## 3 Results and Discussion
### System Identification
#### 3.1.2 Data-Based Mechanistic Modelling of CIVD  
Among the 33 test subjects, 27 showed the hunting reaction with cycles of CIVC and CIVD. Figure 7 shows an example of the resulting hunting reaction to continuous localised cooling using Peltier's element temperature at 2 \({}^{\circ}\)C.  
Figure 6: Schematic representation of the heat transfer between the finger and the Peltier element through three different layers, namely, deep skin layer, surface skin layer and the Peltier’s element. The AVA controls the direct flow between the artery and the surface vein. Within the cooling period of 60 min, all test subjects exhibiting hunting reactions showed three consecutive cycles of CIVC/CIVD (an example is shown in Figure 7). A second-order DTF with two poles (\(n=2\)) and two zeroes (\(m=2\)) was the best model structure (with average \(R_{2}^{T}=0.92\pm 0.04\) and \(YIC=-10.61\pm 3.45\)) to describe the CIVD response to localised cooling for all test subjects with the hunting reaction:  
\[y(k)=\ \frac{b_{1}z^{-1}+b_{2}z^{-2}}{1+a_{1}z^{-1}+a_{2}z^{-2}}u(k-\delta)+ \xi(k) \tag{6}\]  
The structure of the identified model, represented by the triad [\(n=2\), \(m=2\), \(\delta\)], showed an average time delay \(\delta\) of 10.23 min (\(\pm 3.50\) min). In other words, for all test subjects that showing the hunting reaction, the first episode of CIVD started, on average, after \(10.23\pm 3.50\) min from the start of applying the localised cooling. Table 3 shows the estimation results of the identified model (6) for all test subjects showing the hunting reaction. This relatively long time delay, compared to the CIVC case, was somehow misleading in terms of system causality, since during this lag of time between the cause (step cooling) and the modelled effect (CIVD), there was another effect, namely the CIVC. Therefore, the model input, represented by the step decrease in the Peltier's element temperature, cannot fully explain the CIVD effect. However, the rate of heat dissipation (\(\dot{Q}_{dis}\), Watt), from the finger to the Peltier's elements, shows more dynamics in relation to both CIVC and CIVD. The heat dissipation rate (\(\dot{Q}_{dis}\)) is calculated based on [PERSON]'s law of cooling as follows [34]:  
\[\dot{Q}_{dis}=-h_{P}A_{P}(T_{sk}-T_{\infty}) \tag{7}\]  
Figure 7: Dynamic response of finger skin temperature to step decrease in Peltier’s element temperature from 20 \({}^{\circ}\)C to 2 \({}^{\circ}\)C, showing an example of the hunting reaction in a test subject. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(A(z^{-1})\) & \multicolumn{3}{c}{\(B(z^{-1})\)} & \multicolumn{3}{c}{\(\delta\) (min)} & \(R_{T}^{2}\) & \(YIC\) \\ \hline \(a_{1}\) & \(a_{2}\) & \(b_{0}\) & \(b_{1}\) & \(b_{2}\) & & & \\ \hline \(-1.989\) & \(0.980\) & \(0.0\) & \(-0.0047\) & \(0.0045\) & \(10.23\) & \(0.92\) & \(-10.61\) \\ \(\pm 0.005\) & \(\pm 0.002\) & \(\pm 0.0\) & \(\pm 0.0010\) & \(\pm 0.0010\) & \(\pm 3.50\) & \(\pm 0.04\) & \(\pm 3.45\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Average and standard deviation of parameter estimates, \(R_{2}^{T}\) and \(YIC\) of the second-order DTF model describing the CIVD reaction to continuous localised cooling experiments obtained from the test-subjects that shown hunting reaction (i.e., cycles of CIVC/CIVD episodes). where \(h_{P}\) is the heat transfer coefficient between the finger's skin and [PERSON]'s element (=15 W m\({}^{-2}\)\({}^{\circ}\)C\({}^{-1}\), from the datasheet of the [PERSON]'s element) and \(A_{P}\) is the total surface area of the [PERSON]'s elements in contact with the skin (=5.76 \(\times\) 10\({}^{-4}\) m\({}^{2}\)). An example of the simulated finger skin temperature during the CIVD reaction using the identified second-order DTF (6) is shown in Figure 8. Additionally, Figure 8 shows the calculated dissipated heat rate (\(\dot{Q}_{dis}\)) using Equation (7), while the area-under-the-curve represents the amount of the dissipated heat (\(Q_{dis}\), Joules) from the finger to the [PERSON]'s elements.