## 2 Materials and Methods
### Mass Transport  
#### 2.3.1 The Blood-Sided Domain (DOM1)  
On the blood tube side (DOM1), convective and diffusive mass transport of a solution of toxins with the proteins is active. Although the toxins permeate the membrane, this is not the case for the proteins that are rejected from the membrane due to the selective-layer cut-off. The interaction of the toxins and the albumin in the domain is described by an effective reaction rate term in the transport equation. The corresponding transport equation for each chemical species on the blood side is as follows:  
\[\frac{\partial C_{protein,blood}}{\partial t}-\
abla\cdot\left(D_{bulk, protein,blood}\
abla C_{protein,blood}\right)+\mathbf{v}\cdot\
abla C_{protein,blood}=r_{A_{toxin,blood}}, \tag{5}\]  
\[\frac{\partial C_{complex,blood}}{\partial t}-\
abla\cdot\left(D_{bulk, complex,blood}\
abla C_{complex,blood}\right)+\mathbf{v}\cdot\
abla C_{complex,blood}=-r_{A_{toxin,blood}}, \tag{6}\]  
\[\frac{\partial C_{toxin,blood}}{\partial t}-\
abla\cdot\left(D_{bulk,toxin, blood}\
abla C_{toxin,blood}\right)+\mathbf{v}\cdot\
abla C_{toxin,blood}=r_{A_{toxin,blood}}, \tag{7}\]  
where \(r_{A_{i}}\) is the reversible, bimolecular binding reaction rate of the protein-bound _toxin_ to albumin and the formation of the protein-bound toxin-albumin complex. For most toxins, the molar ratio to albumin is typically low, resulting in an excess of albumin binding sites. In this case, the influence of albumin concentration on the free fraction depends mainly on the binding constant [16]. However, the concentration of those toxins in patients with advanced chronic kidney disease becomes relevant, due to their elevated levels, and results in reduced protein binding caused by the competitive toxin binding on albumin available sites. The identification of the binding sites of PCS and IS on HSA has been extracted with the site marker displacement experiments [17], showing that numerous uremic toxins bind specifically to the [PERSON]'s sites I in subdomain 2A and II of human serum albumin [18]. The binding procedure was investigated and explained thoroughly by [PERSON] [4], where it was found that there is a high affinity of the PBTs towards the [PERSON]'s site II in subdomain IIIA on HSA [5], and that there are low-affinity sites for both toxins on HSA. The concentration equilibrium association constant of the PCS and the IS with human serum albumin was determined to be \(K_{eq,PCS}=1\times 10^{5}\) M\({}^{-1}\)[4] and \(K_{eq,IS}=0.98\times 10^{2}\) M\({}^{-1}\)[4], respectively, for the high-affinity sites. The low-affinity sites exhibit much lower binding constants for both toxins, such as \(K_{eq,2,PCS}=8\times 10^{3}\) M\({}^{-1}\) and \(K_{eq,2,IS}=19.6\times 10^{3}\) M\({}^{-1}\)[4]. Moreover, the binding affinity of PCS for HSA has been investigated by [PERSON] [6], who reported a moderate affinity for PCS on HSA that translated to a binding constant of \(K_{eq,PCS}=3.3\times 10^{2}\) M\({}^{-1}\). From the literature review, it is observed that there is a discrepancy in the affinity constants of the toxins on HSA; however, due to the fact that the affinity constants of the toxins are directly correlated to the toxins' bound and unbound fractions, a closer inspection of the fractions may facilitate the indirect extraction of the affinity constants. In another study [6], the measured bound percentage of PCS ranged from 13% to 30%. The bound fraction of various toxins in the human plasma is reported in the work of [PERSON] [19], where the bound fraction of the total indoxyl sulfate is 97.7 \(\pm\) 0.2, while the total bound fraction of PCS is 95.1 \(\pm\) 0.6. The bound fraction of the PBT spiked with uremic concentrations of 40 mg/L, yielding the values of 88.7% and 92% for IS and PCS, respectively. Those findings set the concentration equilibrium association constant of IS and PCS with HSA to \(K_{eq,IS}\) = 0.34 \(\times\) 10\({}^{2}\) M\({}^{-1}\) and \(K_{eq,PCS}\) = 0.54 \(\times\) 10\({}^{2}\) M\({}^{-1}\), respectively.