frac{b_{1}}{1+a_{1}z^{-1}}\]  
\[TF_{2}=\frac{b_{2}}{1+a_{2}z^{-1}}\]  
\[y(k)=[TF_{1}+TF_{2}]\cdot u(k-\delta) \tag{4}\]  
The time constants \(\tau_{1}\) and \(\tau_{2}\) of \(TF_{1}\) and \(TF_{2}\), respectively, were calculated and compared for all test subjects, where \(\tau=1/\ln{(-a)}\). Table 2 shows the mean and standard deviation of the calculated time constants obtained from all test-subjects during the two sets of experiments (Exp.1 and Exp.2, Phase I). The aforementioned results suggested two parallel underlying processes taking place during the CIVC action. However, by comparing the calculated time constants, it became clear that the process represented by \(TF_{2}\) was, on average, 10 times faster (\(\tau_{2}\approx 0.73\) min) than the one represented by \(TF_{1}\) (\(\tau_{1}\approx 7.34\) min). Physical and physiological insights into the resulting mathematical model were investigated using first principles (e.g., [PERSON]'s law and [PERSON]'s law of cooling) and other mechanistic models (e.g., [12]). The heat flow through any object, including the finger, can be described conventionally by lumped quantities such as thermal resistance (or conductance) and thermal capacitance [27]. The heat balance \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(\tau_{1}\) (**min**) & \(\tau_{2}\) (**min**) \\ \hline Mean & 7.34 & 0.73 \\ Standard deviation & 3.2 & 0.56 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mean and standard deviation of the calculated time constants, of the decomposed two first-order TF, obtained from all test subjects (33) during the two sets of experiments (Exp.1 and Exp.2, Phase I).