## 3 Results and Discussion
### Effect of Carbon Loading on Toxin Removal  
The correlation of the membrane activated carbon loading to the total protein-bound toxin removal was studied next. It is important to note that particle loading is an important parameter, affecting not only membrane morphology, but also the number of adsorptive sites in the mixed-matrix membrane per unit volume. Assuming the negligible effect of the particle loading on the transport properties of the membrane, the total protein-bound toxin removal per fiber was estimated for IS (Figure 7a) and for PCS (Figure 7b), respectively.  
The results show a strong positive linear relationship between the activated carbon loading in the membrane and the protein-bound-toxin removal. The y-intercept in Figure 7a (that is, at zero particle loading) is 1.7 \(\times\) 10\({}^{-7}\) mol/fiber, whereas the slope reveals that 2.1 \(\times\) 10\({}^{-10}\) mol of IS are removed in each fiber per kg/m\({}^{3}\) of AC loading. In Figure 7b, the y-intercept is 2.28 \(\times\) 10\({}^{-7}\) mol/fiber with the slope indicating that 1.63 \(\times\) 10\({}^{-10}\) mol of PCS are removed in each fiber per kg/m\({}^{3}\) of AC loading. The simulations show that removal of both toxins is greatly enhanced by the increase of particle loading in the membrane matrix. Comparison of the cases with zero carbon loading and with 411.13 kg/m\({}^{3}\) carbon loading (corresponding to the loading of the MMM, reported in ref [30]) shows that the adsorptive layer achieves a 52% improvement in IS removal, and a 31% improvement in PCS  
Figure 6: (**a**) Total removal curve for IS as predicted by the present model, compared with experimental data found in the literature [30] **(b**) Total removal curve for PCS as predicted by the present model, compared with experimental data found in the literature [30].  
Figure 7: (**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 \(k_{dis, PCS,load}\) at 1.81 s\({}^{-1}\).  
removal in comparison to diffusion alone. In addition, our model seems to estimate well the values for the removal of the PBTs solely by diffusion with commercial membranes. In fact, the Fresenius F8 HPS(r) (Fresenius Medical Care, Bad Homburg, Germany), removed 187 mg/m\({}^{2}\) of IS and 225 mg/m\({}^{2}\) of PCS, while the predicted removal for the M6 membrane was 240 mg/m\({}^{2}\) and 300 mg/m\({}^{2}\), respectively [30]. Considering that the MMM had a three times lower ultrafiltration coefficient than the F8 HPS, we can confirm the finding from the literature that there are no significant differences between the low-flux and the high-flux membranes [19] in the clearance of the PBTs in the absence of activated carbon in their matrices.  
In this fashion, the additional driving force attributed to the presence of the AC sorbents can be quantified and its extent evaluated. Moreover, the dependence of protein and toxin concentrations on spatial coordinates and time can also be estimated. In this way, the resulting apparent dissociation rates can be used in order to estimate the free fraction of the protein-bound toxins in a hemodialysis process accounting for toxin dissociation from the albumin.