The present research used particular microorganisms for fermentation on different concentrations of wheat and soy-flours. The LAB used during the whole experiment were L. plantarum ATCC 8014 (Lp) and L. casei ATCC 393 (Lc) received from the University of Agricultural Science and Veterinary Medicine Cluj-Napoca, Romania. For both LAB, the used medium was MRS broth (enzyme digested casein 10 g/L, meat extract 10 g/L, yeast extract 5 g/L, glucose 20 g/L, dipotassium hydrogen phosphate 2 g/L, sodium acetate 5 g/L, di-ammonium citrate 2 g/L, magnesium sulfate 0.2 g/L, manganese sulfate 0.05 g/L, and tween 80 1.08 g/L) with a final pH of 6.4 ± 0.2 at 25°C.

The first step consisted of microorganism activation on model media. The initial step was the introduction of 9 ml of MRS broth medium in special vials, after which followed the sterilization in autoclave and inoculation with 1 ml of pure L. plantarum or L. casei culture (Figure 1A). These vials with 9 ml of medium and 1 ml of pure culture were incubated at 30°C for 18–24 h. After the introduction of activated bacteria (10 ml) in 90 ml MRS broth and cell viability establishment (Figure 1B), the bottles were re-incubated for 18–24 h (Paucean et al., 2013; Teleky et al., 2020). Each step was performed under sterile conditions.

The incubated media (100 ml) with each LAB was centrifuged for 10 min at 4°C and 7,000 rpm. The supernatant was removed, and the pellet was resuspended in physiological serum (Figure 1C). This washing process was repeated twice. After this phase, the LAB concentration was measured with a UV-VIS spectrophotometer (NanoDrop 1,000; Figure 1D), and the transfer in the dough substrate was effectuated separately. The determination of bacterial concentration was performed by measuring the optical density at 600 nm (OD600) as reported in (Sieuwerts et al., 2018; Dulf et al., 2020). The dough inoculation adapted after (Gerez et al., 2012; Mitrea et al., 2019) from 100 ml media occurred with 10% (20 ml) of microorganisms at the concentration of 8 log¹⁰ CFU/mL (Figure 1E). The incubation time during the fermentation process was 24 h at 37°C.

Sample prelevation during fermentation took place at regular intervals (0, 2, 4, 6, 8, 10, and 24 h). For cell viability, the pour plate method was used for LAB and expressed using logarithmic values of the colony-forming units per mL of sample (log¹⁰ CFU/mL) (Călinoiu et al., 2019b). The inoculated agar plates were incubated for 48 h at 37°C, after which cell counting was performed (Figures 1F, G).

The first step implied using a response surface method to evaluate various operating conditions and determine the optimal parameters for the fermentation process. The multiple regression model used at the beginning considers three factors contributing to each response. Experimental data processing implies the use of polynomial equations: y = a 0 + ∑ i a i x i + ∑ i a i i x i 2 + ∑ i , j a i j x i x j + ∑ i a i i x i 3 + ∑ i , j , k a i j k x i x j x k (1) where y is the response variable, a ₀ , a _i , a _ii , a _ij are the model constant term, the coefficient of linear terms, the coefficient of quadratic and cubic terms, and the coefficient for the interaction between variables i and j, respectively x _i are the independent variables. The model allows the drawing of surface response curves, and through their analysis, the optimum conditions can be determined. The adequacy of the model is determined by evaluating the root mean square error, the regression coefficient (R-squared), and the F-value and p-value obtained from the analysis of variance (ANOVA). Matlab (R2018a) ^® software is used to perform the modeling, analyses, and plots.

The second research step consists in establishing a model using neural networks. The neural network model defines a family of possible equations along with a set of data and a strategy to find better rules among the possible ones based on these data. In contrast, the model form is given explicitly in multiple regression methods.

In the training stage are used 75% of experimental data. The remaining 25% of the data should be used in the model validation step. As data splitting method, the systematic stratification semideterministic method is used. The data were first ordered along the output variable dimension in increasing order. The starting point was randomly selected. Training samples were used first, followed by the testing samples.

The model is required to be as simple as possible. To achieve this, the number of layers is considered 1, and the number of neurons is maximized to 10. If the models created with these parameters have significant errors, they can be adjusted using different neural networks and optimization algorithms. As an optimization algorithm, five specific training methods are used: Levenberg-Marquardt, Quasi-Newton, Scaled Conjugate Gradient, Fletcher-Powell, and Polak-Ribiere. The obtained mathematical model has the following form: y = A ⋅ ( 2 e B ⋅ X + b i + 1 − 1 ) + b ii (2) where y is the response variable, A is a row matrix of dimension (1: number of neurons); B is a matrix of weights for each variable, dimension (number of variables: number of neurons); X: variable column matrix of dimension number of variable: 1; b _i : column matrix of coefficients with dimension (number of neurons:1) and b _ii is the bias value. The adequacy of the models is determined by evaluating the root mean squared error, the regression coefficient (R-squared), and the p-value obtained from the analysis of variance (ANOVA). Matlab (R2018a)^® software is used to perform the modeling, analyses, and plots. As software is also used the tool described in (Dulf et al., 2018).

The study’s purpose is to optimize fermentation by utilizing different substrate compositions of WF and SF by using LAB in single or co-cultures. In Figure 2, the viability results for the three WF and SF mixtures for the LAB L. plantarum, L. casei, and the co-cultures of L. plantarum and L. casei are presented.

The viability of the three different substrate mixtures reaches a final concentration of 9.61 ± 0.04 log CFU/mL. With L. plantarum (Figure 2A), the increase is uniform throughout the experiment. At the same time, L. casei at the beginning presents a slower growth, but the final concentration is similar to L. plantarum of 9.54 ± 0.11 log CFU/mL (Figure 2B). With both LAB, the viability is efficient, with the highest results of 9.67 ± 0.10 log CFU/mL (Figure 2C) The data are also available in the Appendix.

The first step is the analysis of multiple regression models derived from the three different experiments presented in the previous section. In the beginning, the used model is on L. plantarum, after which the L. casei model and the third model implied the co-cultures of L. plantarum and L. casei. The considered independent variables are x₁ time, x ₂ concentration of wheat flour, while x ₃ represents the concentration of soybean flour.

The obtained experimental data were fitted using the four high degree polynomial models, such as the linear, interactive (2FI), quadratic, and cubic models. The test results and model summary statistics are depicted in Table 1.

Based on the test results, the adequacy of the proposed models was estimated. To determine whether the model equations developed are significant, the analysis of variance (ANOVA) is used for the chosen models. In Table 2 the results from ANOVA for all three fermentations are presented. ANOVA also showed that the coefficient of variable x ₂ and the interaction terms x ₂ x ₃ are not significant because x ₂ and x ₃ are complementary variables. The sum of these two is 100% in each case. The resulted models are presented above.

Case 1

y = 0.00069266 x 1 3 − 1.4844 e − 05 x 2 3 − 4.0525 e − 08 x 3 3 + 0.019703 x 1 2 + 0.0021154 x 2 2 + 0.06339 x 1 (3)

Case 2

y = − 0.00084662 x 1 3 − 1.4639 e − 05 x 2 3 − 4.0525 e − 08 − 1.5864 e − 07 x 3 3 + 0.026551 x 1 2 + 0.0020173 x 2 2 + 0.015823 x 1 (4)

Case 3

y = − 0.0014505 x 1 3 − 1.8861 e − 05 x 2 3 + 2.614 e − 07 x 3 3 + 0.044862 x 1 2 + 0.0024482 x 2 2 − 0.096252 x 1 (5)

The Cook’s distance and the normal probability plot for the obtained models are presented in Figures 3, 4, where subfigures 1) represent the results for L. plantarum, 2) are for L. casei, while 3) is for the co-culture of L. plantarum and L. casei. Since there were no significant deviations from the straight line in the normal probability plot, it can be concluded that the standardized residuals of the model have a normal distribution. The Cook’s distance implied that the outliers were not present among the data fitted by the regression model.

The three-dimensional plots are used to present the impact of analyzed factors on the fermentation by utilizing different substrate compositions of WF and SF by using LAB in single or co-cultures. The effects of fermentation time, the concentration of wheat flour, and the concentration of soybean flour are depicted in Figure 5, using the same labeling as in previous figures: 1) for the results for L. plantarum, 2) for L. casei and 3) for the co-culture of L. plantarum and L. casei. Based on the presented plots, the relationship between the response and factors can be considered. The extraction time exhibited a linear, quadratic (in the second case) and a cubic effect, reaching a maximum value and later decreasing in value. The concentration of wheat flour effect presents a slight linear increase. The concentration of soybean flour effect presents a slight linear decrease.

Although the regression model exhibits good results, being a more advanced modeling tool, ANN models are also developed. The best optimization algorithm and the optimum number of neurons for the neural network are obtained using experimental data. Analyzing the mean squared error in each case are obtained as competitive the models developed with the Levenberg-Marquardt, Quasi-Newton and Fletcher-Powell algorithm with 6, 8 and 10 neurons. The obtained mean squared errors are: 0.0265 for Levenberg-Marquardt with 6 neurons, 0.0306 for Levenberg-Marquardt with 8 neurons, 0.0377 for Quasi-Newton with 8 neurons, 0.0439 for Quasi-Newton with 10 neurons and 0.0364 for Fletcher-Powell with 10 neurons.

An ANN model is established for each experimental setting case with different training methods and several neurons. The comparison between the 2D simulation results of ANN models with Levenberg-Marquardt training methods with different neuron numbers and the multiple linear regression from the models mentioned above are presented in Figures 6–8, with the labeling of subplots as follows: 1) for the results for L. plantarum, 2) for L. casei and 3) for the co-culture of L. plantarum and L. casei. The experimental data of cell viability for LAB are presented with stars and are expressed using logarithmic values of the colony-forming units per mL of sample (log10 CFU/mL). For a more accurate model validation, one more dataset was realized, with a smaller sampling period. Using more neurons improves the model’s performance, as presented in Figure 7. The disadvantage is the model complexity. Some of the matrices will grow in size from 6 to 8.

Using the Fletcher-Powel method, the obtained models are more complex, having matrices with 10 rows and 10 columns. The performance of the models is worse than the achievements of the models created via Levenberg-Marquardt, being 0.0113, 0.0441, and 0.0513 in the considered cases. The results are plotted in Figure 8.

Using the Quasi-Newton method as a training network, the created models lead to the errors: 0.0523, 0.0467, and 0.0627, with the simulation results presented in Figure 9. Since it has the same number of neurons as the Levenberg-Marquardt method, it is just as complex, but the performances are more accurate for the model created with Levenberg-Marquardt.

The mathematical models obtained in these three case studies have the form: y = A ⋅ ( 2 e B ⋅ X + b i + 1 − 1 ) + b i i (6)

The L. plantarum case model: b i = ( 2.97 1.8 1.58 − 0.03157 − 0.187 2.53 2.3 3.73 ) ; B = ( − 1.49 − 1.97 − 0.73 − 1.78 − 0.88 − 1.84 − 1.78 − 1.79 − 0.36 − 1.61 − 2.02 1.01 3.4 0.16 0.23 2.62 0.91 0.68 3.1 1.08 1.39 3.21 0.99 0.94 ) ; (7) A = ( 0.08 0.39 0.27 − 0.025 0.86 0.9 0.56 − 1.98 ) ; b ii = 0.563. (8)

The L. casei case model: b i = ( 2.81 − 1.88 1.044 0.22 0.47 − 1.86 − 2.64 − 2.81 ) ; B = ( − 2.49 1.21 − 1.12 2.7 − 0.82 0.85 − 0.78 − 0.84 − 2.61 0.16 − 2.58 0.49 1.91 0.44 0.51 − 1.77 − 1.63 − 0.911 − 0.338 − 2.15 − 0.3 − 2.021 − 1.119 1.18 ) (9) A = ( − 0.17 0.33 0.62 0.08 0.79 − 0.36 1.36 0.17 ) ; b ii = 1.0356. (10)

The co-cultures of L. plantarum and L. casei case model: b i = ( 2.82 − 1.99 − 0.49 0.77 1.1 − 0.27 − 2.26 − 1.71 ) ; B = ( − 2.88 1.02 0.9 0.2 1.68 − 2.24 2.24 1.69 − 0.8 − 6.3 3.09 0.43 − 0.66 − 0.27 2.69 2.12 1.12 − 0.2 − 1.12 1.51 − 1.15 − 3.07 − 2.31 1.44 ) ; (11) A = ( 1.38 0.17 − 0.37 − 0.82 0.39 0.98 0.05 − 0.35 ) ; b ii = − 0.72. (12)

The 3D simulation results of these models are presented in Figures 10, 11, 12. Figure 10 presents the 3D model simulation results with the Levenberg-Marquardt method with eight neurons for L. plantarum; Figure 11 is for results with the Levenberg-Marquardt method with eight neurons for L. casei, while Figure 12 show the results with the Levenberg-Marquardt method with eight neurons for co-cultures of L. plantarum and L. casei.

With the use of the 3D plots and the use of the models, an approximation of the optimal concentration is found by searching the highest point. For the first process, the optimum value of the final product is 10.3076 obtained for 97.7083% concentration of WF and 19 h of fermentation.

In the case of the second process, the optimum value is 9.7571, with a concentration of 95.625% WF in 21 h.

The third process has a maximum amount of 12.2755, with 97.7083% WF in 19 h. Under these optimal conditions, repeated validation experiments are performed, obtaining close values to the predicted ones.

From the ANOVA test of the regression model for L. plantarum, the p-value is 2.69·10⁻¹⁴, which means that the model is significant. R squared has the value of 0.99, showing that the regression model fits well for the first bacterium. The root-mean-squared error has the value of 0.129, proving that the quadratic response for this case fits the results obtained in the experiment.

In the case of the second bacterium, L. casei, the p-value is 3.8·10⁻¹³ with the R squared value 0.985. The root mean squared error is 0.188. The values are not so different from the previous results giving almost as good performance as in the first case.

For the final case, with the co-cultures of L. plantarum and L. casei, the p-value is 4.22·10⁻⁹ with R squared value of 0.948. The root means squared error is 0.333. Out of the three cases, this one presents the less agreeable performances, but still a good result. The results obtained with regression are good for all three cases, with mean squared error under the accepted values.

The performance of the developed ANN models is evaluated by mean squared error. The first developed ANN models were with the Levenberg-Marquardt method with six neurons on each layer. The obtained mean squared error for the L. plantarum is 0.028, for L. casei 0.0176, and the mixed culture of this two is 0.0418. Increasing the number of neurons on each layer can lead to better performances. That was the reason for developing models with eight neurons. The obtained mean squared errors are for the first case 0.0014, for the second case 0.0185, and 0.0124 for the third case. However, a great influence on the results has the used optimization algorithm. For example, the performance of the models with 10 neurons with the Fletcher-Powel method is worse than the achievements of the models created via Levenberg-Marquardt with 8 neurons, being 0.0113, 0.0441, and 0.0513 in the considered cases instead of 0.0014, 0.0185, and 0.0124. Using the Quasi-Newton method with 8 neurons as a training network, the created models lead to the errors: 0.0523, 0.0467, and 0.0627.

The synthesis of the considered ANN models performance measures (Table 3), compared with the results obtained with the regression method, highlights the superiority of the ANN models. The best-considered model for this process (with all three used bacterium) is obtained with the Levenberg-Marquardt method using eight neurons.

For the first case, with L. plantarum, the best results are obtained with the Levenberg-Marquardt method using 8 neurons. In the case of the L. casei, the model with Levenberg-Marquardt with 6 and 8 neurons presents similar performances, but because the best model in most cases is Levenberg-Marquardt with 8 neurons, the decision is to use the same parameters in all cases. As a result, the performance for the two methods differs by an insignificant amount. The co-culture of L. plantarum and L. casei leads to the same conclusion: the best model is obtained with Levenberg-Marquardt with 8 neurons. The superiority of the ANN models is proved by the 100 times smaller mean squared error in almost every case.

The optimum process conditions can be found using a standard optimization algorithm for each developed model. For the fermentation process with L. plantarum, the optimum value of the final product concentration could be 10.3076 log CFU/mL, obtained for 97.7083% concentration of WF and 19 h of fermentation. In the case of L. casei bacteria, the optimum value is 9.7571 log CFU/mL, with a concentration of 95.625% WF in 21 h. The case of co-culture of these two bacteria leads to the best results. The optimum value is 12.2755 log CFU/mL, with 97.7083% WF in 19 h. These results are of great interest for such fermentation processes. Using co-cultures of different bacteria can improve the process performances, as proved in this research.