# American Institute of Mathematical Sciences

September  2012, 17(6): 1903-1937. doi: 10.3934/dcdsb.2012.17.1903

## Modeling high flux hollow fibers dialyzers

 1 Università degli Studi di Firenze, Dipartimento di Matematica "Ulisse Dini”, Viale Morgagni 67/A, I-50134 Firenze

Received  March 2011 Revised  May 2011 Published  May 2012

In hollow fibres dialyzers blood flows in the fibres channel and plasma filtrates through their permeable wall to feed the flow of the permeate (dialyzate), which takes place among the fibres in the opposite direction. We investigate this fluid dynamical problem exploiting the existence of two separate scales: the one in the fibres direction ($\sim \,20\,cm$), and the one along the fibre radius ($\sim \,0.1\,mm$). We formulate a mathematical model based on a two-scale approach providing a full description of the flows of the blood, of the dialyzate and the of the plasma through the membrane, as well as of the progressive increase of the hematocrit. The problem is characterized by various rather unusual features like the slip condition of blood on the membrane and the feedback loop of boundary data for the hematocrit. Blood rheology is assumed to be of shear-thinning type, with hematocrit dependent coefficients. Under some simplifications explicit solutions are found. We show how the necessity of respecting several constraints and of reaching some specific targets influences the selection of the geometrical and physical parameters of the system. Once the fluid dynamical problem has been solved, the removal from blood of chemicals like urea, etc. has been studied.
Citation: Antonio Fasano, Angiolo Farina. Modeling high flux hollow fibers dialyzers. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1903-1937. doi: 10.3934/dcdsb.2012.17.1903
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