The Donnan equilibrium is still valid in high-volume HDF

Abstract

Clinical studies have shown that hemodiafiltration reduces morbidity and mortality of dialysis patients compared to hemodialysis alone. This is attributed to its superior middle molecule clearance compared to standard hemodialysis. However, doubts arose as to whether a high convective flux through the dialyzer membrane has an influence on the equilibrium concentration of small ions, especially that of sodium. Due to the presence of negatively charged impermeable proteins on the blood side, the Gibbs-Donnan effect leads to an asymmetric distribution of membrane permeable ions on both sides of the membrane. In thermodynamic equilibrium, the concentrations of those ions can easily be calculated. However, the convective fluid flow leads to deviations from thermodynamic equilibrium. In this article, the effect of a convective flow on the ion distribution across a semipermeable membrane is analyzed in a theoretical model. Starting from the extended Nernst-Planck equation, including diffusive, convective, and electrostatic effects, a set of differential equations is derived. An approximate solution for flow speeds up to 0.1 ms^-1 as well as a numerical solution are given. The results show that in any practical dialysis setting the convective flow has negligible influence on the electrolyte concentrations.

Keywords: Donnan effect; Hemodialysis; Nernst-Planck equation; hemodiafiltration; hemofiltration; sodium balance; solute kinetics in dialysis.

Figure 1.

Schematic view of the ion distribution on both sides on both sides of a filtration membrane. On the left (retentate) side of the membrane negatively charged protein molecules (yellow) are present which are too large to pass through the membrane pores. Small monovalent ions can pass and are present on both sides. The one-dimensional coordinate x is defined as the distance from the left membrane boundary, the origin has been set at the left boundary of the membrane.

Figure 2.

Na+ and Cl⁻ concentrations at zero transmembrane flow (dashed lines) and a flow speed of 10 ms⁻¹ (solid lines). Retentate Na⁺ and Cl⁻ concentrations at x = 0 are 162 and 146.4 mmol/L, respectively, the temperature is 37°C. Gibbs-Donnan ratio is rD = 0.95. The Debye length is λD = 0.77 nm.

Figure 3.

Deviation of filtrate ion concentrations Δcf from the Donnan equilibrium concentration C_eq = 154 mmol/L as a function of the transmembrane flow speed for three different solutes (NaCl, NaHCO3, KCl). The upper x-axis shows the corresponding Peclet numbers of the cations. Solid lines: numerical solution, dashed lines: approximative solution (32). Retentate cation concentration 162 mmol/L, Gibbs-Donnan ratio r_d = 0.95, temperature 37°C. Note that Δcf for NaHCO3 is negative. In the KCl solution Δcf is negative for vf > 2.7 ms⁻¹ so it cannot be displayed on a log-log scale for flow speeds higher than that value. The insert shows Δcf as a linear-log graph for flow speeds up to 10 ms⁻¹.

Figure 4.

Deviation of filtrate ion concentration ΔC_f from the zero-convection equilibrium value C_eq = 154 mmol/L as a function of the Gibbs-Donnan ratio r_d for convective flow speeds V_f = 0.1, 1, and 10 ms⁻¹. Solid lines: numerical solution, dashed lines: approximative solution (32).