## 2 Materials and Methods
### Data Analysis  
Microbial growth was modeled using the Baranyi Growth Model [30]. For curve fitting, the program DMFFit (IFR, Institute of Food Research, Reading, UK) was used (available at [[https://combasebrowser.errc.ars.usda.gov/DMFit.aspx](https://combasebrowser.errc.ars.usda.gov/DMFit.aspx)]([https://combasebrowser.errc.ars.usda.gov/DMFit.aspx](https://combasebrowser.errc.ars.usda.gov/DMFit.aspx))). The microbial growth is described by an equation as follows:  
\[\begin{array}{l}y(t)=y_{0}+\mu_{max}t+\frac{1}{\mu_{max}}\ln(e^{-y \cdot t}+e^{-h_{0}}-e^{-y\cdot t-h_{0}})-\\ -\frac{1}{m}\ln(1+\frac{e^{\mu_{max}t+\frac{1}{\mu_{max}}}\ln(e^{-y\cdot t}+e^{ -h_{0}}-e^{-y\cdot t-h_{0}})-1}{e^{(\mu_{max}\cdot y_{0})}})\end{array} \tag{1}\]  
where \(y_{(t)}\) is the cell concentration (log CFU/g) at any time \(t\), \(y_{0}\) is the initial cell concentration (log CFU/g), \(y_{max}\) is the maximum cell concentration (log CFU/g), \(\mu_{max}\) is the maximum specific growth rate (d\({}^{-1}\)), \(v\) is the rate of increase in the limiting substrate, assumed to be equal to \(\mu_{max}\), \(\lambda\) is the lag phase duration (d), and \(h_{0}\) is equal to \(\mu_{max}\cdot\lambda\). The growth rates (\(\mu_{\max}\)) and lag phase duration (\(\lambda\)) were estimated, and they are presented with the corresponding standard errors.  
The dependence of the microbial growth rate on storage temperature was estimated according to the Arrhenius equation:  
\[k=k_{ref}\cdot exp[-\frac{E_{a}}{R}.\left(\frac{1}{T}-\frac{1}{T_{ref}}\right)] \tag{2}\]  
where \(T_{ref}\) is a reference temperature, \(R\) is the molar gas constant (8.314 J/K\cdot mol), and \(E_{a}\) is the apparent activation energy (J/mol). The equation was linearized by applying the logarithm to both sides of the equation, and then coefficients \(k_{ref}\) and \(E_{a}\) were estimated using linear regression.  
Sensorial deterioration, as expressed through the overall impression of the studied frankfurter-type sausages, was modeled using a zero-order kinetic equation, as follows:  
\[Overall\ impression_{t}=Overall\ impression_{0}-k_{sensory}\cdot t \tag{3}\]where _Overall impression_t_ is the sensory quality at any time \(t\), _Overall impression_0 is the initial sensory quality of the products (score = 9), and _k_sensory is the sensorial deterioration rate constant (d\({}^{-1}\)) of the samples at each storage temperature.  
The shelf life of frankfurter-type sausages, based on microbiological data and with a limit of acceptance for the TVC load set to 8 log CFU/g, was estimated according to the following equation:  
\[Shelf\ life\ (SL_{microbial})=\left(\frac{\log(N_{limit})-logN_{0}}{k_{ TVC}}\right)+lag\ phase \tag{4}\]  
where _log_(\(N_{limit}\)) is the limit of acceptance for the microbial load (8 log CFU/g), \(N_{0}\) is the initial microbial load of the TVC, and _k_TVC is the growth rate constant (d\({}^{-1}\)) of the TVC at each storage temperature.  
The shelf life of frankfurter-type sausages, based on sensorial data and with a limit of acceptance for the overall impression set to 4.5, was estimated according to the following equation:  
\[Shelf\ life\ \big{(}SL_{sensory}\big{)}=\left(\frac{Overall\ impression_{ initial}-Overall\ impression_{final}}{k_{sensory}}\right) \tag{5}\]  
where the _Overall impression_(_initial_) represents the initial score (=9), the _Overall impression_(_final_) represents the limit of sensorial acceptance (=4.5), and _k_sensory is the deterioration rate constant of sensorial quality (d\({}^{-1}\)) at each storage temperature.