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Competitive inhibition - aspirin?

My textbook says that aspirin "irreversibly inhibits the active site of COX enzymes, preventing the synthesis of prostaglandins and thromboxane, the chemicals responsible for producing pain and fever." What I don't understand is if the binding is irreversible, why aren't the effects of aspirin permanent?
The body will be producing new COX enzymes, so those that are bound with aspirin will be replaced with new enzymes that are not
In addition, the effects of most drugs are not long-lasting, let alone permanent, simply because the drug is metabolized to inactive (or sometimes even active e.g. oxidation of codeine to morphine) compounds, which are then glucuronized, sulphated, etc in the liver, then excreted by the kidneys [water-soluble drugs are mostly excreted unchanged by the kidneys].

This means that very few drugs build up their levels in the body e.g. chloroquine [an anti-malarial agent] is sequestrated in the cornea and other tissues, tetracyclines in growing bones and teeth [hence contra-indicated in pregnant women]; this means a drug has a so-called half-life, the time it takes for half the initial amount in the body to remain - this means that the duration of action is usually limited to a period of about 5 half-lives (from simple maths, only about 3% remains after 5 half-lives, which is very unlikely to have an effect). The same applies to aspirin, which is an acidic compound [acetylsalicylic acid], so is excreted quicker in an alkaline urine [in which it will be neutralized [in terms of pH] and will be ionized, therefore being less prone to reabsorption in the proximal convoluted tubule [charged molecules pass through the cell membrane less readily]].

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