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);

> plot(C1(t),t=0..6);

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);

> plot(C2(t),t=0..6);

> fsolve(C2(t)=0.6,t);

Fri Jan 31 14:58:36 EST 1997