Modeling hemodialysis with albumin retention

Abstract

We develop a new mathematical model for cross filtration in a dialyzer, taking into account the membrane fouling caused by progressive protein deposition. We focus on a single fiber and, starting from the basic laws of fluid dynamics and exploiting the smallness of the aspect ratio of fibers, we obtain significant simplifications of the flow equations. We end up with a hyperbolic system that describes the temporal evolution (specifically the decrease) of the filtering efficiency caused by the albumin deposit. Such an effect is then described quantitatively by means of numerical simulations.

Keywords:

• Hemodialysis
• mathematical modeling

2020 Mathematics Subject Classification:

• 76D08

Disclosure statement

The Authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Notes

1 The Dalton (Da) is the atomic mass unit, defined as 1/12 of the C${}_{12}$ atom mass.

2 Since albumin is a very expensive substance it is not possible to replenish it in the patient. A cheap substitute is used, which however may have negative consequences in the long run [29].

3 Symbols with * refer to dimensional quantities.

4 A rough way of estimating $R_b^{\ast }$ is imposing that $N\pi R_b^{\ast 2}=\pi R_d^{\ast 2}$, i.e. $R_b^{\ast }=R_d^{\ast }{N}^{-1/2}$. This estimate is actually a bit hasty because it assumes that the fibers occupy all the available space. In reality it can be done better, considering that the equivalent fibers have an ideal arrangement that leads to the maximum possible occupation (the porosity of these systems is known). Then the corresponding corrective factor can be applied, that is $N{R}_b^{\ast 2}=(1-\nu )R_d^{\ast 2}$, where ν is the porosity of the tangle of the fiber (a technical data provided by the manufacturer).

5 For the full derivation from Navier-Stokes and continuity equations we refer the readers to [2, 13, 15, 24].

6 Either human serum albumin or one of its substitutes, as hydrophobically derivatized hyperbranched polyglycerol (see, e.g. [30, 31]). There are, indeed, many not expensive artificial alternatives to albumin for plasma expansion which include crystalloids, alternate protein colloids, and non protein colloids.