Mathematical modelling of profiled haemodialysis

A problem presented at the UK MMSG Nottingham 2000.

Presented by:
Dr Steve Baigent (Centre for Nonlinear Dynamics and Its Applications, University College London)
Prof Robert Unwin (Middlesex Hospital)
S Baigent, SJ Chapman, PD Howell, JR King, A Marks, M Penney, R Unwin, JAD Wattis, M Williams

Problem Description

The main function of the kidney is to regulate fluid and electrolyte balance in order to maintain intracellular (IC) and extracellular (EC) fluid volumes and ion compositions within narrow limits. When the kidneys fail to function normally, fluid is retained and several ions and solutes (mainly potassium and hydrogen ions) accumulate (mainly in the EC fluid compartment) and may become life threatening. While changes in diet and the administration of diuretic drugs can help to offset some of the important fluid and electrolyte problems due to reduced kidney function, in severe cases, dialysis treatment is necessary. Typically, haemodialysis patients will dialyse three times a week, with each session lasting 4–6 hours. During each session, 2–3 litres of fluid (or more) is removed, together with catabolic end-products, and osmotically active solutes.

Haemodialysis is essentially an equilibration between the patient's blood and a special fluid across an artificial membrane. The patient's blood is pumped around a continuous circuit from an artery to a dialysis machine and then back to a vein. On reaching the dialysis machine, the blood is fed into a hollow-fibre dialyser, a small cylinder containing many thousands of minute hollow fibres with small pores. On the other side of these pores is pumped fluid of prescribed composition known as the dialysate. In standard dialysis treatment the composition of the dialysate is maintained constant. As the blood traverses the dialyser, water and solutes (ions and nitrogenous products of normal metabolism) both diffuse due to concentration gradients and ultrafiltrate under an applied hydraulic pressure. On reaching the end of the dialyser, the blood is pumped back into the body.

The short duration and intermittent nature of haemodialysis can be problematic. The patient experiences large and acute swings in body fluid volume, and ion and solute composition not normally encountered in health. These changes can result in significant intra- and interdialytic morbidity. In a significant number of patients, the rapid removal of water and osmotically active sodium chloride can lead to hypotension or overhydration and swelling of brain cells, and/or painful muscle cramps. The chief cause of these complications is wide fluctuations in plasma osmolarity. One approach to preventing these fluid shifts and their resulting morbidity is profiled haemodialysis, in which the rate of water removal and/or the dialysis machine sodium concentration are varied according to a predetermined profile. For example, some profiles use an initially high dialysate sodium concentration that has the net effect of osmotically shifting water from the IC fluid into the EC volume, thus maintaining EC volume.

The benefits and effectiveness of profiled dialysis are still under dispute, and its clinical use is far from routine. Used incorrectly, the procedure can lead to a dangerous build-up of EC sodium and fluid overloading. At present profiles are chosen mostly on a trial and error basis. Some simple models have been developed that predict the blood chemical and body compartment fluid volume profiles for a given choice of sodium profile. The aim of this project is to extend and improve these models and to develop mathematical methods for analysing their performance in the more complicated case for which both the sodium dialysate concentration and the ultrafiltration rate are profiled.

Study Group Report

Early discussions highlighted the plasma refilling process as the rate-limiting step of the haemodialysis process. This process was thus investigated in greater detail by decomposing the extracellular fluid into interstial and plasma compartments, and modelling inter-compartmental protein and water transport. By non-dimensionalisation and time scale analysis we arrived at a simplified model that comprises 3 differential equations.

Although the slow refilling process is known to be an important constraint on haemodialysis, little mathematical modelling has yet been done to incorporate osmolyte removal, colloidal osmotic pressure, and volume changes. The new model combines these effects and focuses on the refilling process.

The model must now be fitted with realistic parameter values, tested and refined where necessary to match data from actual profiled haemodialysis sessions. There is scope for using the model to provide clinicians with profiles that achieve given performance targets whilst minimising the likelihood of intradialytic morbidity.

Download the full report