Mathematical model for fluid-glucose-albumin transport in dialysis and its steady-state solutions

Mathematical description of fluid and solute transport between blood and
dialysis fluid in the peritoneal cavity has not been formulated fully yet,
in spite of the well known basic physical laws for such transport. Recent
mathematical, theoretical and numerical studies introduced new concepts on
peritoneal transport and yielded better results for the transport of fluid
and osmotic agent [1, 2, 3]. However, the problem of a combined
description of osmotic ultra filtration to the peritoneal cavity,
absorption of osmotic agent from the peritoneal cavity and leak of
macromolecules (proteins, e.g., albumin) from blood to the peritoneal
cavity has not been addressed yet. Therefore, we present here a new
extended model for these phenomena
and investigate its mathematical structure [4]. The model is based on a
three-component nonlinear system of two-dimensional partial differential
equations with the relevant boundary and initial conditions.
The non-constant steady-state solutions of the model are studied.
The restrictions on the parameters arising in the model were established
with the aim to obtain exact formulae for non-constant steady-state
solutions. As the result, exact formulae for the density of fluid
flux from blood to tissue and the volumetric flux across the tissue were
constructed and two linear autonomous ordinary differential equations for
glucose and albumin concentrations were derived. The analytical results
were checked for their applicability for the description of fluid-
glucose-albumin transport in peritoneal dialysis.

References

[1] Cherniha, R., Waniewski, J.: Exact solutions of a mathematical model
for fluid transport in peritoneal dialysis. Ukrainian Math. J., 57,
1112{1119 (2005)
[2] R. Cherniha, V.Dutka, J.Stachowska-Pietka and J.Waniewski.
Fluid transport in peritoneal dialysis: a mathematical model and numerical
solutions. //Mathematical Modeling of Biological Systems, Vol.I. Ed. by
A.Deutsch et al., Birkhaeuser, P.291-298, 2007
[3] Waniewski J, Dutka V, Stachowska-Pietka J, Cherniha R:
Distributed modeling of glucose-induced osmotic flow.
Adv Perit Dial 2007;23:2-6.
[4] Cherniha, R., Waniewski, J.:New mathematical model for
fluid-glucose-albumin transportin peritoneal dialysis. ArXiv:1110.1283v1 5
oct.2011