In the one-compartment model, the patient's body fluid is considered as one unit; toxins, here represented by creatinine, are removed to the dialyzer by diffusion. The two-compartment model assumes the body fluids are divided into two compartments, one directly accessible to the dialyzer (extracellular) and the other indirectly accessible (intracellular); creatinine is generated in the extracellular compartment [1]. Assuming constant volume and no residual kidney function, the one-compartment model is described by an ordinary differential equation:

Here

,

, and

are the creatinine concentrations in the total, extracellular, and intracellular compartments;

is the mass transfer coefficient;

is the dialyzer clearance; and

,

, and

are the total, extracellular, and intracellular volumes. Total body volume in liters is taken as 57% of body weight in kilograms, and the intracellular and extracellular volumes as 2/3 and 1/3 of total body volume [2].

is the rate of creatinine generation and

is time. The initial condition for the dialysis time period is

, a constant, in the recovery period (no dialysis)

, and the initial values of

,

, and

are equal to the values of

,

, and

at the end of dialysis. These equations are solved using

*Mathematica*'s built-in function

NDSolve. The solution to the two-compartment model exhibits the phenomena of post-dialysis rebound, the observed fast increase in creatinine concentration after dialysis caused by disequilibrium between the compartments.

[1] J. Waniewski, "Mathematical Modeling of Fluid and Solute Transport in Hemodialysis and Peritoneal Dialysis,"

*Journal of Membrane Science,* **274**(1–2), 2006 pp. 24–37. doi:

10.1016/j.memsci.2005.11.038.

[2] K. B. G. Sprenger, W. Kratz, A. E. Lewis, and U. Stadtmüller, "Kinetic Modeling of Hemodialysis, Hemofiltration, and Hemodiafiltration,"

*Kidney International*,

**24**, 1983 pp. 143–151. doi:

10.1038/ki.1983.138.