## 3 Results and Discussion  
### Aerobic Plate Count (APC) of Pasteurized Orange Juice  
The parsimonious polynomial model below with only four parameters (4P) best describes the APC growth rate (\(N\)) over the storage time (\(t\)) at various temperature levels (\(T\)) along with the statistical parameters of the model (Table 1):  
\[N=N_{0}+\exp(b_{1}+\ b_{2}\ast Ln(T))\ast t \tag{1}\]  
The strong linear shape of standardized and Cox-Snell residuals indicated a very reliable model (AICc = 607.55). Figure 1 depicts the population of APC (cfu/mL) versus storage time at each storage temperature. As can be seen, at lower temperatures (10 \({}^{\circ}\)C and 15 \({}^{\circ}\)C), the shelf life is longer than expected, and more specifically, 15 and 10 days are needed, respectively, until the rise of the population to the industrial internal limit of 1000 cfu/mL APC. The APC values \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Estimate** & **Std Error** & **–95\%** & **95\%** \\ \hline b0 & \(-\)0.238380 & 0.40665308 & \(-\)1.035405 & 0.558645 \\ b1 & \(-\)3.550622 & 0.22067326 & \(-\)3.983134 & \(-\)3.118111 \\ b2 & 1.182725 & 0.06742677 & 1.050571 & 1.314880 \\ Sigma & 1.130722 & 0.12053435 & 0.894479 & 1.366965 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistical information for APC model. converge at 20-25 \({}^{\circ}\)C and at 30-40 \({}^{\circ}\)C, indicating that at the temperature range of 20-25 \({}^{\circ}\)C, 6.5 days are needed to reach 1000 cfu/mL, whereas, at the same time, for the temperature range of 30-40 \({}^{\circ}\)C, only 3.5 days are needed to reach 1000 cfu/mL. The steep increase in the growth rate following the rise in the storage temperature was also observed in work by [PERSON] et al. (2021) [15], where the Weibull equation was used to model the shelf life of orange juice at different temperature levels. At 5 \({}^{\circ}\)C and 22 \({}^{\circ}\)C, the inactivation rate, b, of the equation was 0.01 and 3.88, respectively, while in the present study, the same values were 2.18 and 5.67, respectively. Although a direct comparison between the two studies is not feasible due to the different equations (Weibull and polynomial), a rough estimation could be informative. The premature and rapid increase in APC at the higher temperature levels (20-40 \({}^{\circ}\)C) is explained by the diminishing of the lag phase at storage temperatures above 15 \({}^{\circ}\)C, meaning that this decrease in the lag phase, where the microbial growth rate is zero, consequently leads to a reduction in shelf life due to faster spoilage [17]. More specifically, in the study performed by [PERSON] and [PERSON] [17], it was observed that the storage of orange juice at different temperature levels (0, 4, 10, and 15 \({}^{\circ}\)C) reduced the APC lag phase period from 9.5 days to 2.8 and 0 days for 10 and 15 \({}^{\circ}\)C, respectively. In the present study, this decrease in the APC lag phase, along with the increase in the storage temperature, was also apparent. As a matter of fact, in Figure 1, the lag phase at temperatures of 10, 15, 20, and 25 \({}^{\circ}\)C is 8, 4, 2, and 1 day, respectively [19]. The advantage of the polynomial model is the perception of shelf life as a continuous and not categorical variable, enabling the prediction of the shelf life at different temperature values than the experimented ones. This is very important when considering that the usual storage temperatures of pasteurized orange juice do not normally exceed 7 \({}^{\circ}\)C. Thus, using the degradation profiler (Figure 2), which depicts the APC growth at different time points (days) and temperature levels (\({}^{\circ}\)C), the required amount of time for achieving 1000 cfu/mL (internal limit of expiration) at different temperature levels is smoothly obtained. Briefly, if the temperature level of Figure 2 is adjusted at different levels, the predictive table (Table 2) can be created, presenting the shelf life of pasteurized orange juice.