## 3 Results
The low slope condition also occurs when the maximum adsorption capacity is small relative to the slope of the diagonal, as is often the case for the data investigated here and in much of the cited literature [7; 10; 11; 16; 17; 34; 36; 38]. Thus, the use of traditional non-linear regression is justified in those settings. Finally, after fitting of the traditional non-linear and linear regressions, and the improved non-linear regression for the Langmuir, Dubinin-Radushkevich, and Hill isotherms, the models, their parameters, and errors were tabulated and shown in Table 3 and in Figures 3-5. Figure 4: Langmuir, Dubinin-Radushkevich, and Hill isotherms of quercetin derivatives adsorption onto \(\beta\)-glucan obtained with novel improved non-linear regression (with reported \(\beta_{\text{\tiny{6-model}}}\) at the lowest \(c_{\text{\tiny{c}}}\) value) (the confidence intervals are \(\pm 1\) standard errors of the model fits). All three models of adsorption isotherms could be applied to experimental data with these three types of modeling (Figures 3-5). According to the model standard errors, usually the Langmuir models fit a little bit better to the experimental data (errors in non-linear modeling 0.248-0.330, in improved non-linear modeling 0.235-0.324, in linear modeling 0.156-0.410) (Table 3). This is similar to earlier studies [39]. However, errors in the Dubinin-Radushkevich (non-linear 0.246-0.339, improved non-linear 0.232-0.321, linear 0.256-0.360) and Hill models (non-linear 0.253-0.335, improved non-linear 0.240-0.331, linear 0.271-0.499) are similar (Table 3). This allow us to suggest that the parameters of all three models could be used in theoretical description of the adsorption process. Earlier studies of adsorption of polyphenols onto resins [40] or apple polyphenols onto apple cell walls [39] were also described by several adsorption isotherms, like in this study. Furthermore, comparing three different ways to model (non-linear, improved non-linear and linear regression), it can be noticed that improved non-linear regression gave models with the lower error ([PERSON] 0.235-0.324, [PERSON] 0.232-0.321, [PERSON] 0.240-0.331) in comparison to standard non-linear ([PERSON] 0.248-0.330, [PERSON] 0.246-0.339, Hill 0.253-0.335) or linear ([PERSON] 0.156-0.410, [PERSON] 0.256-0.360, [PERSON] 0.271-0.499) (Table 3). Therefore, this way of modelling potentially can give more accurate and precise models with more accurately determined model parameters. To check the models, the values of \(q_{\rm e}\) were calculated by using models obtained with improved non-linear modeling and the results \(q_{\rm e,calc}\) are shown in Table 2. The \(q_{\rm e,calc}\) gave relatively good approximation of the measured \(q_{\rm e}\). Parameters obtained from these adsorption isotherms may theoretically be used to give some insight into the mechanism of the adsorption process, and, in our case, into the mechanism of interactions between quercetin derivatives and the dietary fiber, \(\beta\)-glucan. In particular, it is natural for the adsorption process to be described according to parameters obtained with the improved non-linear regression (Table 3) since improved fitting showed the lowest error. It should also be said that the parameter values are apparent and theoretical, but they allowed us to at least suggest the adsorption mechanism. According to the apparent, theoretical adsorption capacity (\(q_{\rm m}\)) from Langmuir and Hill isotherm or (\(q_{\rm s}\)) from Dubinin-Radushkevich isotherm, we can suggest which derivatives adsorbed more onto \(\beta\)-glucan or which have more tendency to adsorb. Glucoside and rhamnoside of quercetin could theoretically adsorb in higher amount than quercetin-3-galactoside (Table 3), which is similar to experimental results (Table 2). Furthermore, according to the mean free energy of adsorption (\(E\)), the adsorption could be a physical process (since E \(<\) 8000 J/mol) with creation of physical bonds between quercetin derivatives and \(\beta\)-glucan, such as Van der Waals bonds or H bonds.