Modeling blood filtration in hollow fibers dialyzers coupled with patient’s body dynamics

We develop a mathematical model for cross filtration in a hollow fibers dialyzer, taking into account not only the phenomena occurring within the machine, but also the redistribution of chemicals between intra- and extracellular compartments in the patient’s body. The scheme for the cross flow is derived with reference to a single fiber, starting from the basic laws of fluid dynamics and exploiting the smallness of the ratio between the fiber radius and the fiber length to obtain significant simplifications. We end up with a system of integral and partial di¤erential equations in which the input data are in principle unknown. Indeed, the blood composition is considerably altered while going through the fiber and the body reacts redistributing urea, sodium ions, etc. between intra- and extracellular compartments with its own dynamics, thus updating the various concentrations at the dialyzer inlet. Such a coupling is an essential feature of the model. We present numerical simulations, showing a reasonable agreement with the data for a specific patient taken from the literature.