1. © Nuffield Foundation 2011 Nuffield Mathematics Activity Drug clearance

2. © Nuffield Foundation 2011 …. is concerned with the rate at which an active drug is removed from your body. ‘For most drugs, the model that works best is that this rate is proportional to the quantity of the drug remaining in your body.’ Can you write this statement as a differential equation? Can you solve it? Drug clearance …

3. © Nuffield Foundation 2011 The solution of the differential equation gives a graph like that shown below.

4. © Nuffield Foundation 2011 The full dose is present at the start. The amount of drug present diminishes over time.

5. © Nuffield Foundation 2011 The amount of drug in the blood can be easily measured and hence what proportion of the original dose is present It is easy to see when the drug has reached half the original quantity This time is called the half- life of the drug

6. © Nuffield Foundation 2011 After you establish the half-life of a drug, there is an equation from which you can calculate the amount of drug present at any time. Half-lives vary from drug to drug, and from person to person. Ibuprofen has a half-life of 2 to 4 hours, depending on the size and other characteristics of the person. With caffeine, the variation is even greater.

7. © Nuffield Foundation 2011 Reflection The rate at which your body removes drugs is proportional to the quantity of the drug that remains in your body. What is the differential equation that models this situation? What is the typical form of the solution? Can you sketch a typical graph? What is meant by the half-life of a drug? Is the half-life of a drug the same for all people?

8. © Nuffield Foundation 2011 Explore more at http://sonet.nottingham.ac.uk/rlos/bioproc/ha lflife/index.html