Numerical and experimental investigation of solute transports through different types of dialyzer membrane

Abstract

The flow and mass transport through different membrane types from different hemodialyzers are investigated in a co-current direction to emphasize the effect of solute diffusion through the dialyzer membranes. The numerical model consists of the blood flowing in a hollow fiber surrounded by a dialysate flow, where the mass transport and fluid flow were simultaneously calculated. The high flux dialyzers considered in the present study are FINEFLUX FIX-210S eco, ELISIO-210HR, and PEPA FDY-21B, which differ mainly in characteristics of the membrane structure and surface. Urea and maltodextrin solutions are used as model solutes to consider the effect of molecular size difference. The numerically predicted outlet concentrations closely align with experimental values, where the variation between predicted and measured values remain below 10% across all dialyzer types for urea solutions, and specifically below 8% for maltodextrin solution. Among the various dialyzers tested, FINEFLUX membrane could provide the highest maltodextrin clearance (83.09 ml/min) and overall mass transfer area coefficient (KoA) (119.56 ml/min) potentially due to its fast-diffusion characteristic. In this co-current flow study, the results suggest that urea transport is primarily influenced by flow control with minimal impact from boundary layers, while maltodextrin transport is predominantly governed by diffusion control.

Article Highlights

The mass transport simulation of different hemodialysis membranes (in co-current flow) closely aligns with the experimental results.

FINEFLUX membrane could provide the highest maltodextrin clearance due to its fast-diffusion characteristic.

Urea clearance is controlled by dialysate flow; maltodextrin clearance is limited by membrane diffusion.

Introduction

Hemodialysis is the process to remove uremic toxins and fluid volume from the blood primarily through diffusion across a semipermeable membrane. It uses a dialyzer that acts as an artificial kidney for chronic kidney disease patients [1]. The dialyzer contains a large number of hollow fibers in a cylindrical dialyzer module, where each fiber wall is a membrane with different porous characteristics depending on the membrane types. Currently, conventional dialyzer membranes are made of synthetic polymers. High flux membranes are commonly used for treatment, characterized by high diffusive permeability, and increased efficiency of medium or large molecules clearances [2, 3]. The membrane characteristics affect not only the flow and mass transfer in hemodialysis, but also other important phenomena in the blood and dialysate phases [4]. In hemodialysis, toxin removal mechanism mainly involves molecular diffusion and internal filtration process to a smaller extent [5, 6]. While internal filtration occurs due to water flux across a semipermeable membrane, diffusion is induced by the concentration gradient of solutes between the blood and dialysate phases [7]. In addition to the concentration difference, the molecular sizes of solutes also determine the diffusion rate. A low molecular weight substance diffuses at a faster rate and results in a higher removal rate [8, 9].

The common term referring to the volumetric flow rate at which a specific substance is removed is called ‘clearance’. Clearance is one of the most important factors to evaluate the dialyzer membrane performance. Improving the solute clearance by diffusion was achieved by considering parameters that characterize flow condition and structure of dialyzers [8]. It was found from dialysis experiments that the increasing blood and dialysate flow rates resulted in a higher clearance due to a sustained large concentration gradient [2, 10, 11]. Further, increasing the dialyzer membrane surface area, length, and fiber radius could improve the clearance of small and middle molecules [11, 12]. Another important parameter indicating the rate of solute transport across the semipermeable membrane is the mass transfer coefficient. This parameter was usually derived using a correlation equation [13, 14]. Fukuda et al. [14] examined how internal filtration influenced the mass transfer coefficient of dialysate fluid. They employed Michaels’ equation [15] to develop a correlation for the mass transfer coefficient on the dialysate side.

Several prior studies utilized Michaels’ equation to model solute transport from the blood side across the membrane to the dialysate side [8, 13–18]. Michaels’ equation was originally developed under the assumption that the overall mass transfer coefficient remains unaffected by the flow rates of both blood and dialysate. However, subsequent investigations revealed a significant influence of dialysate flow on the overall mass transfer coefficient [10]. Furthermore, internal filtration, which further complicated the mass transfer analysis, was also not considered by Michaels’ equation [14, 19]. Although Kunikata et al. [13] proposed a mass transfer correlation equation to address this impact of dialysate flow, the influences of blood flow and internal filtration remain uncertain and require further investigation.

The Kedem-Katchalsky (K-K) model is a widely recognized framework employed to describe the transport dynamics of solutes and water across permeable membranes [23]. Despite its extensive utilization and success in numerous numerical studies, an acknowledged limitation of the K-K model lies in its assumption of negligible resistance to solute transport at the external boundary layers on the membrane sides [26, 27]. Nevertheless, empirical evidence demonstrates that the K-K equation has yielded good results with relatively small discrepancies when compared against experimental measurements [16, 17, 20–22]. This equation was used to model transport phenomena of sufficiently dilute and well-stirred solution in a single hollow-fiber dialyzer [17, 23–25]. This equation was used to calculate clearances of small-molecule solutes including urea and protein-bound uremic toxin out of a dialyzer [21]. In other works [16, 20], the fluid transport on the shell-side of hollow fiber bundles was described by Darcy’s equation, while the clearance was modeled by the Kedem-Katchalsky model.

In the present manuscript, we investigate the solute clearance induced by concentration-gradient diffusion by applying insights from fluid dynamics and diffusive-convective mass transfer. While the K-K model describes variances in membrane characteristics based on hydraulic permeability and diffusive coefficient, our adopted transport model, derived from the K-K concept, integrates factors from membrane characteristics such as membrane porosity, thickness, and other structural properties. This comprehensive approach aims to more effectively account for their collective impact on solute transport [30]. A single hollow fiber is modeled rather than considering the entire bundle of fibers within the dialyzer. This modeling approach enables us to focus on fundamental transport phenomena through an individual fiber which represents the overall dialysis process [17, 21, 28]. A mathematical model is employed to simulate both mass transfer and flow behaviors throughout a hemodialysis process. The changes in solute concentrations are numerically simulated throughout the domain. Moreover, our investigation extends beyond theoretical simulations to incorporate dialysis experiments conducted in parallel. To emulate substances in the blood and focus on individual substance, urea and maltodextrin are chosen in this study as representative substances. Although, presently, maltodextrin is considered a small middle molecule, the difference in molecular weights (M) between urea (M = 60) and maltodextrin (M = 1700) is sufficient to effectively demonstrate the impact of molecular size on mass transport through the membranes. The computed outlet concentration, clearance and the overall mass transfer area coefficient (KoA) are compared against experimentally obtained data, providing a comprehensive validation of the proposed model. Diffusion is the primary mechanism for the removal of small solutes such as urea, creatinine, and electrolytes (e.g., potassium, sodium). Specifically, in the case of uremia, which is characterized by high levels of urea and other waste products in the blood, hemodialysis effectively manages this condition through diffusion, thereby reducing toxin levels. Additionally, in advanced stages of chronic kidney disease (CKD), the kidneys’ ability to filter and remove waste products is severely compromised. In such cases, hemodialysis heavily relies on diffusion to clear these accumulated toxins from the blood stream. To examine the ability of solutes to transport through different membranes via diffusion, we minimized internal filtration through the membrane by setting up the experiment in the co-current dialysis configuration. In contrast to counter-current flow, co-current flow conditions provide more uniform pressure difference along the length of the dialyzer membrane, thereby reducing the likelihood of internal filtration. The results obtained in the present study could provide a basis for future assessment of convection in both co-current and counter-current operating conditions [29].

Methods

Numerical modeling

The dialyzer module consists of a large number of hollow fibers bundled together as depicted in Fig. 1a. In the current work, a single hollow fiber is considered for numerical simulation. The cross-section view with dimensional parameters of each fiber is shown in Fig. 1b. The dialyzers chosen in the present investigation are FINEFLUX FIX-210S eco, ELISIO-210HR, and PEPA FDY-21B. The parameters related to the dialyzer characteristics in Table 1 include the inner radius of the hollow fiber (R_HF), membrane thickness (L_M), effective length of the hollow fiber (L) and the ultrafiltration coefficient of the membrane (Kuf). All the dialyzers considered here have the effective surface area of 2.1 m². The width of the concentric dialysate channel (L_DC) can be calculated using Eq. (1). In our model development, we aimed for a comprehensive representation of membrane resistance to fluid and mass transport across the entire membrane, rather than solely focusing on skin layers. The L_M in Table 1, obtained from the manufacturers’ technical datasheets, represents the total membrane thickness. While acknowledging the complexity introduced by multiple skin layers and their potential implications for fluid and mass transport, our current study focuses on developing a model that captures general membrane behavior based on available data and assumptions. The parameters related to the solute characteristics including the solute diffusion coefficients (D) and the blood-side inlet concentration (C_{0_B}) are given in Table 2, while the inlet concentrations of the dialysate flow (C_{0_D}) is zero. In the present study, the effective diffusion coefficient of solute in membrane (D_M) accounts for the bulk transport of solute through both membrane pores and the solid structure of the membrane. The D_M depends on both the solution and the membrane type [10]. It is determined by using the pore diffusion model (PDM), which assumes straight pores, as shown in Eqs. (2) [29–31]. The PDM model makes it possible to estimate diffusive permeability of dialysis membranes from simple structure parameters such as pore radius, porosity. The average blood velocity $(U_{avg\_B})$ and the average dialysate velocity $(U_{avg\_D})$ are obtained by Eqs. (6) and (7), respectively.

1

$$L_{\mathit{DC}}=R-(R_{\mathit{HF}}+L_M)$$

2

$$D_M=D\times f(q)\times S_D\times A_k$$where

3

$$q=r_s/r_p$$

4

$$f(q)=\frac{1-2.1050q+2.0865q^3-1.7068q^5+0.72603q^6}{1-0.7587q^5}$$

5

$$S_D={(1-q)}^2$$

6

$$U_{avg\_B}=\frac{Q_B}{N\times \pi {(R_{\mathit{HF}})}^2}$$

7

$$U_{avg\_D}=\frac{Q_D}{\pi {(R_{\text{module}})}^2-N\pi {(R_{\mathit{HF}}+L_M)}^2}$$where R_HF is the radius of fibers, D is the diffusion coefficient of solute in solution, f(q) is the friction coefficient that represents the resistance of the solute molecules flowing through the membrane pore walls [32], $S_D$ is the steric hindrance factor which accounts for the resistance of molecules diffusing through the pores, A_k is the membrane porosity, q is the ratio of solute radius $r_s$ to pore radius r_p, R_module is the radius of the dialyzer module, and N is the number of the hollow fibers. The determination of the porosity value, A_k, presented a challenge due to its elusive nature and status as the only unknown variable in our calculations. Given that all other parameters were known, we employed a trial-and-error methodology to ascertain the appropriate values of A_k for both solutes. Our investigation spanned a range of A_k values from 0.1 to 0.5. Our findings indicated that, while A_k exhibited minimal impact on the predicted outlet concentration for urea, it exerted a noticeable influence in the case of maltodextrin. Consequently, we selected the value of A_k that minimized the error across both urea and maltodextrin scenarios. As expected, the optimized porosity value that minimized the error was the same for both urea and maltodextrin for each type of dialyzer.

Fig. 1 [Images not available. See PDF.]

Diagrams of a dialyzer module, and b single hollow fiber domain

Table 1. The technical data of dialyzers used

Dialyzers	FINEFLUX FIX-210S eco	ELISIO-210HR	PEPA FDY-21B
Manufacturer	NIPRO	NIPRO	NIKKISO
Membrane type	Asymmetric Triacetate (ATA)	Polyethersulfone (PES)	Polyester-Polymer Alloy (PEPA)
RHF (mm)	0.1	0.1	0.105
LM (mm)	0.025	0.04	0.03
L (mm)	254	290	290
LDC (mm)	0.049	0.046	0.051
KoA (Urea) (ml/min)	N/A	1976(1)	1010(2)
Kuf (ml/hr/mmHg)	81(3)	82(4)	64(5)
Membrane surface area (m2)	2.1	2.1	2.1
Priming volume (ml)	125	130	129

⁽¹⁾evaluated at Q_B = 300 ml/min, Q_D = 500 ml/min, Q_F = 10 ml/min

⁽²⁾evaluated at Q_B = 300 ml/min, Q_D = 500 ml/min, Q_F = 0 ml/min

⁽³⁾evaluated at Q_B = 250 ml/min

⁽⁴⁾evaluated at Q_B = 300 ml/min

⁽⁵⁾evaluated at Q_B = 200 ml/min

Table 2. The parameters related to solute characteristics and mass transport

Property	Solute		Unit	Description
	Urea	Maltodextrin
$D$	1.81 × 10–9	2.2 × 10–10	m2/s	Diffusion coefficient, solutes [33, 34]
$K$	0.99	0.99	–	Partition coefficient [2]
$C_{0\_B}$	30	0.4412	mol/m3	the blood-side inlet concentration
$U_{avg\_B}$ (QB)	13.81 (300)	13.81 (300)	mm/s (ml/min)	Average inlet velocity of blood flow (vol. flow rate)
$U_{avg\_D}$ (QD)	4.94 (162)	7.632 (250)	mm/s (ml/min)	Average inlet velocity of dialysate flow (vol. flow rate)
rs	0.24	0.695	nm	Solute radius [35]
ρ	997		kg/m3	Density
μ	0.89		mPa.s	Viscosity
T	24–26		oC	Temperature

The solute radii (r_s) for urea and maltodextrin are listed in Table 2, whereas the geometric parameters of membranes (r_p, A_k) and the computed values of diffusion coefficients DM are shown in Table 3. The blood flow rate Q_B represented by aqueous solutions was set at 300 ml/min (U_{avg_B} = 13.81 mm/s), while the dialysate flow rate Q_D was the optimized value to achieve minimum convection through the dialyzer membrane by balancing the inflow and outflow of dialysate. The outlet concentrations of solutions are numerically computed.

Table 3. The membrane parameters and the diffusion coefficient of the solutes passing through the membranes

Dialyzer	Pore radius (rp) (nm)	Membrane porosity (Ak)	Diffusion coefficient of solute through membrane (DM) (m2/s)
			Urea	Maltodextrin
FINEFLUX	360	0.4	7.22 × 10–10	8.73 × 10–11
ELISIO	460	0.15	2.71 × 10–10	3.28 × 10–11
PEPA	485	0.25	4.52 × 10–10	5.47 × 10–11

A single hollow fiber is implemented for the numerical simulation. The blood represented by urea or maltodextrin solutions flows through the lumen (inner cavity) of the fiber, while the dialysate (RO water) flows outside. The model domain, consisting of three separated phases: blood, membrane, and dialysate with flow directions, is shown in Fig. 2.

Fig. 2 [Images not available. See PDF.]

Schematic diagram of the computational domain

The hollow fiber is modeled as a 2D axisymmetric domain, where the mass transport and the fluid flow are simultaneously calculated. Steady incompressible laminar Newtonian flows are assumed. We employ constant fluid properties, where the fluid density and viscosity are assumed to be equivalent to those of water for diluted solutions as in our case [12, 16, 18, 21]. Moreover, the sieving effect is considered negligible in our scenarios, where convective transport, such as internal filtration across a membrane, is absent [19]. The model for mass transport of diluted species, as represented by Eq. (8), incorporates both diffusion and convection terms, where J_i denotes the diffusive flux vector of species i. The solute transport in the blood and dialysate phases is driven by convection and diffusion processes, whereas within the membrane, it is solely driven by diffusion. Consequently, in the membrane domain, the second term of Eq. (8) is neglected. The single-phase laminar flows of the blood and dialysate phases are governed by Eqs. (10) and (11) to compute the velocity field. The fluid flow is described by the continuity and Navier–Stokes equations, which are expressed in the vector form.

8

$$\mathrm{\nabla }·{\mathbf{J}}_i+\mathbf{u}·\mathrm{\nabla }·c_i=0$$which

9

$${\mathbf{J}}_i=-D_i\mathrm{\nabla }{c}_i$$

10

$$\mathrm{\nabla }·u=0$$

11

$$\rho \left(\mathbf{u}·\mathrm{\nabla }\right)\mathbf{u}=\mathrm{\nabla }·\left[-p\mathbf{I}+\mathbf{K}\right]$$where

12

$$\mathbf{K}=\mu (\mathrm{\nabla }\mathbf{u}+{(\mathrm{\nabla }\mathbf{u})}^T)$$

J_i is the diffusive flux vector ($mol/(m^2·s)$), u is the velocity vector (m/s), $c_i$ is the concentration of species i ($mol/m^3$), D_i is the diffusion coefficient of species i $(m^2/s)$, $\rho$ is the density ($kg/m^3$), p is the pressure (Pa), K is the viscous stress tensor (Pa), $\mu$ is the dynamic viscosity ($Pa·s$).

The domain boundary conditions are modeled as shown in Fig. 3 consisting of the transport of diluted species and single-phase laminar flow conditions. The axial symmetry boundary is imposed, where the flows are not allowed to cross a symmetrical line ($\mathbf{u}·\mathbf{n}=0$) with no concentration gradient $\mathrm{\nabla }{c}_i=0$ of species across this center line. Solute concentrations at the inlet boundaries ($c_{0,i}$) are constant as specified by the Flux inflow boundary condition ($\mathbf{n}·({\mathbf{J}}_i+\mathbf{u}{c}_i)=\mathbf{n}·(\mathbf{u}{c}_{0,i})$) with no concentration gradient at the outlet boundaries ($\mathbf{n}·D_i\mathrm{\nabla }{c}_i=0$). No flux condition ($-\mathbf{n}·{\mathbf{J}}_i=0$) is imposed at the outer boundary and at the membrane boundaries. While this condition prohibits any diffusive transport of solute across the boundary, it does not prevent solute movement parallel to the boundary. Thus, within the confines of the boundary conditions, solute can still redistribute along the surfaces of the boundary without crossing it.

Fig. 3 [Images not available. See PDF.]

Descriptions of boundary conditions

At the blood-membrane and the dialysate-membrane interfaces, the partition coefficient of solute species i (K_i) is defined as the ratio of the bulk solute concentration in the membrane phase (c_i,u) to that in the liquid phase (blood and dialysate) (c_i,d) ${\text{K}}_{\text{i}}=\frac{{\text{c}}_{\text{i,u}}}{{\text{c}}_{\text{i,d}}}$. The value of c_i,u is lower than c_i,d due to the decrease in the occupied liquid volume in the membrane phase caused by the presence of solid membrane tissue (partition effect). The applied constant diffusive flux at interface (${\mathbf{J}}_{i,u}={\mathbf{J}}_{i,d}$) implies that the ratio of a solute concentration in two phases under an equilibrium state can be determined from K_i [36].

The constant velocity in the flow direction is applied at the inlet of the blood and dialysate flows with no flow in the transverse direction: $u·t=0$ (t is a unit tangent vector of the boundary), while the pressure at the outlet is set to be an atmospheric pressure. No-slip condition is imposed at the walls, meaning that the fluid velocities at all walls are zero.

The overall mass transfer area coefficient (KoA) is derived from the concept of mass balance through a dialysis process as [14]

13

$${\text{Q}}_{\text{B}}\left({\text{C}}_{\text{BI}},-,{\text{C}}_{\text{BO}}\right)={\text{K}}_{\text{o}}\text{A}{(\mathrm{\Delta }\text{C})}_{\text{av}}$$

For a co-current flow operating condition, the average concentration difference ${(\mathrm{\Delta }C)}_{\mathit{av}}$ through the process is given by

14

$${\left(\mathrm{\Delta },C\right)}_{\mathit{av}}=\frac{\left({\text{C}}_{\text{BI}},-,{\text{C}}_{\text{DI}}\right)-\left({\text{C}}_{\text{BO}},-,{\text{C}}_{\text{DO}}\right)}{\text{ln}\left[\frac{\left({\text{C}}_{\text{BI}},-,{\text{C}}_{\text{DI}}\right)}{\left({\text{C}}_{\text{BO}},-,{\text{C}}_{\text{DO}}\right)}\right]}$$

Therefore, KoA is

15

$$\text{KoA}=\frac{{\text{Q}}_{\text{B}}({\text{C}}_{\text{BI}}-{\text{C}}_{\text{BO}})}{\left({\text{C}}_{\text{BI}},-,{\text{C}}_{\text{DI}}\right)-({\text{C}}_{\text{BO}}-{\text{C}}_{\text{DO}})}\text{In}[\frac{\left({\text{C}}_{\text{BI}},-,{\text{C}}_{\text{DI}}\right)}{\left({\text{C}}_{\text{BO}},-,{\text{C}}_{\text{DO}}\right)}]$$where Q_B is blood flow rate (ml/min). C_BI is inlet blood concentration (mol/m³). C_BO is outlet blood concentration (mol/m³). C_DI is inlet dialysate concentration (mol/m³). C_DC is outlet dialysate concentration (mol/m³).

Experimental implementation

In the dialysis experiment, the aqueous urea and maltodextrin solutions were chosen as test solutions to simulate contaminants in the blood, while RO water was used as a dialysate. The close-up images of the inner surfaces, the outer surfaces, and the cross-sectional planes of the dialyzer membranes used in this study were taken by the scanning electron microscope (SEM) at 3000× magnification as depicted in Fig. 4. The experiments were conducted at ambient temperature, where the urea solution and RO water flew through a dialyzer in the same direction (co-current flow) to avoid internal filtration across the membrane [17]. An actual picture and the schematic drawing of the experimental setup are shown in Fig. 5a and b, respectively, where the flows of solutions are driven and regulated by peristaltic pumps. The precision of the flow sensors and the pressure sensors is $\pm$ 1% of Rdg (Displayed value) [37] and $\pm$ 1.0% of F.S. (Full scale) or less [38], respectively.

Fig. 4 [Images not available. See PDF.]

The SEM images of the inner, outer, and cross-sectional membrane surfaces of a FINEFLUX FIX-210S eco, b ELISIO-210HR, and c PEPA FDY-21B

Fig. 5 [Images not available. See PDF.]

Experiment setup: a actual equipment b Schematic representation

The flow rate of the aqueous solutions was set at 300 ml/min (13.81 mm/s). The dialysate flow rate was adjusted using the flow meter to optimize it for minimal convection through the dialyzer membrane, achieved by balancing the inflow and outflow of dialysate. This equilibrium resulted in nearly equal pressures, which were monitored, on both the blood and dialysate sides. The optimized values of the dialysate flow rate matched those utilized in the simulations presented in Table 2. The outlet solutions (both the aqueous solution and the dialysate flows) were collected after a steady state is reached. The urea concentration was measured by the same method used to analyze blood urea nitrogen (BUN), so called a UV enzymatic method for BUN analysis [39]. The concentration of maltodextrin was quantified by measuring the absorbance of the iodine-maltodextrin complex at a wavelength of 540 nm using a UV–VIS spectrophotometer. A calibration curve was created to establish a relationship between absorbance values and known concentrations of maltodextrin. Each experiment was repeated three times to ensure reliability. The average values of outlet concentration ± SD for urea and maltodextrin obtained experimentally (Exp.) are reported in Tables 4 and 5, respectively.

Table 4. The urea outlet concentrations, clearances and KoAs obtained from the experiments (Exp.) and the numerical simulation (Numer.)

Dialyzers	Outlet concentrations (mol/m3)			Clearance (ml/min)			KoA (ml/min)
	Exp	Numer	Diff. (%)	Exp	Numer	Diff. (%)	Exp	Numer	Diff. (%)
FINEFLUX	21.74 ± 4.19	19.69	− 9.43	87.62	103.12	17.69	142.15	192.47	35.40
ELISIO	20.40 ± 2.20	19.53	− 4.28	98.95	104.70	5.80	178.13	199.11	11.78
PEPA	20.14 ± 3.66	19.60	− 2.71	101.57	104.30	2.42	188.12	197.40	4.93

Table 5. The maltodextrin outlet concentrations, clearance and KoA obtained from the experiment and the numerical simulation

Dialyzers	Outlet concentrations (mol/m3)			Clearance (ml/min)			KoA (ml/min)
	Exp	Numer	Diff. (%)	Exp	Numer	Diff. (%)	Exp	Numer	Diff. (%)
FINEFLUX	0.319 ± 0.63	0.3297	3.29	86.09	75.82	–11.94	137.43	110.72	–19.44
ELISIO	0.359 ± 1.43	0.3315	− 7.83	60.83	74.59	22.63	81.89	107.98	31.86
PEPA	0.336 ± 0.42	0.3301	− 2.92	76.80	75.54	–1.64	114.84	110.09	–4.13

Results and discussion

Numerical simulation results

The governing equations with the imposed boundary conditions are numerically solved based on constant parameters given in Tables 2 and 3 using COMSOL Multiphysics. The computed streamline and pressure contour maps are shown in Fig. 6a and b, respectively. The velocity profiles of both the blood and dialysate phases manifest fully developed parabolic patterns, with the maximum velocity occurring at the centerline of the domain (r = 0). Based on the given inlet blood flow rate, the dialysate flow rate in the urea case is greater in the maltodextrin case (Table 2) in order to maintain minimal transmembrane pressure. Due to the exceptionally low Reynolds numbers in the present study, the entrance length is considered very short in comparison to the overall fiber length. Consequently, the streamlines are straight lines parallel to the lateral boundaries, with no cross-flow mixing. Furthermore, a constant pressure gradient was specified, resulting in a linear decrease in pressure downstream along the length of the dialyzer [22]. Overall, the streamlines and pressure contours exhibit the same distribution pattern across all different dialyzers.

Fig. 6 [Images not available. See PDF.]

Plots of field variables: a Streamlines and b Pressure contours for Urea and Maltodextrin solutions

The concentration profiles and total flux of urea (small molecule) and maltodextrin (larger molecule) solutions at steady state are shown in Figs. 7a and 8a, respectively, while Figs. 7b and 8b show their concentration profiles along the radial coordinate of the model domain at half the fiber length (blue line) and at the fiber outlet (green line). The concentration difference between the blood and dialysate sides (ΔC) indicates the progress of diffusion process, where the higher ΔC at the same traveling distance indicates the remaining driving force for solute diffusion.

Fig. 7 [Images not available. See PDF.]

Concentration profile and total flux a and concentration distribution at different locations of urea solution for different dialyzer types: FINEFLUX FIX-210S eco, ELISIO-210HR, and PEPA FDY-21B

Fig. 8 [Images not available. See PDF.]

Concentration profile and total flux a and concentration distribution at different locations b of maltodextrin solution for different dialyzer types: FINEFLUX FIX-210S eco, ELISIO-210HR, and PEPA FDY-21B

As the urea solution travels along the fiber length, continuous diffusion occurs. It is evident that urea concentrations drop rapidly after entering the dialyzer fiber for all dialyzer types (Fig. 7a), indicating fast diffusion of urea to the dialysate side of the membrane, as expected due to its very small molecular size. Nevertheless, its diffusion rate still varies among different membrane types. For Fineflux, the urea concentrations in the blood and dialysate sides reach equilibrium very quickly only at half the fiber length (ΔC ~ 0), while the highest ΔC is clearly observed for Elisio. Nevertheless, at the fiber outlet, the ΔCs are minimal for all membranes indicating that the mass transport of urea already approaches equilibrium at the fiber outlet. The results also indicate that the transport of urea is primarily influenced by flow control, with minimal impact from boundary layers on both sides of the membrane, consistent with previous findings [40, 41]. (Note that an abrupt drop in the urea concentration at the membrane interface, as shown in Fig. 7b is a common phenomenon of liquid free volume reduction due to the presence of the solid membrane tissue (partition condition) as mentioned earlier. This phenomenon considerably decreases the overall mass transfer coefficient [19, 42]. It is accounted for by the partition coefficient K_i, defined earlier in Sect. 2.1. The further reduction in urea concentration as the solute permeates through the membrane is due to the mass transfer resistance [43, 44].

On the other hand, maltodextrin diffuses at relatively lower rates due to a larger molecular size (around 30 times larger than urea). As can be seen, maltodextrin concentration decreases more slowly than urea in all types of dialyzers (Fig. 8a) with higher ΔCs both at half fiber length and at the fiber outlet (Fig. 8b). This suggests that the equilibrium is not reached even at the end of the fiber length (outlet), where ΔC of maltodextrin in the case of Fineflux membrane is again the smallest, indicating the fastest diffusion. Notice also that slower flow of fluid along the membrane surface on both blood and dialysate sides creates concentration gradients within the liquid phases, limiting the rate at which solutes can move across the membrane. This observation emphasizes the impact of boundary layers developed near the membrane surfaces. Therefore, it could be projected that a larger molecule would possess even lower mass diffusivity through the membrane and would result in even stronger concentration gradient even at the end of the fiber length. This model would provide a hint for the required ranges of diffusivity through a membrane that could prevent large protein molecules from leaking out to the dialysate side. Note that the drop in the maltodextrin concentration at the membrane interface (partition effect) occurred similarly to the case of urea. However, this drop is less prominent than the concentration gradient caused by a slow mass diffusion making the partition effect more difficult to observe.

The simulated results confirm that, unlike urea, maltodextrin (larger molecule) undergoes transport primarily through diffusion control. The numerically obtained outlet concentrations of urea and maltodextrin solutions will be compared and analyzed in the following section.

Comparisons between experimental and numerical results

Dialyzer clearance for the three dialyzers studied are determined using the modeled and the experimentally-measured inflow and outflow solute concentrations [45]. Based on these concentrations, the overall mass transfer area coefficient (KoA) [10, 15] is computed using Eq. (15).

A comparison between the numerical and the measured values of outlet concentrations, clearances and KoAs for urea solution is presented in Table 4. Overall, the urea outlet concentrations are nearly the same for all membrane types due to its fast diffusion. A lower outlet concentration corresponds to a higher clearance, indicating a high removal rate of urea. While the exit concentrations derived from the simulation closely match the experimental data, there is a notable deviation between the experimental and the modeled urea clearances and KoAs.

This discrepancy primarily arises from a scaling effect of the blood flow rate, where a small difference in solute concentrations results in proportionally larger percentage differences in clearances and KoAs. Among the three dialyzers, the experimental data and the numerical prediction consistently reveal the lowest outlet urea concentration together with the highest urea clearance and KoA for PEPA membrane. The lowest urea clearance for FINEFLUX membrane is potentially because it reaches diffusion equilibrium too quickly. In which case, the increase of dialysate flow would assist in improving its urea clearance (flow-control). Furthermore, the urea KoAs obtained in our study are compared with the manufacturer-provided values for PEPA and ELISIO membranes list in Table 1. It was found that the KoAs in this study are significantly lower than the manufacturer-provided KoAs due to the difference in flow configuration. Here, a co-current flow configuration was utilized, while the manufacturer values were obtained under a counter-current flow configuration. The counter-current flow maintains a consistently high concentration gradient across the dialyzer length, enhancing the driving force for urea diffusion thus resulting in higher KoAs than that of co-current flow systems [29].

For maltodextrin (a larger molecule), a comparison between the numerical values and measured values is presented in Table 5. The simulated results for the outlet maltodextrin concentration also agree with the measured values. For this larger solute molecule, the clearance values and KoAs are observed to be lower than those of urea primarily due to slower diffusion rates and pore size restrictions. FINEFLUX membrane could provide the lowest outlet concentration and the highest clearance and KoA potentially due to its fast-diffusion characteristic as this dialyzer features intermediate pore sizes, where the pores on the surface in contact with the solution are smaller and more compacted compared to those on the surface facing the dialysate (Fig. 4a). From Table 5, the simulated results of PEPA membrane, characterized by a relatively uniform pore distribution (Fig. 4c), yields the most accurate outcomes. This is because the diffusion through the membrane is modeled under the assumption of uniform pore distribution (constant diffusivity throughout the membrane). However, pore tortuosity is particularly evident in asymmetrical membranes, leading to non-uniform porosity along the solute permeation path. Therefore, enhancing the numerical predictions could be achieved by extending the pore diffusion model (PDM) used in this study to accommodate multi-layered membranes and incorporate pore tortuosity [31]. Incorporating such complexities into the PDM model could pave the way for significant advancements in future research.

Conclusions

Our study utilizes a numerical model as a promising tool for optimizing dialysis treatments for membranes with different characteristics. Here, numerical and experimental investigations were conducted to analyze the flow and mass transport behaviors of urea (a small molecule) and maltodextrin (a larger molecule) through various dialyzers. Our numerical model predicted consistent trends of outlet solute concentrations, solute clearances, and KoA values, where the most accurate prediction was achieved in PEPA membrane due to its uniform pore distribution. The results revealed that the choice of dialyzer membrane significantly influences solute clearance, where FINEFLUX membrane could provide the highest maltodextrin clearance and KoA potentially due to its fast-diffusion characteristic. Nevertheless, the urea clearance and KoA of FINEFLUX were lower than those of PEPA and ELISIO membranes regardless of a shorter equilibrium diffusion distance. These results suggest that urea transport is primarily influenced by flow control, with minimal impact from boundary layers, while maltodextrin transport is predominantly governed by diffusion control. This indicates variations in clearance mechanisms for solutes of different molecular sizes. These findings hold promise for improving dialysis efficacy in clinical practice by providing a tool for dialyzer selection and treatment strategies. In future work, the accuracy of our model could be improved further by incorporating the effect of membrane asymmetry and pore tortuosity.

Acknowledgements

This work was financially supported by Research Administration Division, Thammasat University under the grant number: TUFT 20/2566.

Author contributions

T.K.: Conducted the simulations and experiments, Data curation, Created figures, Writing original draft. C.P.: Conducted the literature review, Data analysis, Designed the experiments, Contribute to manuscript writing. T.E.: Provided experiment materials, Provided expertise and advice in medical analysis. W.P.: Supervised the project, Developed the research idea, Implemented mathematical model, Data analysis, Reviewed and edited manuscript, Secured funding.

Funding

This work was financially supported by Research Administration Division, Thammasat, University under grant number: TUFT 20/2566.

Data availability

The numerical and experimental data, including sensitive medical data collected at Chulabhon International College of Medicine, Thammasat University, are not publicly available due to considerations of sensitivity. However, the data can be made available by contacting the corresponding author upon reasonable request.

Declarations

Competing interests

The authors declare no competing interests.

Abbreviations

A_k

Membrane porosity

C_{0_B}

Blood inlet concentration, mol/m³

C_{0_D}

Dialysate inlet concentration, mol/m³

ΔC

Concentration difference between the blood and dialysate sides, mol/m³

c_i

Concentration of species i, mol/m³

c_0,i

Inlet concentration of species i, mol/m³

c_i,u

Solute concentration in membrane phase, mol/m³

c_i,d

Solute concentration in the liquid phase (blood and dialysate), mol/m³

D

Solute diffusion coefficient in solution, m²/s

D_i

Diffusion coefficient of species i, m²/s

D_M

Effective diffusion coefficient of solute in membrane, m²/s

f(q)

Friction coefficient

J_i

Diffusive flux vector, mol/(m².s)

K_i

Partition coefficient of species i

K

Viscous stress tensor, Pa

L_M

Membrane thickness, mm

L

Effective length of the hollow fiber, mm

L_DC

Width of the concentric dialysate channel, mm

N

Number of the hollow fibers

p

Pressure, Pa

q

Ratio of solute radius r_s to pore radius r_p

Q_B

Blood flow rate, ml/min

Q_D

Dialysate flow rate, ml/min

R

Radius of fiber, mm

R_HF

Inner radius of the hollow fiber, mm

R_module

Radius of the dialyzer module, mm

r_s

Solute radius, mm

r_p

Pore radius, mm

S_D

Steric hindrance factor

u

Velocity vector, m/s

U_{avg_B}

Average blood velocity, m/s

U_{avg_D}

Average dialysate velocity, m/s

ρ

Density, kg/m³

μ

Dynamic viscosity, Pa.s

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