In this section we describe an exponential decay model for the concentration of a drug in a patient's body. We assume that the drug is administered intravenously, so that the concentration of the drug in the bloodstream jumps almost immediately to its highest level. The concentration of the drug then decays exponentially. If we use C(t) to represent the concentration at time t, and to represent the concentration just after the dose is administered then our exponential decay model would be given by
For example, suppose that for a particular drug, the following data were obtained. Just after the drug is injected, the concentration is 1.5 mg/ml (milligrams per milliliter). After four hours the concentration has dropped to 0.25 mg/ml. From this data we can determine values of and k as follows. The value of is the initial concentration, so we have
To find the value of k we need to solve the equation
which we get by plugging in t=4 and using the data C(4)=0.25. Maple commands for solving for k and defining and plotting the function C(t) are shown below.
> k1 := solve(0.25=1.5*exp(-4*k),k);
> C1 := t -> 1.5*exp(-k1*t);
Now, suppose we are working with the same drug, but the initial dose is doubled. Under the assumptions we have made, this means that the initial concentration would also double to 3.0 mg/ml, and the concentration C(t) given by our model would be
where k would have the same value, k=0.4479398673 found above.
A problem facing physicians is the fact that for most drugs, there is a concentration, m, below which the drug is ineffective and a concentration, M, above which the drug is dangerous. Thus the physician would like the have the concentration C(t) satisfy
This requirement helps determine the initial dose of a drug and when the next dose should be administered. For example, suppose that for the drug in the experiment the maximum safe concentration is 5 mg/ml, or M=5, and the minimum effective concentration is 0.6 mg/ml, or m=0.6. Then the initial dose must not produce a concentration greater than 5 mg/ml. Suppose that the initial dose is chosen to produce an initial concentration of 4.75 mg/ml. Then how many hours later will the drug concentration reach the minimum effective level? The answer is about 4.6 hours, as shown by the following commands.
> C2 := 4.75*exp(-k1*t);