Cardizemtm is used in treating the symptoms of chronic angina and hypertension. As a calcium channel blocker, it is able to both increase the myocardial oxygen supply and decrease the body's demand for oxygen. It does this by lowering the heart rate and expanding the volume of the circulatory system, which leads to a lower blood pressure. Experiments show that the drug Cardizemtm (diltiazem hydrochloride) has a half life of 20 hours inside the human body. In our model the rate of change of Cardizemtm in the body is assumed to be proportional to the amount present.
Problem 1: Write a differential
equation describing the rate of change of the quantity of Cardizemtm
and determine the constant of proportionality.
Problem 2: Old man Withers receives
Cardizemtm intravenously at a continuous rate of 4 mg/min.
Write a differential equation describing the rate of change of the quantity
of Cardizemtm and compute its solution assuming that there is
no Cardizemtm initially present in old man Withers.
Problem 3: The fluid in the IV bag
is emptied in two hours. How much Cardizemtm is in old man Withers's
blood stream at that point?
Problem 4: Two hours after the bag is emptied, a nurse realizes he has forgotten to replace the bag. To make up for lost time, he triples the rate at which Cardizemtm is administered. He then puts up a larger bag that can feed Cardizemtm at this faster rate for up to 6 hours. He also leaves a note for the next nurse to turn down the rate of flow, but the note is swept off the counter by a breeze from the door after he leaves. The next nurse doesn't notice the note lying on the floor, so she doesn't check the rate of flow.
The lethal amount of Cardizemtm is 40 mg per kg of body mass. Determine if old man Withers will become old dead man Withers if he weighs 100 kg.
Hint: What differential equation describes the rate of change
of Cardizemtm during the 2 hours without an IV? What about after
the new IV is administered? It is helpful to take t=0 to be the initial
time for each new differential equation and determine the quantity of Cardizemtm
using the previous solution. Also take care to be consistent with your units of measure.
for mathematical applications. There are
illustrating how we might use interactive web objects to
help students learn Calculus.