Modeling Hemodialysis Therapy
Kidney function in patients with partial or complete kidney (renal) failure is insufficient to adequately remove fluid (water) intake, resulting in the retention of fluids that is characterized as edema. Kidney function is also insufficient to adequately remove excess electrolytes (salts) and waste metabolites generated through ordinary digestive and metabolic processes. As a result, the concentrations of such metabolites will increase to toxic levels unless something is done to help remove them.
The major source of waste metabolites is the liver since this is where most of the energy conversion processes in the body take place. The end products of carbohydrate and fat metabolism tend to be water and carbon dioxide, both of which can be lost from the body through respiratory processes (breathing). The end products of protein metabolism, however, are generally eliminated through the kidneys. Urea is the largest mass of waste metabolite produced in the liver from protein metabolism. Because it is present in such large quantities in the blood and is easily measured, urea is generally used as a marker of renal function.
Hemodialysis, the first and most successful artificial organ technology, is a life-saving breakthrough for people with renal failure. Hemodialysis is based upon the thermodynamic principle that a difference in concentration (potential) can drive a flux that moves a substance from a region of high concentration to a region of low concentration. During hemodialysis blood is removed from the body and passed through a filter called a dialyzer before returning to the body. The dialyzer is made from thousands of hollow plastic fibers contained in a plastic shell. Unlike the plastic shell, which is made of impermeable material, the hollow fibers are permeable to the passage of low molecular weight molecules across the fiber. Blood flows through the center (lumen side) of the fibers. A fluid, called dialysate, flows on the exterior of the fibers (shell side). Dialysate is a “physiologic” fluid that contains electrolyte (salt) concentrations similar to that found in a healthy person’s blood. The difference in waste metabolite concentration in the blood and the dialysate creates a flux of the metabolite from the blood to the dialysate. Electrolytes are not removed because the dialysate and blood have similar electrolyte concentrations.
The schematic diagram at the right depicts the simplest model for generation and elimination of urea in the body. Urea is generated in the liver and passed into the blood stream. The blood stream is considered a single “compartment” in the body that acts as a reservoir for urea. Urea can be removed from the body by the kidney, eliminated with urine. If the kidneys are not functioning, urea can be eliminated by hemodialysis. The symbols associated with the diagram are:
C : blood urea concentration, mg/mL
t : time, min
G : urea generation rate, mg/min
V : blood volume, mL
Kr : renal (kidney) mass transfer
Kd : dialyzer mass transfer coefficient, mL/min
a) Use conservation of mass, i.e., the change in mass of urea in the body (blood) over a period of time is equal to the mass produced by the liver less the loss through kidney elimination and dialysis over that period of time, to derive a first order rate equation that describes the concentration of urea in the body as a function of time. You may have to use dimensional analysis to find the appropriate groupings of variables to represent the generation and removal rates.
b) What is an appropriate initial condition?
c) What is the steady-state solution to your differential equation?
d) When Kr is zero (no renal function) and G is much, much less than the dialysis removal rate, what is the time constant for your differential equation? What is the physical meaning of this time constant?
e) What is the analytical solution for your equation?
f) For Kr = 0 and dialysis treatment for 6 hours every 48 hours (Kd = 0 when dialysis is not taking place), sketch a graph of urea concentration (relative to an initial urea concentration) for 120 hours. How does the magnitude of G impact your sketch? How does the magnitude of Kd impact your sketch?
Copyright © 2001, John F Patzer II
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