## 1 Introduction  
During the last years, consumers have become much more aware of the importance of nutrition and health, showing interest in purchasing healthier foods. Therefore, the dietary approach aimed at promoting and maintaining health and well-being, in a natural and low-cost way, has triggered the increase in functional food products consumption. It is estimated that up to 2030 the global probiotics market will have a turnover of approximately 73.9 billion dollars, leading the food sector to the highest economic value [1, 2]. Several efforts have been dedicated to the development of functional foods, i.e., foods characterised by additional functionalities, such as health promotion and disease prevention. In particular, all the functions related to the probiotics have gained a growing interest [3, 4]. The production process of functional probiotics foods, typically, involves the fermentation, which is carried out by using a starting proteolytic matrix (substrate) and a bacterial strain. Commonly, probiotics are provided to consumers with fermented products, including fermented vegetables, or meat, yogurt. The majority of the commercially available products employ both the Bifidobacterium (present in the large intestine), and the Lactobacillus (present in the small intestine), due to their well-known safe use [5]. In the strains selection for their use in commercial products, a pivotal aspect is represented by the strain's ability to survive during the processing, product development, and transit through the human gut. These features depend on the species and strains; for instance, Lactobacillus are commonly preferred to Bifidobacteria for industrial applications, due to their higher resistance to low pH [1; 6; 7]. For this reason, the Lactobacillus strains have been industrially employed for the systems production and fermentation. Most of the studies in the literature are performed on dairy products, e.g., ice cream, cheese, fermented milk, and yogurt [1; 8; 9], but a growing interest is attracting non-dairy products, e.g., cereals, fermented vegetables, soy, and fruit juices [3; 5; 10; 11], to respond to the increasing demand of vegetarians, vegans, lactose intolerant and milk protein allergic consumers. In this context, the proposal of new fermentation processes requires the definition of a standard protocol to be adopted for the probiotic and therefore optimized by maximizing the production capability in order to meet the industry needs. Thus, a comprehensive knowledge of both the raw material and every single step of the production process is fundamental for the development of new types of fermented products, with the final aim to comprehend the possible process impacts on the fermentation [12]. According to the literature, in addition to the bacterial strain, the main factors affecting the microbial growth during a fermentation process and the subsequent lactic acid production can be many different, e.g., the substrate [5; 10], the operational parameters such as temperature [8], pH [13], glucose addition [3], etc. Establishing the optimal conditions for the fermentation process are therefore crucial not only to improve the functional effect on consumers' health, but also from an economic point of view due to the increased production capability. In this light, the development of empirical models to predict the bacterial growth kinetics plays a key role for the definition of the best strategy to control and optimize the fermentation process, with the aim of extending the process itself from the laboratory scale to the industrial one [14]. Generally, there are two main types of mathematical models for the description of the microbial growth, i.e., structural and non-structural [15]. The former considers the microbial structure, the function, and the composition. The resulting models are accurate but very complex. While the non-structural models consider only the bacterial concentration without the need to evaluate other physiological characteristics of the cells [16].