## 3 Results and Discussion
### Mathematical Models  
Mathematical models have already been applied in many areas of food technology and various processes, from microbiological to physical-chemical control. The combination of methods and processes, such as the use of bioimpedance and mathematical modelling, has been researched for various purposes, mainly to verify the change in physical-chemicalaspects in food, as in the case of milk adulteration. [PERSON] et al. [18] have proposed statistical models based on conductance and phase angle; i.e., electrical parameters, to assess changes due to milk adulteration with ethanol, sodium chloride, and sodium bicarbonate. According to the authors, the research aimed to establish a correlation between the results of the electrical measurement and the added substances. Table 5 presents the statistical parameters of the proposed mathematical models from the experimental measurements for the frequencies \(10^{1}\), \(10^{3}\), and \(10^{6}\) Hz, i.e., low, medium, and high frequency and Equations (3)-(5) are, respectively, the mathematical models to predict the impedance magnitude as a function of temperature (\(T\)) and dilution (\(D\)). Figure 7 presents the regression surfaces of the models. The possibility of using mathematical models is important because if there is a correlation of factors, a methodology can be proposed to identify adulteration by these substances from these models [18]. \[ \begin{split}&|Z_{L}|~{}=~{}414.73-12.0732\cdot T-4.81235\cdot D+0.149 0\cdot T^{2}+0.434274\cdot D^{2}\\ &~{}~{}~{}~{}-0.0092\cdot D^{3}+7.04196\times 10^{-5}\cdot D^{4}-1. 79283\times 10^{-7}\cdot T\cdot D^{4}\\ &|Z_{L}|~{}=~{}298.425-8.38551\cdot T-4.74573\cdot D+0.103552\cdot T ^{2}+0.413078\cdot D^{2}\\ &-0.0086\cdot D^{3}+6.47784\times 10^{-5}\cdot D^{4}-1.44913 \times 10^{-7}\cdot T\cdot D^{4}\\ &|Z_{L}|~{}=~{}233.488-6.47223\cdot T-3.50796\cdot D+0.0750184 \cdot T^{2}+0.30982\cdot D^{2}\\ &-0.0065\cdot D^{3}+5.02774\times 10^{-5}\cdot D^{4}-1.31422 \times 10^{-7}\cdot T\cdot D^{4}\\ \end{split}  \tag{3}\]  
It should be noted that the equation models for \(10^{1}\) Hz, \(10^{3}\) Hz, and \(10^{6}\) Hz were similar to the significance of the terms and the interactions between the variables. In all of them, the linear, quadratic, cubic, and fourth-order factors of dilution (% water) have statistical significance (\(p<0.05\)), as well as the temperature (T) \(\times\) dilution (D) interaction for the analysis of milk adulterated with water. However, the high-order temperature terms showed no significance in any of the models, leaving only their linear effect. The adjustment of impedance values in the regression models resulted in a coefficient of determination (\(R^{2}\)) of approximately 0.99 for all models. The same value (0.99) was found for the adjusted \(R^{2}\). [PERSON] et al. [18] presented a coefficient of determination for their models with values greater than 0.94 for milk adulterated with ethanol, sodium chloride, and sodium bicarbonate. The value of \(R^{2}\) indicates the ratio between the variation explained and the total variation, representing a measure of the degree of adjustment [47], and values close to unity imply that most of the variability in the magnitude of \(Z\) is explained by the regression model, which indicates that the model has a good degree of adjustment [37]. Given this, the model showed a good fit, which can be seen in the regression surface graph (Figure 11). The probability values were lower than 0.05, indicating that the terms of the model have significant effects on the impedance response variable, i.e., the model is adequate at the 95% confidence level (Table 5).