## 2 Results and Discussion
### Shelf Life Prediction Models of HPP Marinated Lotus Root Slices  
Zero- and first-order kinetic reaction equations (Equations (1) and (2)) are commonly used to construct food quality reaction kinetic models [55]. \[\text{Zero}-\text{order reaction model}:\text{C}=\text{kt}+\text{C}_{0} \tag{1}\]  
\[\text{First}-\text{order reaction model}:\text{C}=\text{C}_{0}\text{e}^{ \text{kt}} \tag{2}\]  
where t is the storage time (days), C is the index value of storage time t, C\({}_{0}\) is the initial value of the index, and k is the decay rate of the index. The hardness, TVC, and \(\Delta\)E data of HPP marinated lotus root slices stored at different storage temperatures and times in Figures 2 and 3 and Table 3 were fitted with Equation (1) or (2). The corresponding zero- and first-order reaction rate constants (k) and the determination coefficients (R\({}^{2}\)) are listed in Table S2. The average R\({}^{2}\) values of the kinetic equations for changes in hardness and TVC under the first-order reaction (0.9688 and 0.9900, respectively) were slightly higher than those of the zero-order reaction (0.9524 and 0.9571, respectively), whereas the average R\({}^{2}\) values of the zero- and first-order reactions for \(\Delta\)E changes were similar (0.9000 and 0.9029, respectively). It has been reported that the changes in the hardness of white button mushrooms and the TVC of grass carp follow the first-order reaction, and the change in \(\Delta\)E of tambaqui fillet follows the zero-order reaction [56; 57; 58]. A quantitative description of the relationship between the temperature and the rate of a chemical reaction was further performed using the Arrhenius equation (Equation (3)) [55]. \[\text{lnk}=\text{ln}\,\text{k}_{0}-\frac{\text{Ea}}{\text{RT}} \tag{3}\]  
where k is the deterioration rate, k\({}_{0}\) is the pre-exponential factor, Ea is the activation energy (J/mol), T is the storage temperature (K), and R is the universal gas constant (8.314 J/(mol\(\cdot\)K)). According to Equation (3), the reaction rate constant (k) for hardness, TVC, and \(\Delta\)E of the HPP marinated lotus root slices in Table S2 were taken logarithmically and linearly fitted to the corresponding 1/T. Ea and k\({}_{0}\) were calculated from the slope and intercept of the fitted equations, respectively, as shown in Table 4.  
By combining the corresponding zero- or first-order kinetic model (Equation (1) or (2)) with the Arrhenius equation (Equation (3)), shelf life prediction models with the variables of temperature (T) and quality factor (C) were established, as shown in Equations (4) and (5):  
The Arrhenius model under the zero-order reaction:  
\[\text{SL}=\frac{\text{|C-C_{0}|}}{\text{k}_{0}\text{e}^{-\frac{\text{kt}}{ \text{RT}}}} \tag{4}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & \multicolumn{3}{c}{**Relative Error 1**} \\ \cline{3-8}  & & \(\text{k}_{0}\) & **Ea (kJ/mol)** & \multicolumn{3}{c}{**Stored at 4 \({}^{\circ}\)C**} & **Stored at 25 \({}^{\circ}\)C** & **Stored at 45 \({}^{\circ}\)C** & **Average** \\ \hline \multirow{3}{*}{Hardness} & Zero-order & 4.50 \(\times\) 10\({}^{6}\) & 25.00 & 38.90\% & 3.98\% & 6.25\% & 16.37\% \\  & First-order & 5.57 \(\times\) 10\({}^{2}\) & 25.33 & 31.03\% & 15.26\% & 1.56\% & 15.95\% \\  & Zero-order & 6.49 \(\times\) 10\({}^{3}\) & 27.92 & 15.21\% & 20.13\% & 16.06\% & 17.14\% \\  & First-order & 1.79 \(\times\) 10\({}^{3}\) & 27.47 & 16.02\% & 8.07\% & 20.27\% & 14.79\% \\ \(\Delta\)E & Zero-order & 1.31 \(\times\) 10\({}^{3}\) & 22.44 & 56.30\% & 18.11\% & 29.44\% & 34.62\% \\  & First-order & 32.54 & 22.15 & 56.83\% & 21.20\% & 29.12\% & 35.71\% \\ \hline \hline \end{tabular}  
* Relative error = (1 Predicted shelf life (day) \(-\) 15 / 15) \(\times\) 100%. HPP, high-pressure processing; TS, thermal-and-soaking; TVC, total viable count; \(\Delta\)E, total color difference.  
\end{table}