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?

Answer 1

Solution

Volume of urine sample (V) = 2000 ml

Diameter of cylindrical container = 10 cm

Radius of the container (r) = 10/2 = 5 cm

Height of the standing urine sample in the cylinder = h

h = 2000 / ( 25 x 3.14 )

h = 25.47 25.5 cm

a ) c = concentration of protein (mg/ml)

y = distance from the base of the cylinder (in cm)

According to the question, c is a function of y, c = f (y)

and, c varies linearly with y

This means the equation of straight line can be applied here.

Equation : c = my + b

where, m = slope of the curve

b = Y-axis intercept

b ) Average protein concentration can be calculated as a definite integral of 'c'.

c = f (y)

=

where, m (protein concentation at the top) = 0.14 mg/ml

n (protein concentration at the bottom) = 0.96 mg/ml

c ) The linear graph obtained:

Calculating the average concentration of protein:

The amount of protein in the urine sample is 1.5g which is greater than the critical amount of 1g. Thus the patient needs active treatment to treat this condition.