Medical Case Study: Testing for Kidney Disease Patients with kidney disease often have protein in their urine.
Question:
Medical Case Study: Testing for Kidney Disease
Patients with kidney disease often have protein in their urine. While small amounts of protein are not very worrisome, more than 1 gram of protein excreted in 24 hours warrants active treatment. The most accurate method for measuring urine protein is to have the patient collect all his or her urine in a container for a full 24-hour period. The total mass of protein can then be found by measuring the volume and protein concentration of the urine. However, this process is not as straightforward as it sounds. Since the urine is collected intermittently throughout the 24-hour period, the first urine voided sits in the container longer than the last urine voided. During this time, the proteins slowly fall to the bottom of the container. Thus, at the end of a 24-hour collection period, there is a higher concentration of protein on the bottom of the container than at the top. One could try to mix the urine so that the protein concentration is more uniform, but this forms bubbles that trap the protein, leading to an underestimate of the total amount excreted. A better way to determine the total protein is to measure the concentration at the top and at the bottom, and then calculate the average protein concentration. Suppose a patient voids 2 liters (2000 ml) of urine in 24 hours and collects it in a cylindrical container of diameter 10 cm (note that 1 cm3 = 1 ml). A technician determines that the protein concentration at the top is 0.14 mg/ml, and at the bottom is 0.96 mg/ml. Assume that the concentration of protein varies linearly from the top to the bottom.
(a) Find a formula for c, the protein concentration in mg/ml, as a function of y, the distance in centimeters from the base of the cylinder.
(b) Write an integral that gives the average protein concentration in the urine sample.
(c) Estimate the quantity of protein in the sample by multiplying the average protein concentration by the volume of the urine collected. Does this patient require active treatment?
Probability & Statistics for Engineers & Scientists
ISBN: 978-0130415295
7th Edition
Authors: Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, Keying