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December 5

Proportional to

Why does the "proportional to" symbol ( $\propto$ ) look like an infinity symbol with the right side cut off? GeoffreyT2000 (talk) 20:51, 5 December 2015 (UTC)[reply]

The resemblance is likely an accident. It's likely

\propto

was chosen because it resembles the medieval notation for cross-multiplication of fractions. I don't have a source for this though. Sławomir
Biały 23:16, 5 December 2015 (UTC)[reply]

Sadly, Florian Cajori's A History of Mathematical Notations fails to mention this sign. However, Wikipedia attributes it here to William Emerson and dates it to 1768. If that is correct, then this page in The Doctrine of Fluxions must be the place where the sign was first used, and as you see, Emerson gives no explanation for it. I think the best guess is just that he wanted something that wasn't near enough to any existing symbol to be confused with it. --76.69.45.64 (talk) 04:43, 6 December 2015 (UTC)[reply]

Long term level of drug in blood based on halflife

Let's say I take a drug once daily that has a 12.4hr half life, and peak plasma level is reached in 90 minutes from the dose. Is there any easy way to calculate, or a website that will do it for me, to find out what my baseline level of the drug will be 23 hrs and 59 minutes after the last dose? In other words, its lowest level in the blood, right before my next dose, assuming I have been taking the drug for a long time. I know this can be expressed as a percentage, but let's assume a dose of 100mg that is fully absorbed if giving a specific amount is more helpful. Simplifying the answer by ignoring the peak plasma level at 90 minutes can also be done for convenience if you like, and we can assume peak plasma level at the time the dose is taken. Thanks. μηδείς (talk) 20:56, 5 December 2015 (UTC)[reply]

If the decay is strictly exponential, as the term half-life implies, then the formulas given here apply. 22 hours 29 minutes is 1.813 times the half-life, so the remaining amount from a single dose will then be e^{−1.813 ln 2} or about 0.285 of the peak amount. However, we must also consider the amount remaining from previous daily doses, whose peak levels were reached 46 hours 29 minutes ago, 70 hours 29 minutes, and so on. In this case adding 5 terms is enough, and gives an answer of 0.385 times the peak level from a single dose. --76.69.45.64 (talk) 04:59, 6 December 2015 (UTC)[reply]

The problem is significantly more complicated if the drug is itself bioactive and has any metabolites with activity, which is not an uncommon situation. In these cases, the minimum level of activity is not directly determined by the mass of drug present but also by the masses of metabolites present and their respective decay rates. Decay profiles that are not strictly exponential (i.e. not first order) are common and the term half-life is still used in chemistry for use in such cases. EdChem (talk) 09:23, 6 December 2015 (UTC)[reply]

Thanks, guys. I am quite happy with the approximated results. I am aware of the various complicating factors, such as bioactive metabolites (not relevant in this case) and the fact that peak serum concentration is delayed. I'm taking two different pills, once a day, that have the same desired effect. The doctor said I could take each of them them whenever I wanted, as long as I took them at the same time each day, and that I should judge the effectiveness on my own. The information above makes me think I will start taking one at lunch and one at bedtime rather than both at lunch. (If the result had been .70, rather than .38 I'd have kept the status quo.) I knew there was a way to calculate an answer, although I couldn't remember it (I could solve it graphically if I needed). I did get so far as series as an undergrad, but haven't needed recourse to anything more complicated than dividing by fractions or calculating interest or volumes since the 80's. Appreciate the help. μηδείς (talk) 17:22, 6 December 2015 (UTC)[reply]

It would be interesting to see how the blood concentrations of drugs taken in pill form are modeled; I'm sure drug companies must have done all sorts of research on this. Pills take time to dissolve, then pass through the various parts of the digestive tract until they are finally absorbed into the blood stream. Then there are the possible effects of coatings and taking the pill on a full stomach versus an empty one. So presumably the 90 min. figure should be taken as an estimate and the actual time can vary according to a number of factors. Injections are probably easier to model but there are several types to consider. I assume that with intravenous injections you can assume the drug reaches peak concentration pretty much instantaneously, while with intramuscular and subcutaneous injections the situation is more complex. --RDBury (talk) 18:13, 6 December 2015 (UTC)[reply]

Yes, I was actually able to get a lot of information of the drugs in their articles and from the manufacturer: the peak serum level time, the half-life, the effect of taking the drugs with food, the means of metabolism, the effects of metabolites. I just couldn't find the baseline level you would have after you'd been using the drug for some time. (There must be a term for that.)

Since my basic question was, would it make sense to take these medications one before lunch and one before bedtime rather than both before lunch (which is more convenient, but apparently less effective based on observation and the math above), a very rough estimate would be enough. μηδείς (talk) 18:55, 6 December 2015 (UTC)[reply]