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Lumped-parameter model for haemodialyser with application to simulation of patient-artificial-kidney system

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Abstract

A simple lumped-parameter mathematical model is developed for analysing the performance of a haemodialyser. Analytical solutions are derived for the exit concentration of solute in blood leaving the haemodialyser. A method of ana priori estimation of the model parameters is suggested. The validity of the model is illustrated by comparing it with the numerical solution of more exact 2-dimensional models and some experimental data on commercial dialysers. The utility of the model is illustrated by an application to the simulation of a patient-artificial-kidney system, wherein compact analytical expressions are shown to describe the whole complex system.

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Abbreviations

a :

parameter defined in eqn. 14

A 11,A 12 :

parameters defined by eqns. 32 and 33, respectively

C b :

concentration of solute in blood, mole/cm3

C bi :

concentration of solute in the blood entering the dialyser, mole/cm3

C bo :

concentration of solute in the blood leaving the dialyser, mole/cm3

C d :

concentration of solute in the dialysate fluid, mole/cm3

C di :

concentration of solute in the dialysate entering, mole/cm3

C do :

concentration of solute in the dialysate leaving, mole/cm3

C dw :

concentration of solute at the wall in contact with the dialysate, mole/cm3

C t :

concentration of solute in the intracellular fluid, mole/cm3

C w :

concentration of solute at the wall in contact with the blood, mole/cm3

D :

molecular diffusivity of solute in blood, cm2/s

D 2 :

molecular diffusivity of solute in dialysate fluid, cm2/s

D E :

axial dispersion coefficient, cm2/s

h :

half-channel width of a parallel plate dialyser, cm

I :

rate of production of urea in the body, mole/s

k :

permeability of the tube wall, cm/s

k d :

effective mass-transfer coefficient for the dialysate side fluid, cm/s

k L :

effective mass-transfer coefficient from the blood to the tube wall, cm/s

K :

exchange coefficient between intra and extracellular compartments, h−1 or s−1

L :

length of the dialyser, cm

N :

rate of removal of urea in the dialyser, mole/s

P :

parameter defined by eqn. 12

P e :

parameter defined by eqn. 13

q :

ratio of flow ratio,Q B/QD

Q B :

volumetric flow rate of blood, cm3/s

Q D :

volumetric flow rate of dialysate, cm3/s

R :

radius of the tubular haemodialyser, cm

R A :

rate of solute removal at the wall, mole cm−2s−1

t :

time elapsed since start of dialysis, s

u b :

average velocity of blood, cm/s

V C :

total volume of intracellular compartment, cm3

V E :

total volume of extracellular compartment, cm3

W :

width of the dialysate groove, cm

x :

axial distance, cm

x * :

dimensionless axial distance,x/L

y :

dimensionless axial length of the dialyser (LD/u b R2 for tubular andLD/u b h2 for flat plate dialysers, respectively)

α:

parameter defined in eqn. 15 or 16

β 1,β 2 :

parameters defined in eqns. 42 and 43, respectively

γ:

geometry factor (=1 for flat plate, 2 for cylinder)

λ 1,λ 2 :

roots of eqn. 38

ν:

dimensionless wall permeabilitykR/D

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  1. National Chemical Laboratory, 411 008, Poona, India

    P. A. Ramachandran & R. A. Mashelkar

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  1. P. A. Ramachandran
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  2. R. A. Mashelkar
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Ramachandran, P.A., Mashelkar, R.A. Lumped-parameter model for haemodialyser with application to simulation of patient-artificial-kidney system. Med. Biol. Eng. Comput. 18, 179–188 (1980). https://doi.org/10.1007/BF02443292

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