## 4. Drug Levels

Now we switch our focus to the questions at hand:

1. Will the maximum amount of drug in the body continue to rise if the medication is take regularly?
1. If so, will the drug level in the body become toxic?
2. If not, what will happen? Will the drug reach some steady state? Is it possible to predict the drug level over a prolonged period?
2. Can we ensure that the drug is present in a high enough quantity so that it will be effective? A low enough quantity so that it will not be toxic?
3. Suppose a patient decides to double the dose (take twice as many pills), but only half as often (say, every 8 hours instead of 4). Does this make any difference to the long-term levels?

To answer these questions, we will apply what we have learned in the previous sections to develop a model for drug levels in the body.

Consider taking one dose of a drug every four hours. One dose consists of 1 pill with 400 mg of active medical ingredient. The drug has a half-life of 4 hours.

Assume that there is no drug in the body before the initial dose is taken, and that the initial dose is taken at time 0. Then, at , we have 400 mg of drug in the body. This can be seen in Figure 6 below.

Figure 5: Initial dose.

Before the next dose, what will happen to the amount of drug in the body?

Moreover, using the equation we can predict the amount of drug in the body for any value of . Below, you can see the decay of the initial dose in the body according to the half-life of the drug.

Figure 6: Exponential decay of the initial dose.

But at this time, , we take another dose, as four hours have elapsed. What happens to the amount of drug in the body now?

At , another dose of the drug is taken. Thus, the amount of the drug in the body will increase by one dose, which in this case gives . You can see this in Figure 8 below.

Suppose this process continues. At , before the next dose is taken, what is the amount of drug in the body?

Then at , we take another dose. At this point, the amount of drug in the body is

This process continues, as shown below:

Figure 8: The process continues.

So far we have a continuous representation of the behaviour of the drug levels in the body, and we could start answering our guiding questions. However, the continuous representation shown in Figure 9 is fairly complicated, and it is a representation of one specific example. Thus, we shall switch to a discrete-time model that is not only simpler, but also that can handle more than just our example.

For our discrete-time model, should we sample the amount of drug in the body before or after taking the dose?

This results in Figure 10, which gives a visual representation of the behaviour that we will be modeling.

Figure 9: Discrete-time model of drug levels in the body.

In the example above, the interval between the doses was 4 hours and the half-life of the drug was also 4 hours. That is, the half-life and the time between doses was the same. Let us consider an example where the half-life and the interval between doses are different.

Consider, and , that is, the half-life of some drug is 2 hours and the change in time, or interval between doses, is 6 hours. We again assume that there is no drug in the body before the initial dose, and that the initial dose is take at time . Let the initial dose contain 800 mg of drug. You can see this graphically below.

Figure 10: The initial dose.

Then, using , we have

At this time, another dose is taken, so the amount of drug in the body after the dose is taken is 900 mg. You can see this below.

Using a calculator, complete the following table:

FLASH TABLE

Using this information, we can produce the following graphical representation of drug levels in the body for this drug.

Figure 12: Discrete-time model of drug levels in the body.

Thus, we can follow the same procedure as above to model the drug levels in the body for an arbitrary drug.

In the next section, we will formulate an expression for the y-values in Figures 9 and 12.