ESTIMATION OF DONNAN FACTOR FOR MULTI-ION SOLUTIONS WITH DIFFERENT CONCENTRATIONS OF MACROMOLECULES

The Gibbs-Donnan theory describes the equilibrium of ionic solutions separated by a semipermeable membrane, when only one solution contains charged macromolecules that cannot pass the membrane. The problem was originally formulated for simple solutions, such as dissociated NaCl, which allowed to calculate the ratio between ions concentrations at the sides of the membrane (Gibbs-Donnan factor). Solutions like plasma and interstitial fluid, and dialysate, contain however ions with different valence, and proteins in each compartment have a different concentration (and therefore charge); the classic theory needs an extension to describe these real-world scenarios. In this study we propose a new method to calculate Gibbs-Donnan factors for such more complex cases. From electroneutrality considerations and the definition of Nernst potential it is possible to derive a new expression of Gibbs-Donnan factor which depends on the total concentrations of ions of the same valence. This allows, for two solutions A and B, to calculate the concentration of all ions in solution B, given: 1) macromolecule concentrations in A and B, 2) solutes concentrations in A. The transitivity of Donnan factors was proved: for 3 solutions at equilibrium (A, B, C) the factor of A to C is equal to the ratio of the factors from A to B and B to C. As a demonstration, the theory was applied to the calculation of the concentrations in interstitial fluid (Na+, K+, Ca2+, Mg2+, Cl-, HCO3-, HPO42-, SO42-, C3H5O3-) starting from values reported in literature. The calculated interstitial concentrations were close to the reported ones, with a global RMSE equal to 6%. The Donnan factors for small ions in biological fluids have relatively small deviation from unity, but their precise value may be of importance for ions whose concentrations are similar at both sides of the membrane, as for example sodium and calcium transport in hemodialyzers and peritoneal dialysis.