Mass Transport in High-Flux Hemodialysis

Diffusive Transport in Blood and Dialysate

In HD, both blood and dialysate flow tangential to the membrane surface. Although this flow appears unrelated to the solute diffusion, it plays a very important role in this mode of transport. Figure 2A shows a typical tangential velocity profile for blood in a fiber tube. The velocity profile is highest at the tube center and zero at the membrane wall (12). For relatively high flow rates (i.e., , where U_B is the average velocity in the tube, d_in is the tube’s inner diameter, and D_B is the solute diffusivity in blood), solutes at the center are not able to diffuse to the wall but are, instead, swept away by the bulk flow. Only solutes near the wall (where the flow is nearly zero) are able to diffuse to the membrane surface. This effect creates a thin region near the membrane wall where the solute concentration varies substantially. This region—termed the concentration boundary layer—exists on both the blood and the dialysate sides due to tangential flow (13). The thickness of the boundary layer is denoted as δ_B on the blood side and δ_D on the dialysate side (Figure 2B).

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Figure 2.

Effect of blood flow on diffusive mass transfer. (A) Velocity profile of blood flow in a tube. For sufficiently high flow rates, solutes near the tube center (“bulk region”) are swept away by flow, and only those near the wall are able to reach the membrane. (B) This phenomenon causes the concentration to vary in a thin region near the wall (the concentration boundary layer). Depicted here are concentration boundary layers on both the blood and dialysate sides. (C) Clearance level of three different toxins by an AN69 dialyzer with increasing blood flow rate at zero net ultrafiltration rate. Increasing the blood flow rate significantly improves small molecule clearance through attenuation of boundary layer effects while marginally affecting transport for larger toxins. The molecular masses of creatinine, vitamin B12, and myoglobin are 113, 1355, and 16,700 Da, respectively. (C) is reprinted from ref. 24, with permission.

The major consequence of concentration boundary layers is alteration of the mass transfer rate of solutes in their passage from blood to dialysate. To get a qualitative picture of this effect, an examination of Fick's law is helpful (14). This law states that the flux of solute (rate of moles transferred per unit area) entering the membrane from the blood side is equal to the solute diffusivity times the slope in concentration. Because the majority of concentration change occurs across the boundary layer, an approximate expression for the magnitude of the flux is (1)where D_B is the diffusivity of the solute, is the change in concentration across the boundary layer, and is the boundary layer thickness. The mass transfer resistance is defined as the ratio of concentration drop to the solute flux. On the blood side, this quantity is , with a similar expression for the dialysate side: . This analysis indicates that reducing the boundary layer thicknesses on the blood and dialysate sides decreases their mass transfer resistances and, hence, increases the overall rate of solute removal. Thus, controlling boundary layer thicknesses is a key design consideration for dialysis engineering (15), especially with regard to small solutes.

Boundary layers have been studied extensively in the engineering community for over a half century. For the given dialyzer geometry (Figure 2B), one sees that the boundary layer develops nonuniformly from the flow inlet and is determined by the tangential flow rate (either blood or dialysate), dialyzer geometry, and solute diffusivity. Theoretical expressions and empirical correlations are given in many classic papers (16⇓⇓⇓⇓⇓⇓–23). The key point to make from a clinical standpoint is that the average boundary layer thickness can be tuned by the external flow rate—the higher the flow rate, the smaller the boundary layer thickness.

Figure 2C shows a typical example of how increasing flow rate alters the diffusive removal rates for different surrogate toxins (24). For small solutes, increasing the flow rate increases the rate of toxin removal because the mass transfer resistance of the blood and dialysate makes up a significant fraction of the overall resistance. However, this strategy only works to a certain extent. For larger molecular mass toxins (e.g., myoglobin; molecular mass, 17 kDa), boundary layer effects play only a minor role in mass transport, as the dominant mass transfer resistance for a compound of this size is the membrane itself. In conclusion, the benefit of increasing blood flow rate, a common clinical approach used to augment diffusive solute clearance, is limited largely to small solutes and is mediated by mitigation of boundary layer effects.

One important distinction between blood and dialysate is that the former is a suspension composed of red blood cells and plasma, creating additional considerations. The distribution of some solutes, such as urea and creatinine, includes the intracellular space of red blood cells, and transfer from this region to the plasma has to occur before dialytic removal is achieved. As such, the transfer rate for such solutes across cellular membranes affects dialytic removal rates. Furthermore, red blood cells may influence solute mass transfer rates both negatively (by effectively increasing diffusive path length) and positively (by augmenting diffusivity through their rotational motion). Finally, the nonaqueous constituents of plasma (especially proteins) affect diffusive mass transfer resistance by influencing the plasma viscosity.

Diffusive Membrane Transport

Solutes must diffuse through a porous membrane in addition to the fluid in the blood and dialysate compartments. In the following analysis, it is assumed that pores provide the only pathway by which solute and fluid transfer can occur through the membrane. For larger solutes, the pore structure substantially hinders the diffusivity of the solute compared with the bulk fluid phases; Figure 3A describes what membrane-related factors give rise to this effect (25).

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Figure 3.

Solute diffusion through a porous membrane. (A) The effective diffusivity of solute moving through a porous membrane is hindered by four factors discussed in the text. Blue spheres represent solute molecules. (Note that because this figure is intended to highlight pore tortuosity, it does not depict the interconnectivity that exists between pores in a membrane.) (B) Overall mass transport coefficient from blood to dialysate measured for dextrans with varying molecular mass. Three types of high-flux dialyzers are tested: SOLACEA-25H (cellulose triacetate membrane), Optiflux F250NR (polysulfone membrane), and CT190 (cellulose triacetate membrane). For all three dialyzers, the overall mass transport coefficient is dominated by the membrane diffusivity for the relatively high molecular mass dextrans studied. The exponential decay with molecular mass is indicative of mass transfer limited by the membrane due to the equilibrium partitioning of solute. (B) is reprinted from ref. 29, with permission.

The membrane physical properties that hinder diffusion include the porosity of the membrane (i.e., the volume fraction occupied by the pores), tortuosity of the pores (defined as the average length of the pore divided by the membrane thickness), the frictional forces exerted on the solute by the membrane pores (26), and the equilibrium partition coefficient of the membrane (∅). For solutes that do not interact with the pore walls, this latter parameter represents the ratio of pore volume accessible for the solute to the total pore volume. This thermodynamic property drastically reduces the concentration of solute inside the membrane, and hence, it reduces the driving force (concentration gradient) for mass transfer. Generally, the most important factor controlling the effective diffusivity within the membrane is the equilibrium partition coefficient.

Giddings et al. (27) used statistical analysis to find the equilibrium partition coefficient for different geometries. They found that for membranes having pores formed by a network of randomly intersecting planes, the partition coefficient exponentially decays as a function of the ratio of solute size to pore size (27,28). Thus, effective solute diffusivity within the membrane is exquisitely sensitive to the relationship between solute and pore dimensions.

As an illustration of membrane diffusion, experimental data from Kim et al. (29) are shown in Figure 3B. These researchers determined an overall mass transport coefficient (30) for three commercially available dialyzers, where J_s is the solute flux through the membrane and () is the concentration change of solute between the bulk blood and dialysate compartments. Figure 3B shows the trend for mass transfer coefficients of dextrans having different molecular masses. For the large molecular mass dextrans studied (>5 kDa), is dominated by the membrane diffusivity, where Dm is the membrane diffusivity and is the membrane thickness. The overall mass transport coefficient is seen to decrease exponentially with respect to the molecular mass (i.e., solute size) due to the equilibrium partitioning of solute, as predicted by the above analysis.

Clinical Implications

For clinical correlation, one source of confusion pertains to dialyzer KoA, a manufacturer-derived in vitro value for a specific solute/dialyzer combination used to estimate in vitro clearance when blood and dialysate flow rates are known (30). In this expression, KoA is the product of the overall mass transfer coefficient (Ko) and the membrane surface area (A). The term Ko is the inverse of the overall diffusive mass transfer resistance (Ro), which is the sum of the resistances provided individually by the blood, membrane, and dialysate. Although KoA traditionally has been considered to be independent of blood and dialysate flow rates, data from Leypoldt et al. (31) demonstrated a dependence of in vitro KoA on dialysate flow rate most likely due to dialysate boundary layer effects, challenging the validity of this assumption. Furthermore, several experts have raised fundamental questions about the validity of translating in vitro KoA data to the clinical environment, challenging the assumption that in vivo KoA is independent of blood flow rate (31⇓–33). These questions have focused specifically on the characteristics of blood that influence diffusive mass transfer resistance (especially viscosity and hematocrit). Furthermore, because KoA is by definition a “pure” diffusive parameter, it is not clear how the convective contribution to solute clearance by modern high-flux dialyzers is incorporated as a result of internal filtration (a ubiquitous phenomenon when such a dialyzer is used in the HD mode). It is worthwhile noting that the original KoA derivation by Michaels (30) applied only to low-flux dialyzers with no internal filtration, for which “pure diffusion” conditions could be achieved. A corollary relevant point is that internal filtration rates are influenced by blood flow rate itself. (Internal filtration is developed more fully in the next section.) Thus, although data are inconclusive at present, the validity of the original assumption that blood flow rate does not influence Ko (and therefore, overall mass transfer resistance) is highly questionable on the basis of the characteristics of contemporary high-flux HD. This alternative viewpoint is an important consideration in light of the axiom that the blood compartment provides the most significant (“controlling”) resistance to diffusive mass transfer for small solutes in modern dialyzers.

Another clinical correlation relates to the relationship between solute clearance and blood flow rate, which has traditionally divided into “blood flow–limited” and “membrane-limited” regions. However, the former term largely reflects blood flow delivery per fiber, which is determined by the relationship between membrane surface area and total blood flow rate to the filter. In turn, per fiber blood flow rate (rather than total blood flow rate to the filter) is the primary parameter influencing boundary layer phenomena for small solutes in the blood compartment, consistent with the above discussion.

Membrane Hydraulic Permeability

The ultrafiltration velocity at which fluid moves through a membrane is (35) (2)where J_F is ultrafiltrate flux (rate of volume transferred normalized to surface area), L_p is the hydraulic permeability of the membrane, is the hydrostatic (i.e., measured) pressure difference across the membrane, and is the osmotic pressure difference. The permeability L_p depends on the pore geometry, membrane thickness, and fluid properties. For example, for cylindrical (nonconnecting) pores of uniform radius and moderate tortuosity, on the basis of the Poiseuille law, where s is the pore radius, ε is the membrane porosity, is the membrane thickness, τ is the membrane tortuosity, and μ is the fluid viscosity (25). It should be noted here that all membranes have some degree of pore interconnectivity (36⇓–38), which plays a significant role in determining L_p by influencing the number of pathways that fluid can take to traverse the membrane. Nevertheless, a general rule of thumb is that a strong dependence exists between hydraulic permeability and the square of average pore size (). (Although not accounting for osmotic pressure effects, the hydraulic permeability surrogate used in clinical practice, ultrafiltration coefficient [Κ_uf: milliliters per hour per millimeter Hg], is approximately equivalent to L_p times the nominal surface area of the membrane [5].)

The osmotic pressure depends on the total concentration of impermeable solutes in both the blood and dialysate. In HD and hemodiafiltration, the osmotic pressure difference (Δπ) is largely a function of blood protein concentration through its nonlinear effect on plasma oncotic pressure (39,40). Lastly, concentration polarization plays a significant role in determining the osmotic pressure of a protein-containing solution subjected to membrane ultrafiltration (see next section).

For a high-flux dialyzer (operated in the HD mode), the net transmembrane pressure gradient at a given point along the length of a dialyzer varies considerably, mostly due to the large axial pressure drop in the blood compartment. At the approximate midpoint of the filter, the combination of hydrostatic and osmotic pressures in the blood compartment actually produces a “reverse transmembrane pressure,” resulting in ultrafiltration from dialysate to blood (“back filtration”) in the distal filter segment (35). This flow, in combination with the blood to dialysate ultrafiltration in the proximal segment, creates an “internal filtration” circuit in which the proximal ultrafiltration rate may be as high as 80 ml/min. (As suggested previously, on the basis of the direct relationship between blood flow rate and pressure drop, increased blood flow rate leads to a greater degree of internal filtration.) In fact, the convective solute removal occurring in association with this circuit is typically responsible for the majority of middle molecule elimination during high-flux HD (15). On the other hand, ultrafiltration occurs only from blood to diafiltrate along the entire length of the hollow fiber under typical hemodiafiltration operating conditions (i.e., blood flow rate >350 ml/min; ultrafiltration rate ∼100 ml/min).

Sieving Coefficient and Concentration Polarization

When a fluid passes through a membrane, some dissolved solutes may pass freely, while others may be partially rejected. The fraction of the solute that can penetrate through the pores and exit the other side of the membrane is termed the sieving coefficient (41,42), S, ranging from zero to unity. (Note that S is used as an abbreviation for sieving coefficient and s for mean pore size.) Sieving coefficients are typically measured in steady-state ultrafiltration mode and are used clinically in the estimation of convective clearance of middle molecular uremic toxins. In its most rigorous form, the blood compartment concentration comprising the sieving coefficient is on the basis of a plasma water determination.

The value of S for a particular solute depends on the relative size between the molecule and the pore (i.e., ). For particles much smaller than the average membrane pore size (e.g., urea and creatinine), sieving coefficients are close to unity, whereas particles much larger than the membrane pore size have sieving coefficients close to zero. When the size of the solute corresponds to the membrane molecular mass cutoff, the value is . Dialyzer manufacturers typically target S >0.6 for β₂-microglobulin (43,44) and S <0.008 for albumin (66.5 kDa) after blood or plasma exposure that modifies the membrane’s effective permeability through fouling (45).

To understand the physics underlying sieving, a typical concentration profile for a partially rejected solute can be evaluated using “stagnant film theory” (Figure 4A) (46). When fluid moves from the blood to the filtrate side, solute accumulates at the membrane entrance due to rejection from the membrane. This local increase in concentration is called concentration polarization and can increase the wall concentration to several times that in the bulk (47⇓–49). The exact shape of the concentration profile in the polarized solute region is set by the steady-state competition between convection pushing the solute toward the wall and “back diffusion” of the solute into the bulk blood solution. This phenomenon is more pronounced at higher ultrafiltration rates. Although increasing blood flow rate may diminish polarization (by decreasing the boundary layer thickness over which polarization occurs), the effect is very small due to the large size of molecules for which concentration polarization is relevant.

Figure 4A shows that stagnant film theory yields three different sieving coefficient parameters. The observed sieving coefficient (S_o) is the ratio between the bulk concentrations in the filtrate and the blood compartments (). This parameter is what one measures experimentally or clinically. The actual sieving coefficient S_a is the ratio of concentrations just outside the membrane (S_a = C_F/C_B*), reflecting the effect of solute polarization. The values of S_o and S_a do not differ greatly at low ultrafiltration rates (i.e., when concentration polarization is not prominent). In fact, at zero ultrafiltration, no solute polarization occurs, and the blood and filtrate concentrations are equal (because diffusion across the membrane abolishes any concentration gradient), such that . As ultrafiltration rate increases, these two parameters diverge () due to solute accumulation at the wall (C_B*>C_B). Finally, reflects the value of S_a at very high ultrafiltration rates (i.e., maximal concentration polarization).

Figure 4B (50) reinforces the important clinical observation that observed (measured) sieving coefficient is a strong function of ultrafiltration flux. Although a flat-sheet polyethersulfone membrane was used in the study generating these data, its molecular mass cutoff is consistent with membranes used for clinical therapy. Furthermore, the filtration flux range studied includes values typically seen in high-flux HD and, especially, hemodiafiltration (approximately 10⁻⁶ m/s). The observed sieving coefficient is numerically high at low ultrafiltration rates, decreases at intermediate rates, and increases back to S_o = 1 at high rates. The practical learning here is that the prescribed ultrafiltration rate in high-flux HD, and especially hemodiafiltration, has a significant effect on membrane selectivity, along with the size of the solute and membrane pore geometry. The reason for the loss of selectivity at high ultrafiltration rates stems from the fact that the increased concentration at the wall amplifies the concentration drop across the membrane. This effect augments diffusive mass transfer of species across the membrane and concomitantly reduces the effectiveness of membrane sieving (i.e., selectivity). Although proximal ultrafiltration rates may be as high as 80 ml/min in high-flux HD as part of an internal filtration circuit, this loss of selectivity is particularly important in hemodiafiltration due to the use of higher ultrafiltration rates that operate over the entire length of the diafilter.

The physical arguments discussed above demonstrate the need for careful consideration in choosing ultrafiltration rates that balance toxin removal against albumin losses, especially in hemodiafiltration (51,52). On one hand, prescription of net ultrafiltration rates substantially greater than those used typically in high-flux HD (<20 ml/min) (53) is expected to enhance the removal of larger molecular mass toxins (54). However, the ultrafiltration rate range typically applied in hemodiafiltration (>80 ml/min) may lead to a self-defeating scenario in which membrane selectivity is lost (as discussed above) and albumin losses are substantial (55,56). Prescription of relatively high blood flow rates (typically at least 350 ml/min) may be helpful in this context by mitigating concentration polarization and loss of selectivity, at least to some extent.

A specific study clearly demonstrating this loss of membrane selectivity in hemodiafiltration was performed by Gayrard et al. (56), who measured mass removal of several uremic surrogate molecules and albumin over a 20- to 25-L infusate volume range. Although an increase in infusate volume from 20 to 25 L per session resulted in an essentially linear increase in the mass removal of β2-microglobulin (12 kDa) and retinol binding protein (21 kDa), mass removal of α-1-antitrypsin (55 kDa) demonstrated an exponential increase. However, the mass removal curve for albumin showed a similar exponential increase in this range, demonstrating a loss of membrane selectivity between a high molecular mass uremic surrogate and albumin. The authors suggested in their conclusion that the higher ultrafiltration rates effectively increased the molecular mass cutoff of the membrane, consistent with an effect on membrane sieving properties. Although a quantitative score encompassing the removal of uremic solutes across a wide molecular spectrum has been developed by Maduell et al. (57), more clinical data are required to assess the effect of ultrafiltration rate on membrane selectivity.

In addition to loss of membrane selectivity, concentration polarization can lead to other detrimental effects. The increased concentration of solute at the membrane wall increases the blood osmotic pressure, which decreases fluid flow across the membrane through a reduction in the net transmembrane pressure gradient. Maintenance of the prescribed ultrafiltration rate can only be achieved through an increase in hydrostatic pressure, potentially degrading dialyzer performance (especially in hemodiafiltration) (55,56,58). The accumulation of proteins at the wall can eventually lead to the formation of a permanent “gel-like” layer, effectively fouling the membrane (by pore blockage or narrowing) and increasing its hydraulic resistance (48,59).