Mathematical Models for Peritoneal Transport Characteristics

Abstract

Four mathematical models and for the description of peritoneal transport of fluid solutes are reviewed. The membrane model is usually applied for (1) separation of transport components, (2) formulation of the relationship between flow components and their driving forces, and (3) estimation of transport parameters. The three-pore model provides correct relationships between various transport parameters and demonstrates that the peritoneal membrane should be considered heteroporous. The extended threepore model discriminates between heteroporous capillary wall and tissue layer, which are assumed to be arranged in series; the model improves and modifies the results of the three-pore model. The distributed model includes all parameters involved in peritoneal transport and takes into account the real structure of the tissue with capillaries distributed at various distances from the surface of the tissue. How the distributed model may be applied for the evaluation of the possible impact of perfusion rate on peritoneal transport, as recently discussed for clinical and experimental studies, is demonstrated. The distributed model should provide theoretical bases for the application of other models as approximate and simplified descriptions of peritoneal transport. However, an unsolved problem is the theoretical description of bi-directional fluid transport, which includes ultrafiltration to the peritoneal cavity owing to the osmotic pressure of dialysis fluid and absorption out of the peritoneal cavity owing to hydrostatic pressure.

Membrane Transport of Several Ions During Peritoneal Dialysis: Mathematical Modeling

“…Practical applications of the theory of membrane transport are typically based on the formulas that relate the solute flux across the membrane to the boundary concentrations and do not involve explicit modeling of the intramembrane processes (12–14). However, these formulas are actually derived from models of the membrane structure and solute distribution within the membrane, and this approach was applied for transport of neutral solutes, see for example (12–14), and in our current study for ion transport. Some assumptions are necessary to derive such formulas, and one of them is the stability of the concentrations outside the membrane (19).…”

“…The applied formulas are valid only for dilute, ideal solutions of noninteracting molecules (10,11). In particular, ion transport is sometimes analyzed without electrostatic interactions taken into account (12–14), that is, assuming that the chemical potential of the ion is the only driving force and neglecting the electrostatic interactions. An example of such an approach can be found in modeling of ion transport during PD, where phenomenological, compartmental models are commonly applied in clinical and laboratory research (12–14).…”

“…These models are based on phenomenological (lumped) parameters for an apparent transport barrier between blood and dialysis fluid, and the main advantage of this approach is that it decreases substantially the number of estimated parameters. More advanced models take into account spatially distributed capillary bed and different transport characteristics for the capillary wall and tissue between capillaries (13,15–17). However, none of these models takes into account the impact of electric charge and membrane potential on transport characteristic of small ions, as sodium, potassium, chloride, bicarbonate, charged amino acids, etc.…”

“…In mathematical modeling of solute transport in PD, each solute is typically considered as a neutral (i.e., not charged), independent particle, which means that its transport does not depend on any other solute (12–14). However, small ion transport parameters cannot be predicted by such mathematical models (10).…”

Transport asymmetry in peritoneal dialysis: application of a serial heteroporous peritoneal membrane model

“…18 or more recently Ref. 29). The proposed models usually describe the peritoneal exchanges as fluid flow and solute fluxes between two well-mixed compartments (namely, the patient's blood and the peritoneal cavity content) through a "peritoneal membrane."…”