## 4 Discussion  
adsorption process can be helpful in studying the interactions between bioactive compounds like polyphenols and food constituents such as dietary fiber. This can be seen in earlier studies [16; 17; 34; 36; 37]. Adsorption can be modeled with various adsorption isotherms. Particular adsorption isotherms, Langmuir and Dubinin-Radushkevich, are chosen for this study due to their use in earlier studies. Namely, the applicability of Langmuir models for adsorption of polyphenols onto \(\beta\)-glucan [16; 17], polyphenols from apples onto resin [38], apple procyanidins onto apple cell wall material [34; 36; 37], various polyphenols onto cellulose and xylan [9], various polyphenols onto cellulose [10] has been reported. Then Dubinin-Radushkevich isotherm was used to model the adsorption of polyphenols onto tannery shavings [18]. In addition, we also chose to use Hill adsorption isotherm. adsorption isotherms could give some theoretical information about adsorption process and finally, about interactions. Namely, the Langmuir model describes a monolayer adsorption at the surface of the adsorbent and when the molecules occupy the specific site on the surface, no further adsorption on that place is possible [17; 18; 41]. Furthermore, it assumes that the adsorbent has a limited adsorption capacity and that all binding sites at adsorbent have the same affinity toward molecules from solution [41]. The Dubinin-Radushkevich model describes an adsorption process on a heterogeneous surface and it can be used to calculate mean free energy to distinguish between physical and chemical adsorption [28]. Hill isotherm can describe the adsorption of different molecules onto  
Figure 6: Chemical structures of quercetin derivatives. some adsorbents and help to describe the differences between molecules in terms of cooperative or non-cooperative bonding [28]. To obtain information from isotherms and their parameters, isotherms must be properly fitted to experimental results. The fitting of experimental results (\(q_{\text{e}}\)_vs._\(c_{\text{e}}\)) to isotherms often depends on the modeling methodology. Proper modeling can give lower error between experimental results (\(q_{\text{e}}\)) and the fitted adsorption isotherms (\(q_{\text{e,model}}\)). If results are fitted with lower errors, the \(q_{\text{e,model}}\) values will be closer to experimental values and adsorption can be better interpreted. Quality fitting and interpretation of adsorption can be achieved by non-linear and linear regression. In some papers, the adsorption data were analyzed with non-linear regression [7; 9; 10; 17; 34; 36; 37] and in some with the use of linear regression [18]. Between those two types of modeling, non-linear has shown advantages, since linear fitting have already shown many constrictions [28]. That is why the use of non-linear fitting is more common [28]. A potential difficulty with the linearized forms is that the transformed forms of \(c_{e}\) and \(q_{e}\) do not retain the structure of variability of the original forms, and consequently there can be a loss of statistical efficiency with the use of the least squares procedure with the linearized forms. This can be observed in larger errors in the fitted parameters and the fitted isotherm curves. We investigated and compared both types of modeling, non-linear and linear. To improve the fitting and to get lower error and more reliable isotherms and their parameters, we also developed and used an improved non-linear modeling for the first time. Novel, improved non-linear fitting gave lower errors (Table 3), as theoretically predicted (Figure 2). Namely, the theory predicted that the error of the model should be lower if the fitted \(q_{\text{e,model}}\) value is on the \(-45\) degrees line corresponding to the conservation of mass. Indeed, the improved non-linear fitting predicts the \(q_{\text{e,model}}\) value on this diagonal line and does give lower error (Table 3).