## 1 Introduction  
There is increasing scientific evidence regarding the concept of food not only as a source of energy and vehicle of essential nutrients for the body, but also as an instrument that is able to beneficially influence some biological functions and reduce the risk of disease development [1]. In recent years, the scientific research on functional foods has mainly focused on the identification of biologically active components in foods that have the potential to provide specific beneficial effects for human health and reduce the risk of diseases. Several techniques were developed in order to enhance the content of these bioactive compounds or promote their production in food products. Lactic acid fermentation is one of the most important food processing technologies used to produce functional probiotic foods. Currently, most probiotics available on the market are dairy based, consisting of several types of fermented milks, yogurts, and cheeses [2]. The choice of different fermenting substrates, such as cereals, fruits, vegetables, and legumes, as alternatives to dairy-based foods, is one way to meet the needs of consumers affected by lactose intolerance, milk protein allergy, high blood cholesterol content, or requiring a low-fat diet [3]. The fermenting strain used for this work was _Lactobacillus paracasei_ (LP) CBA L74, and its probiotic nature was already demonstrated by in vivo studies [4]. Fermentation of cereal-based substrates using LP CBA L74 as a starter culture was widely studied in previous works for aqueous rice, oat, and wheat flour suspensions [5; 6], and for aqueous mixtures of oat and rice flours [7], highlighting how the addition of different concentrations of glucose to the fermenting medium could affect bacterial growth and lactic acid production during the process. Moreover, the microbial activity of LP CBA L74 was also investigated for legume-based substrates [8] and for minimal fermenting broths as a rice flour supernatant [9]. The functional ability of cereals fermented with LP CBA L74 was demonstrated by in vitro studies reported by [PERSON] et al. [10] and [PERSON] et al. [11], where the effect of inhibiting gliadin peptide P31-43 entry into the epithelial cells of fermented rice was studied in order to propose an alternative therapeutic approach to the gluten-free diet for celiac patients. Moreover, the experimental results showed that the inhibiting effect was evidently improved when pH control was applied during the process [11]. It is important to establish the optimal operating conditions for fermentation, which would be beneficial not only in terms of improving the functional effect of fermented products on consumer health, but also from an economic point of view, in terms of improved production. For this reason, knowledge of the fermentation process kinetics through the development of mathematical models plays a vital role in designing and developing improved strategies for optimization and in controlling and scaling up biological process systems to ensure their economic viability [12; 13]. For this purpose, kinetic modeling for _Lactobacillus helevticus_, _Lactobacillus casei_, _Lactobacillus delbrueckii_, _Lactobacillus fermentum_, and _Lactobacillus bulgaricus_ growth and acid production has been reported by several researchers [14; 15; 16; 17; 18; 19]. To describe a microbial process, two kinds of models can be developed, structured and unstructured models. Structured models take into account cell structure, their function, and composition. Although they have been reported to accurately describe lactic acid fermentation [16; 20], they are very complex. In unstructured models, only total cellular concentration is considered, and hence, no physiological characterization of the cells is needed. The simplest non-structural growth model was proposed by [PERSON] in 1942, who described the bacterial growth rate based on the limiting substrate concentration during the fermentation process [21].