## 2 Brief Literature Review on Predictive Microbiology  
Predictive Microbiology assumes the reproducibility of the response of microorganisms to environmental factors based on previous observations of their behaviours [2]. Each microorganism has different growth stages, making it necessary to use tools that integrate predictive microbiology in the kinetic study of microbial growth to help in predicting the shelf life of foodstuffs in which those microorganisms multiply [3]. Some authors suggest the year 1922 as the origin of predictive models, highlighting the tests carried out by [PERSON] and [PERSON] [4], who resorted to a linear model to characterize the thermal processing of _Clostridium botulinum type A_ neurotoxin spore destruction. The microbial growth curve describes the density of cell populations in the liquid environment, over time (minutes or hours), by measuring the optical density of those populations [5]. The typical microbial growth curve is subdivided into four distinct phases, as shown in Figure 1, namely:  
1. Lag or latency phase, characterized by the absence of microbial replication. The cells adapt to the environment, rich in nutrients, in order to initiate replication;
2. Logarithmic or exponential phase, characterized by accelerated cell division;
3. Stationary phase, characterized by the decline of the metabolic rate due to the depletion of nutrients in the environment;
4. Death or logarithmic decline phase, characterized by the exponential decline of living cells and their growth. Figure 1: Generic microbial growth curve [5]. Predictive models characterize microbial growth in each phase as a function of time (\(t\)), presenting different essential variables, such as the maximum specific growth rate (\(\mu_{\max}\)), the logarithmic increase in the microbial population (\(A\)) and the maximum microbial population reached (\(N_{\max}\)). Estimating the length of lag phase (\(\beta\)) may be regarded as a good indicator of food shelf life, being influenced by factors such as the physiological state of bacteria or the conditions of the environment in which food is stored or displayed [6]. [PERSON] and [PERSON] [7] suggested that predictive microbiology models could be categorized in three different forms, depending on the variables to consider:  
* Primary models, in which variations of environmental factors are not considered. They only describe the concentration of microorganisms as a function of time;
* Secondary models, which consider variations in environmental factors, such as temperature (\(T\)), pH (potential of hydrogen) or water activity (\(a_{w}\));
* Tertiary models: integrated models enhanced through computational modeling. Among the existing predictive microbiology models, it is imperative to highlight the model proposed by [PERSON] [8] and [PERSON] and [PERSON] [9]. It is one of the most used primary models, due to its ability to adjust to data and its easily interpretable parameters. The model describes an approximate movement to a sigmoid curve, despite presenting an intermediate phase close to a straight line, unlike classic sigmoid curves. One of the main limitations pointed out in primary models is the lack of biological foundations. Previously proposed algebraic models were deduced from insufficiently solid mechanistic considerations. There was a biological gap in these models, making it difficult to mechanically interpret the proposed parameters [10]. Based on this assumption, the model in question was developed in order to consider the biological mechanism of microorganisms' growth. It introduced the concept of cellular physiological state (\(t_{0}\)), (defined as the effect of the history of cells, capable of influencing lag phase), providing the models hitherto proposed with greater biological suitability [10]. Figure 2 shows the adaptation of the four parameters adopted by [PERSON] and [PERSON] [9] for the microbial growth curve.