## 2 Materials and Methods
### Microbial Growth Modelling  
Mathematical models that describe microbial behavior (growth, inactivation, survival) are generally divided into two types: (i) primary and (ii) secondary [28]. Primary models are developed and used to describe microbial behavior as a function of time and estimate microbial kinetic parameters. Over the years, several primary models (e.g., logistic, Gompertz, Richard equation) have been developed and used to predict and describe sigmoid microbial growth curves [29,30].  
The proper fitting of these experimental and mathematical models, used in addition to the estimated kinetic parameters, is critical for selecting the primary model. The model proposed by [PERSON] and [PERSON] (1994) [31], as given by Equations (1) and (2), is one of the most-used models due to its ability for proper kinetic parameter determination even from a limited amount of data; and, as will be described in the following paragraphs, it gives a good fitting of the experimental data of microbial growth for all different bacon categories investigated.  
\[y(t)=y_{o}+\mu_{max}\cdot A(t)-\ln\left[1+\frac{\exp(\mu_{max}\cdot A(t))-1}{ \exp(y_{max}-y_{o})}\right] \tag{1}\]  
and  
\[A(t)=t+\frac{1}{\mu_{max}}\cdot\ln[\exp(-\mu_{max}\cdot t)+\exp(-\mu_{max} \cdot\lambda)-\exp(-\mu_{max}\cdot(t+\lambda))] \tag{2}\]  
where \(y_{o}\) (logCFU/g) describes the decimal logarithm of initial microbial population for t = 0; \(y_{max}\) (logCFU/g) is the decimal logarithm of maximum microbial population; \(\mu_{max}\) (logCFU/g/((logCFU/g)d\({}^{-1}\))) is the maximum specific growth rate; (logCFU/g/((logCFU/g)d\({}^{-1}\))), t (d) is time; A(t) is an adjustment function that ranges from 0 to t and is related to the physiological state of microbial density; and \(\lambda\)(d) is the lag phase.  
To determine the effects of environmental conditions, such as the temperature, secondary models are used. The most-known and frequently used secondary models are Arrhenius-type models and square root-type models. In this research, to study the effect of temperature on the microbial growth of different bacon samples, both of these secondary models were implemented, comparing preliminary results of the sum of square errors (SSE) between the two secondary models. The Arrhenius model presented the least SSE and was used as the secondary model for this analysis. The temperature dependence of the maximum specific growth rate and lag phase were modelled by the Arrhenius equation, shown in Equations (3) and (4), respectively, and the evaluation of SSE was conducted using Equation (5) [32]. Data analysis and mathematical modeling was performed in Microsoft, Excel 365.  
\[\mu_{max}(T)=\mu_{max,ref}\cdot\exp\left(\frac{E_{\alpha,\mu}}{R}\cdot\left( \frac{1}{T}-\frac{1}{T_{ref,\mu}}\right)\right) \tag{3}\]  
where \(\mu_{max,ref}\) (d\({}^{-1}\)) describes the pre-exponential factor for maximum specific growth rate; \(E_{\alpha,\mu}\) (J/mole) is the activation energy for maximum specific growth rate of the investigated microorganism; (J/mole), R (8.314 J/mole.K) is the universal gas constant; T (K) is the absolute temperature; and Tref (K) is the reference temperature, here assumed to be equal to 4 \({}^{\circ}\)C.  
\[\lambda(T)=\lambda_{\text{ref}}\cdot\exp\left(-\frac{\text{E}_{\text{a}}\lambda}{ R}\cdot\left(\frac{1}{T}-\frac{1}{\text{T}_{\text{ref},\lambda}}\right)\right) \tag{4}\]  
where \(\lambda_{\text{ref}}\) (d) is the pre-exponential factor for the reference temperature, and E\({}_{\text{a},\lambda}\) (J/mole) is the activation energy to express the temperature dependency of the lag phase of the investigated microorganism.  
\[\text{SSE}=\sum_{i=0}^{n}\left(\mu_{\text{obs}}-\mu_{\text{pred}}\right)^{2} \tag{5}\]  
where \(\mu_{\text{obs}}\) (d\({}^{-1}\)) is the maximum specific growth rate of the experimental measurements; \(\mu_{\text{pred}}\) (d\({}^{-1}\)) is the maximum specific growth rate calculated in the investigated model; n (-) is the number of observed data; and i = 0, 1, 2, 3..., n.