## 3 Results and Discussion
### Protein-Bound Toxins Removal  
Following validation of the model for creatinine by an ex-vivo study, a parameter study has been performed for the investigation of the removal efficiency of the PBTs under AC loading in  
Figure 4: Comparison of numerically calculated removal of creatinine with the experimental dynamic adsorption-diffusion experiment for a 4-h treatment. Experimental data were extracted from [30]. Red and grey curves are obtained by utilizing Langmuir constants from [PERSON] et al. [22]; the black dashed curve is obtained by a fitting procedure.  
the membrane. Due to the fact that the specific association and dissociation rate constants were not available in the literature, the toxins' apparent dissociation rate constant was considered as a fitting parameter.  
Using the mathematical model and the optimization routine previously described, we adopted the Langmuir adsorption constants that were calculated in the previous section. The error function depends on the model parameters and was represented by an n-dimensional surface, the minimum of which was sought. In order to ensure that a given set of \"optimal\" model parameters represented the true global minimum of the error surface and not a local minimum, we varied the initial guesses and compared the resulting model for several parameters. Surface plots of the relationship between \(q_{toxin,0t}\), \(Kl_{toxin,i}\) and the relative error regarding the experimental data for several apparent dissociation rates were produced.  
It was observed that the dissociation rate constant value affected drastically the removal and the error function value. This was attributed to the fact that for low values of the dissociation rate constant, the concentration gradient to drive the toxin transport through the membrane was weak, and the values of \(q_{toxin,0i}\), \(Kl_{toxin,i}\) became irrelevant as the sorption was negligible. On the other hand, high values of the dissociation rate constant increased the concentration gradient significantly, resulting in a total toxin removal that overestimated the experimental data to a great extent. The values of \(k_{dis,IS,blood}\) = 0.7278 s\({}^{-1}\) and \(k_{dis,PCS,blood}\) = 1.81 s\({}^{-1}\) provided the most appropriate concentration gradient for the removal of both toxins. The surface plots for the best-fitted apparent dissociation rate for IS and PCS are shown in Figure 5a,b, respectively.  
The optimal parameter combinations are located in the valleys of both surfaces with an average error of 4.4%. The final results for each toxin are shown in Table 2. The parameter set for IS gives an error value of 4.8%, while the parameter value set for PCS gives an error value of 3.9%.  
In Figure 6a,b, the total removal curve for the optimal combination of parameters is shown for each toxin and compared to the experimental data in [30]. The curve fit is satisfactory, and the final toxin removal for both toxins was accurately reproduced.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Toxin** & \(k_{dis,i}\) **(1/s)** & \(q_{toxin,oi}\) **(mol/KgAC)** & \(Kl_{toxin,oi}\) **(m3/mol)** \\ \hline IS & 0.73 & 0.073 & 41 \\ PCS & 1.82 & 0.069 & 25 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Effective transport properties of the membrane layers.  
Figure 5: (a) Error function surface for IS removal for fixed \(k_{dis,IS,load}\) at 0.72875 s\({}^{-1}\); (b) Error function surface for PCS removal for fixed \(k_{dis,PCS,blood}\) at 1.81 s\({}^{-1}\).