## 3 Results and Discussion
### Rancimat Trials--Concentration-Dependent Effects in a Lipid Matrix (Olive Kernel Oil)  
The first approach to evaluate the effect of the addition of increasing amounts of antioxidants in OKO was to examine the correlation between the induction period (IP) and antioxidant concentration (\(C\)). To this end, IP was determined using increasing concentrations of antioxidants, and plotted as a function of \(C\) (Figure 2). \begin{table}
\begin{tabular}{c c c} \hline
**Antioxidant** & \(n_{\text{t}}\) & **IC\({}_{50}\) (\(\upmu\)M)** \\ \hline BHT & 0.31 & 160.6 \\ CA & 2.63 & 20.0 \\ FA & 1.10 & 47.8 \\ CouA & 0.28 & 187.7 \\ \hline \end{tabular}
\end{table} 
Table 1: Total stoichiometry (\(n_{\text{t}}\)) and IC\({}_{50}\) values determined for the hydroxycinnamates considered. Each compounds was tested at \(C=16\)\(\upmu\)M, with [DPPH\({}^{\bullet}\)]\({}_{0}=102.3\)\(\upmu\)M.  
Figure 2: Changes in induction period (IP) as a response to increasing antioxidant concentration (C). Assignments: BHT, butylated hydroxytoluene; CA, caffe acid; CouA, \(p\)-coumaric acid; FA, ferulic acid. Model fitting was performed with SigmaPlot(tm) 12.5 software, and revealed that IP may be correlated with C following a three parameter power function:  
\[\text{IP}_{(C)}=\text{IP}_{(0)}+kC^{a} \tag{6}\]  
where \(\text{IP}_{(C)}\) represents the IP recorded with any C, and \(\text{IP}_{(0)}\) represents the one determined without antioxidants added. The terms \"\(k\)\" and \"\(a\)\" are fitting parameters. For all antioxidants tested, both \(k\) and \(a\) were determined from the non-linear regression and are provided in Table 2.  
The rearrangement of Equation (6) would give the following:  
\[\text{IP}_{(C)}-\text{IP}_{(0)}=kC^{a} \tag{7}\]  
The term \(\text{IP}_{(C)}-\text{IP}_{(0)}\) represents the extension in IP provoked by a given concentration of antioxidant under the Rancimat conditions employed, and it is assigned as \"induction period extension\" (IPE) from now on. Thus, Equation (7) may be written as the following:  
\[\text{IPE}=kC^{a} \tag{8}\]  
The parameter \(k\) is a constant that might be associated with the nature of the antioxidant. The parameter \(a\) shows the sensitivity of IPE as a response to modifying \(C\), that is, the higher the \(a\) for a given antioxidant, the lower the concentration required to achieve a certain IPE for a given oil. Thus, \(a\) may be regarded as a measure of antioxidant effectiveness. Furthermore, to shed more light regarding the stabilizing potency of the antioxidants tested, the mean rate (\(R_{\text{m}}\)) of inhibitor (antioxidant) consumption was also determined [42] as follows:  
\[R_{\text{m}}\left(\text{mg kg}^{-1}\text{h}^{-1}\right)=\frac{C}{\text{IP}} \tag{9}\]  
If \(R_{\text{m}}\) is plotted as a function of \(C\), a linear correction may be obtained, as illustrated in Figure 3.  
The analytical data derived from the linear regressions are given in Table 3. According to these correlations, \(R_{\text{m}}\) is directly proportional to \(C\). Therefore, increasing \(C\) would provoke a monotonous effect on \(R_{\text{m}}\). The slope of the linear function depicted in Figure 2 has units of \(\text{h}^{-1}\) and represents a first-order reaction constant, termed as \(k_{\text{s}}\) (instability constant) from now on. The higher the \(k_{\text{s}}\), the higher the rate of antioxidant depletion. Therefore, an antioxidant that exhibits low \(k_{\text{s}}\) might be more stable under the conditions employed, provoking higher IPE. In this regard, the ranking of the compounds tested would be CA \(>\) CouA \(\approx\) FA. Additionally, based on the \(a\) values presented in Table 2, the ranking would be CA \(>\) CouA \(>\) FA. Considering these findings, it was evident that CA was a superior antioxidant compared to the other two hydroxycinnamates tested, FA and CouA.