Membrane Transport of Several Ions During Peritoneal Dialysis: Mathematical Modeling

A Mathematical Model for Transport in Poroelastic Materials with Variable Volume:Derivation, Lie Symmetry Analysis, and Examples

Fluid and solute transport in poroelastic media is studied. Mathematical modeling of such transport is a complicated problem because of the volume change of the specimen due to swelling or shrinking and the transport processes are nonlinearly linked. The tensorial character of the variables adds also substantial complication in both theoretical and experimental investigations. The one-dimensional version of the theory is less complex and may serve as an approximation in some problems, and therefore, a one-dimensional (in space) model of fluid and solute transport through a poroelastic medium with variable volume is developed and analyzed. In order to obtain analytical results, the Lie symmetry method is applied. It is shown that the governing equations of the model admit a non-trivial Lie symmetry, which is used for construction of exact solutions. Some examples of the solutions are discussed in detail.

Kinetics simulation of transmembrane transport of ions and molecules through a semipermeable membrane

We have developed a model to study the kinetics of the redistribution of ions and molecules through a semipermeable membrane in complex mixtures of substances penetrating and nonpenetrating through a membrane. It takes into account the degree of dissociation of these substances, their initial concentrations in solutions separated by a membrane, and volumes of these solutions. The model is based on the assumption that only uncharged particles (molecules or ion pairs) diffuse through a membrane (and not ions as in the Donnan model). The developed model makes it possible to calculate the temporal dependencies of concentrations for all processing ions and molecules at system transition from the initial state to equilibrium. Under equilibrium conditions, the ratio of ion concentrations in solutions separated by a membrane obeys the Donnan distribution. The Donnan effect is the result of three factors: equality of equilibrium concentrations of penetrating molecules on each side of a membrane, dissociation of molecules into ions, and Le Chatelier's principle. It is shown that the Donnan distribution (irregularity of ion distribution) and accordingly absolute value of the Donnan membrane potential increases if: (i) the nonpenetrating salt concentration (in one of the solutions) and its dissociation constant increases, (ii) the total penetrating salt concentration and its dissociation constant decreases, and (iii) the volumes ratio increases (between solutions with and without a nonpenetrating substance). It is shown also that only a slight difference between the degrees of dissociation of two substances can be used for their membrane separation.

Quantifying citrate-enhanced phosphate root uptake using microdialysis

AIMS

Organic acid exudation by plant roots is thought to promote phosphate (P) solubilisation and bioavailability in soils with poorly available nutrients. Here we describe a new combined experimental (microdialysis) and modelling approach to quantify citrate-enhanced P desorption and its importance for root P uptake.

METHODS

To mimic the rhizosphere, microdialysis probes were placed in soil and perfused with citrate solutions (0.1, 1.0 and 10 mM) and the amount of P recovered from soil used to quantify rhizosphere P availability. Parameters in a mathematical model describing probe P uptake, citrate exudation, P movement and citrate-enhanced desorption were fit to the experimental data. These parameters were used in a model of a root which exuded citrate and absorbed P. The importance of soil citrate-P mobilisation for root P uptake was then quantified using this model.

RESULTS

A plant needs to exude citrate at a rate of 0.73 μmol cm^-1 of root h^-1 to see a significant increase in P absorption. Microdialysis probes with citrate in the perfusate were shown to absorb similar quantities of P to an exuding root.

CONCLUSION

A single root exuding citrate at a typical rate (4.3 × 10^-5 μmol m^-1 of root h^-1) did not contribute significantly to P uptake. Microdialysis probes show promise for measuring rhizosphere processes when calibration experiments and mathematical modelling are used to decouple microdialysis and rhizosphere mechanisms.

Sequential peritoneal equilibration test: a new method for assessment and modelling of peritoneal transport

BACKGROUND

In spite of many peritoneal tests proposed, there is still a need for a simple and reliable new approach for deriving detailed information about peritoneal membrane characteristics, especially those related to fluid transport.

METHODS

The sequential peritoneal equilibration test (sPET) that includes PET (glucose 2.27%, 4 h) followed by miniPET (glucose 3.86%, 1 h) was performed in 27 stable continuous ambulatory peritoneal dialysis patients. Ultrafiltration volumes, glucose absorption, ratio of concentration in dialysis fluid to concentration in plasma (D/P), sodium dip (Dip D/P Sodium), free water fraction (FWF60) and the ultrafiltration passing through small pores at 60 min (UFSP60), were calculated using clinical data. Peritoneal transport parameters were estimated using the three-pore model (3p model) and clinical data. Osmotic conductance for glucose was calculated from the parameters of the model.

RESULTS

D/P creatinine correlated with diffusive mass transport parameters for all considered solutes, but not with fluid transport characteristics. Hydraulic permeability (L(p)S) correlated with net ultrafiltration from miniPET, UFSP60, FWF60 and sodium dip. The fraction of ultrasmall pores correlated with FWF60 and sodium dip.

CONCLUSIONS

The sequential PET described and interpreted mechanisms of ultrafiltration and solute transport. Fluid transport parameters from the 3p model were independent of the PET D/P creatinine, but correlated with fluid transport characteristics from PET and miniPET.