## 3 Results and Discussion
### Kinetics
#### 3.5.1 Kinetics in the Simulated Digestion Process  
Simulated digestion processes can have a different behavior. A concentration of a compound can increase with time as the compound is released from the food matrix in the digestive tract or created from some bigger molecule as a degradation product. In this case, the curve in the diagram \(c_{t}\) vs. \(t\) follows an increasing function. Kinetic Equations (2) and (3) can be adjusted to correspond to increasing concentrations of observed compounds. The adjustment can be seen in the replacement of the parameter \(c_{0}\) with \(c_{\infty}\), which is the concentration of the compound at the endpoint of the reaction, and \(c_{t}\) with \(c_{\infty}-c_{t}\). Now \(c_{t}\) in the digestion process is increasing with time to the endpoint where it reaches maximum \(c_{\infty}\), and, accordingly, the difference \(c_{\infty}-c_{t}\), which represents the part that still needs to be released, will decrease in time as expressed in the following equation. \[c_{\infty}-c_{t}=c_{\infty}e^{-kt} \tag{6}\]  
Rearranging this equation, we can write the equation for the first order as:  
\[c_{t}=c_{\infty}(1-e^{-kt}) \tag{7}\]  
This equation was used for the kinetic models of starch digestion [32; 33; 34; 35]. Such expressions are suitable for the setting in which the initial concentration is 0. Likewise, replacing \(c_{t}\) with \(c_{\infty}-c_{t}\) and \(c_{0}\) with \(c_{\infty}\) in Equation (3), Equation (3) for the second order becomes:  
\[c_{\infty}-c_{t}=\frac{c_{\infty}}{1+c_{\infty}kt} \tag{8}\]After rearranging Equation (8), this equation can be written as:  
\[c_{t}=c_{\infty}(1-\frac{1}{1+c_{\infty}kt}) \tag{9}\]  
Equations (7) and (9) are usually used for the simulated digestion processes to study the kinetics of these processes. 5.2 The Modification of Equations of the First and Second Order for the Gastrointestinal Digestion of Phenolic Compounds  
When studying the release of phenolic compounds in digestion, their concentration can increase or decrease. In the stomach, their concentration increases since they are liberated from the matrix of food. The \(c_{t}\) vs. \(t\) is an increasing function. We could apply Equations (7) and (9) to study the kinetics of the phenolic compound in the stomach. However, in the small intestine, phenolic compounds can be degraded or changed due to different pH, and consequently, their concentration in the natural form can decrease over time. The \(c_{t}\) vs. \(t\) can be a decreasing function. In this case, we could apply Equations (2) and (3) to study the kinetics of the phenolic compound in the small intestine. Instead, we modified Equations (2) and (3) to be able to use them in both cases when \(c_{t}\) vs. \(t\) is an increasing or decreasing function, now with an allowance of non-zero initial and final concentrations. Equation (2) describes the first order that was modified by replacing \(c_{0}\) with \(c_{0}-c_{\infty}\) and by replacing \(c_{t}\) with \(c_{t}-c_{\infty}\), which represents the part of a compound that still needs to react, either for liberation or degradation. After rearranging and expressing \(c_{t}\), Equation (2) becomes:  
\[c_{t}=(c_{0}-c_{\infty})e^{-kt}+c_{\infty} \tag{10}\]  
Likewise, Equation (3) describes the second-order was modified by replacing \(c_{0}\) with \(c_{0}-c_{\infty}\) and by replacing \(c_{t}\) with \(c_{t}-c_{\infty}\). After rearranging and expressing \(c_{t}\), we have the equation:  
\[c_{t}=\frac{c_{0}-c_{\infty}}{1+|c_{0}-c_{\infty}|kt}+c_{\infty} \tag{11}\]  
These modifications lead the curve in the diagram \(c_{t}\) vs. \(t\) from the value \(c_{0}\) at the \(t=0\), and level it toward the value \(c_{\infty}\). In addition, when \(c_{\infty}<c_{0}\), the curve is decreasing, and when \(c_{\infty}>c_{0}\), the curve is increasing. In this way, the modified Equations (10) and (11) can adjust to the experimental data that show the increase in \(c_{t}\) with time or the decrease in \(c_{t}\) in time.