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*C*1(*t*) = *C _{0}*

We also said that a drug would have a minimum effective concentration,
*m* and a maximum safe concentration *M*. The problem of devising a
safe and effective treatment program is to come up with a dose of the
drug and an interval between doses, so that the drug concentration is
always above the minimum effective level and below the maximum safe
level. For simplicity, we will assume that the doses all supply the
same amount of the drug and that the time between doses is fixed.

The first problem is to figure out a reasonable time between
doses. One way to get a handle on this is to calculate the maximum
time between doses. That is, suppose a dose is given such that the
concentration immediately after the dose is given is *M*, the maximum
safe dose. If we calculate the time at which the concentration has
decayed to *m*, then this gives the maximum time interval between
doses. For example, consider the drug from the exercises in part 1 of
this lab. To help us later on, we'll refer to this drug as drug B. The
experimental data given said that the initial
concentration was 4.8 mg/ml and the concentration 3 hours later was
2.23 mg/ml. From this data we can determine the value of *k* with the
following command.

> k2 := solve(2.23=4.8*exp(-3*k),k);

Then we can calculate the maximum time between doses with the command shown below to be about 5.8 hours.

> solve( 8*exp(-k2*t)=1.8,t);

This calculation tells us that we can try to use any time interval less than 5.8 hours between doses. Many factors could be important in determining the value to use, including practical concerns like hospital schedules and shift changes.

In the first part of this lab, we presented the expression

*C*1(*t*) = *C _{0}*

Below, we describe how one can find the concentration at any later
time, assuming that the doses are equal and that the time between
doses is fixed, but it turns out to be rather complicated. To help you
visualize what is going on, a Maple procedure called `drug` has
been written. If you give this procedure values for *C _{0}*, the
concentration jump caused by each dose,

> readlib(drug):

> ?drugTo see how to use the

> C := drug(C0=7,L=3,k=k2);

> C(0);

> C(2.999);

> C(3);

> plot(C(t),t=0..12);

Notice that the concentration just after a dose increases with each
successive dose. Notice also that the concentration soon exceeds the
maximum safe concentration *M*=8. This means that the dose is too high,
and this is not a safe treatment program.

You can go back and plot the concentration for a longer time
interval. You should notice that the values of the peak concentrations
just after a dose rise rapidly for the first few doses and then appear
to level
off. It can be shown that this always happens, so the maximum
concentration eventually levels off as more doses are given. The
limiting value for an infinite number of doses can be calculated
explicitly, using information on geometric series, but that is a
topic for Calc 3.
To help you understand what is involved in the `drug` program,
we describe how to obtain formulas for the concentration after the
first few doses below.

We can calculate the
concentration just before the second dose is administered by setting
*t*=*L* in our equation

*C*1(*t*) = *C _{0}*

*C*1(*L*) = *C _{0}*

*C _{0}* +

*e*^{-k(t-L)}

*C*2(*t*) = *C _{0}* (1 +

Now, suppose that a third dose of the drug is given at *t*=2*L*. The
concentration just before the third dose is given is *C*2(2*L*), which
is

*C*2(2*L*) = *C _{0}* (1 +

*C*2(2*L*) = *C _{0}* (

*C _{0}* (1 +

*C*3(*t*) = *C _{0}* (1 +

This process can be continued to get expressions for the concentration for any number of equal, regularly-spaced doses.

1/31/1998