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## 'Drug Stabiliser' printed from http://nrich.maths.org/

Drugs which are to be taken regularly by patients (such as anti-depressants) are often described as having a half-life : a time taken for the body to clear half of the remaining levels of the initial dose of drug. For example, after one half-life, one half of the initial dose of drug remains in the body; after two half-lives, one quarter of the initial dose of
drug remains in the body, and so on. As drugs are taken on a regular basis the levels in the body build up until steady minimum and maximum levels are reached.

This problem contains various questions concerning these levels of drugs in the body. The various parts can be addressed numerically on a spreadsheet or using algebra. The first parts are relatively simple, whereas the latter parts build in complexity.

The effective half-life of the drug venlafaxine is about $12$ hours. Suppose that a single dose of $100$ mg of venlafaxine is administered on Monday morning. On which morning will the level of the drug first have dropped below $10$mg?

Another tablet is given on Wednesday morning. What levels of the drug will be in the body on Friday morning?

To be effective, drugs need time to reach steady minimum levels within the blood.

If one of these tablets is given each morning, what will be the final steady minimum level?

If one of these tablets is given each morning and each evening, what will be the final steady minimum level?

Determining the correct dosages of drug for individuals can be a difficult business, especially since it takes time for the drug levels in the body to reach stable levels: changes to dose will only take full effect several days later. In this second part, we look at the effects of Fluoxetine (otherwise known as Prozac) on the body.

Fluoxetine has a half-life of between $4$ and $6$ days, depending of the individual.

What would be the stable, long term peak level of drug of a patient taking a regular dose of $20$mg of fluoxetine per day?

To match this peak level, what equivalent weekly dose would need to be taken?

In each case, what are the lowest and highest long-term levels of drug in the body?

What issues might this raise for the patient?

Would missing a tablet lead to problems?