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Journal of Theoretical Medicine
Volume 3, Issue 2, Pages 143-160

Mathematical Modelling of Profiled Haemodialysis: A Simplified Approach

1Center for Nonlinear Dynamics and Its Applicantions, University College London, Gower Street, London WC1E 6BT, UK
2Center for Nephrology and Institute of Urology and Nephrology, Royal Free and University College Medical School Middlesex Hospital, Mortimer Street, London W1N 8AA, UK
3Formerly Department of Mathematics, Imperial College of Science, Technology and Medicine, Imperial Collge, 180 Queen's Gate, London SW7 2BZ, UK

Received 1 April 1999; Accepted 11 July 2000

Copyright © 2001 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


For many renal patients with severe loss of kidney function dialysis treatment is the only means of preventing excessive fluid gain and the accumulation of toxic chemicals in the blood. Typically, haemodialysis patients will dialyse three times a week, with each session lasting 4-6 hours. During each session, 2-3 litres of fluid is removed along with catabolic end-products, and osmotically active solutes. 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. Profiled haemodialysis, in which the rate of water removal and/or the dialysis machine sodium concentration are varied according to a predetermined profile, can help to prevent wide fluctuations in plasma osmolality, which cause these complications. The profiles are determined on a trial and error basis, and differ from patient to patient. Here we describe a mathematical model for a typical profiled haemodialysis session in which the variables of interest are sodium mass and body fluid volumes. The model is of minimal complexity and so could provide simple guidelines for choosing suitable profiles for individual patients. The model is tested for a series of dialysate sodium profiles to demonstrate the potential benefits of sodium profiling. Next, using the simplicity of the model, we show how to calculate the dialysate sodium profile to model a dialysis session that achieves specified targets of sodium mass removal and weight loss, while keeping the risk of intradia-lytic complications to a minimum. Finally, we investigate which of the model profiled dialysis sessions that meet a range of sodium and fluid removal targets also predict extracellular sodium concentrations and extracellular volumes that lie within “safe” limits. Our model suggests that improvements in volume control via sodium profiling need to be set against potential problems in maintaining blood concentrations and body fluid compartment volumes within “safe” limits.