The pool of fluid in the body of a patient undergoing dialysis has been modeled by Enderle et al. (2005) as a two-compartment system, as shown diagrammatically in figure, where R is the rate of production of urea by the patient's body; V₁, is the volume of the intracellular fluid; V₂ is the volume of the extracellular fluid (blood and interstitial fluids); C₁ and C₂ are the concentrations of urea in the fluids of the two compartments, respectively; k₁₂ and k₂₁ are the mass transport parameters between the two compartments; and k₂ is the clearance rate constant for the dialysis unit.
The attached figure shows a two-compartment model of the fluid of a patient undergoing dialysis.
An unsteady-state mass balance of urea on each of the compartments yields the following two differential equations:
V₁ dC₁/dt = R - k₁₂C₁ + k₂₁C₂
V₂ dC₂/dt = k₁₂C₁ - k₂₁C₂ - k₂C₂
For Patient X, the following parameters apply:
R = 100 mg/h
k₁₂ = 33 liters/h
k₂₁ = 33 liters/h
V₁ = 10 liters
V₂ = 25 liters
The dialysis unit clearance rate constant is k₂ = 8 liters/h.
When Patient X arrives at the dialysis unit, his blood urea nitrogen (BUN) is 150 mg/liter. Integrate the differential equations (1) to obtain answers to the following:
(a) How many hours of dialysis will the patient require in order to reduce the level of BUN to 75 mg/liter?
(b) After the completion of the treatment, how long will it take for the BUN of the patient to rise back to the 150 mg/liter level?
(c) Experiment with setting the values of k₁₂ and k₂₁ to be unequal to each other
(say k₂₁ = 0.7 k₁₂, i.e., slower transfer from the extracellular pool to the cellular one) and interpret the results.
Show clearly how you obtain your answers, and illustrate this by showing the concentrations vs. time profiles of C₁, and C₂ in all parts of the problem.