## 2. Introduction

By using sequences and the mathematics associated with the concept of half-life you will develop a simple mathematical model to explore Drug Levels in the Human Body.

Consider the following:

Imagine that you have a headache, and wish to take a pain-relieving drug. One dose of the drug is 1 pill with 400 mg of active medical ingredient. The drug's container directs you to take 1 dose every 4 hours. Since your headache is not relieved in 4 hours, you decide to take another dose, and in another 4 hours, you take another dose. Suppose this process continues, with doses being taken at regular time intervals; in this case, every 4 hours.

A few questions that aries immediately:

- Will the maximum amount of drug in the body continue to rise if the medication is taken regularly?
- If so, will the drug level in the body become toxic?
- If not, what will happen? Will the drug reach some
*steady state*? Is it possible to predict the drug level over a prolonged period?

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

In the coming sections, we will learn the mathematical background required to develop a simple discrete model which we can use to answer the questions above!