### A1. *In vitro* experiment for identification of model parameters

A schematic of the experiment is shown in Figure 6. The experiments were performed for a pulsatile pump (T-PLS; BHK Inc., Seoul, Republic of Korea) and a non-pulsatile pump (AK95 roller pump; Gambro Inc., Hechingen, Germany) under various flow rate conditions from 100 to 400 mL/min in increments of 50 mL/min. Figure 7 shows the pressure waveforms at the dialyser inlet generated by the two pumps. The AK95 pump generated a very weak pulsatile flow that can be considered as almost non-pulsatile flow. Each test was performed using bovine whole blood (heparin of 16 000 IU/L, Hct = 31%, T = 37°C). In the dialysate circuit, we used the AK95 roller pump at a constant flow rate of 300 mL/min for all test cases. A polysulphone dialyser (FX60; FMC Inc., Frankfurt, Germany) was used with normal saline as the dialysis solution. An ultrasonic flow sensor (T109; Transonic Systems Inc., Ithaca, NY) was placed in the blood circuit before the dialyser to measure the flow rate. Four pressure sensors (Pressure transducer; Sensys Inc., Seoul, Republic of Korea) were inserted before and after the dialyser in the blood and dialysate circuits. In the experiment, according to the pump type and flow rate, we measured the four pressures (at the dialyser inlet and outlet of the blood circuit, and at the dialyser inlet and outlet of the dialysate circuit) and UF coefficient of the dialyser. Before starting the test, we recirculated bovine blood through the dialyser at 200 mL/min for 2 h to develop a sufficient protein layer as suggested in a previous study [13].

The goal of the *in vitro* experiment was to establish reference data for the variation in TMPm and UF coefficient according to pump type and flow rate. Thus, the parameters such as TMPm and the UF coefficient for a specific condition of flow rate and pump type can be obtained by interpolating the reference data.

### A2. Governing equations of the mathematical model

The time derivatives of the intracellular fluid (*V*
_{
ic
}), interstitial fluid (*V*
_{
is
}) and plasma (*V*
_{
pl
}) can be expressed as

where *k*
_{
f
}is the osmotic filtration coefficient at the cellular membrane, *O*
_{
ic
}and *O*
_{
is
}are the osmotic concentrations of the intracellular and interstitial compartments, respectively, *F*
_{
a
}is fluid filtration rate at the arterial capillaries, *R*
_{
v
}is the fluid reabsorption rate at the venous capillaries and *Q*
_{
f
}is the UF rate.

The time derivatives of the solute concentrations in the intracellular (*M*
_{
s,ic
}) and extracellular (*M*
_{
s,ex
}) compartments are expressed as follows:

where *η*
_{
s
}is the mass transfer coefficient of solute *s* at the cellular membrane, *β*
_{
s
}is the equilibrium ratio of solute *s*, *C*
_{
s,ic
}and *C*
_{
s,ex
}are the intracellular and extracellular concentrations of solute *s*, respectively, *G*
_{
s
}is the generation rate of solute *s* and *J*
_{
s
}is the solute transfer rate through the dialyser membrane. We assumed *G*
_{
urea
}to be 6.24 mg/min [2], and *J*
_{
s
}is derived below:

where *α*
_{
s
}is the Gibbs-Donnan equilibrium ratio of solute *s*, *r* is the plasma water fraction and *K*
_{
s
}is the clearance of solute *s.* HDF or HFx treatments use simultaneous diffusion and convection for toxin removal. Thus, toxin clearance is calculated as the sum of diffusive clearance (*Kd*
_{
s
}) and convective clearance (*Kc*
_{
s
}) as follows:

The diffusive clearance is expressed as

where *KoA*
_{
s
}is the diffusive mass transfer coefficient of the dialyser for solute *s* and *Q*
_{
b
}and *Q*
_{
d
}are the blood and dialysate flow rates through the dialyser, respectively. The convective clearance is expressed as

where *Si*
_{
s
}is the membrane sieving coefficient of solute *s. T*, termed the transmittance, represents the mL/min increase in clearance for each mL/min of filtration and can be calculated as follows:

The HF treatment uses only convection for toxin removal, and thus the convective clearance is simply calculated as follows:

More detailed equations were described previously by Ursino *et al*. [6] and Depner and Garred [2].

### A3. Derivation of the internal UF rate for the HFx treatment

To calculate the internal UF rate in the HFx treatment model, we introduce the UF coefficient per unit length, which is determined by dividing the UF coefficient by the dialyser length as follows:

where *K*
_{
ufl
}is the UF coefficient per unit length and *L* is the dialyser length. As a simplified approximation, the local pressure distributions along the dialyser fibres in the blood and dialysate circuits are described as follows:

where *x* indicates the local position along the dialyser, *P*
_{
b
}(*x*) and *P*
_{
d
}(*x*) are the mean pressures at local position *x* in the blood circuit and dialysate circuit, respectively. The UF rate, which is the primary determinant of the convective clearance of the toxin, is calculated as follows:

where *M* is a local point where neither UF nor backfiltration is generated. The internal UF rate is directly related to the convective toxin clearance in the HFx treatment, and therefore, the convective clearance for the HFx treatment is calculated using Eq. (A9) in the Appendix.

### A4. Definition of criterion indices

The time-averaged concentration of solute *s* (*TAC*
_{
s
}) is calculated by integrating the concentration profile of the solute in the plasma compartments with respect to time as follows:

Equivalent renal clearance (*EKR*
_{
s
}) is calculated as follows:

*EKRc*
_{
s
}is the corrected value of *EKR*
_{
s
}for the normalised water volume, 40 L, as follows:

where *V* is the total body fluid.

The mean pre-treatment concentration of solute *s* (*MPC*
_{
s
}) is acquired by averaging the pre-treatment concentrations in the plasma compartment at steady state, and the weekly std *Kt/V* is then calculated as follows:

where *t* is the total time. All equations are adapted from Ursino *et al*. [6] and Depner and Garred [2].