## 8. Long-term behaviour

At the beginning of the module, we asked questions about the drug levels in the body over a longer period of time. Does the amount of drug in the body grow infinitely, or is there a point at which it levels off at some steady state?

You already know the answer to this question, as you have already interacted with the simulator. The simulator depicts the steady state as a green line, so we can infer that a steady state must be some specific value, like toxic level for instance. We consider the definition of a steady state with respect to our study. It is a level of drug in the body, at which after more doses are taken, the level of drug in the body remains at the steady state. In other words, once the steady state has been reached, and another dose is taken, the amount of drug in the body will remain at the steady state. That is, when .

There are two ways to determine the steady state, one requires a knowledge of limits, and the other does not. We start with the method that does not require knowledge of limits:

Let denote the steady state. At the steady state, . Consider the recursive definition of our sequence,

and substitute:

Solve the expression above for .

Using limits and the general form of our sequence,

it is also possible to solve for the steady state.

Thus, the steady state, , for any drug is given by:

Answer the first question without the simulator, using it only to check your work. Please use the simulator for the remaining two questions.

Suppose the effective level for a particular drug in a particular patient is 500 mg, and dosages of 300 mg are given every 4 hours, with an initial dosage also of 300 mg. Given that the drug has a half-life of 4 hours, what is the steady state? Is the steady state high enough to ensure that the drug reaches its effective level?

Using the simulator, set , the number of pills to 1 and let the pill size be 200 mg. Also set .

Find the steady state.

Confirm this result mathematically.

One of our guiding questions was: Suppose a patient decides to double the dose (take twice as many pills), but only half as often (say, every 4 hours instead of 2). Does this make any difference to the long-term levels?

We now have the tools required to answer this question.

Using the simulator, set the time between doses to 4, the pill size to 650 mg and the number of pills to 2 and the half-life to 4 hours.

Take note of the steady state.

Now, double the number of pills, so that the number of pills is 4. Also, adjust such that the pills are being taken only half as often. In other words, instead of taking the pills every 4 hours, we will take the pills every 8 hours.

Does this make any difference?

Congratulations! You have answered another of our guiding questions.