Thread: Calculate decay of medicine in human body

This is basically a repost to Dr. Math over a week ago where I didn't receive an answer. Maybe they thought it was for homework.

A friend of mine is an internist who is going to carry out some studies where he wants to take extra care not to accumulate certain medicines in the bodies of his patients, eventually poisoning them.

That has happened sometimes with gold salts which have a "half life" of 15 hours. That means, after each 15 hours there is still half of what was there 15 hours ago. So if he gives the patient another dosage after, say, 12 hours, there is still quite a good amount left in the body, but I don't know how much.

The question is, how could he calculate the right amount of medicine to bring the total in the body up the the original level - no more, no less?

The formula should depend on the initial dosage (which will be achieved again with those additional "maintenenace" dosages every so many hours), time elapsed since the last dosage, and the "half time" of the medicine. Only for these gold salts is it 15 hours, but there are others.

I know how to plot this curve, and it seems to be similar to the discharge curve of a capacitor (in electronics), but have no clue how to develop the formula. If somebody could provide me with a simplified version that is easy to convert into an Excel spread sheet, I would appreciate even more.

Originally Posted by Uwe

This is basically a repost to Dr. Math over a week ago where I didn't receive an answer. Maybe they thought it was for homework.

A friend of mine is an internist who is going to carry out some studies where he wants to take extra care not to accumulate certain medicines in the bodies of his patients, eventually poisoning them.

That has happened sometimes with gold salts which have a "half life" of 15 hours. That means, after each 15 hours there is still half of what was there 15 hours ago. So if he gives the patient another dosage after, say, 12 hours, there is still quite a good amount left in the body, but I don't know how much.

The question is, how could he calculate the right amount of medicine to bring the total in the body up the the original level - no more, no less?

The formula should depend on the initial dosage (which will be achieved again with those additional "maintenenace" dosages every so many hours), time elapsed since the last dosage, and the "half time" of the medicine. Only for these gold salts is it 15 hours, but there are others.

I know how to plot this curve, and it seems to be similar to the discharge curve of a capacitor (in electronics), but have no clue how to develop the formula. If somebody could provide me with a simplified version that is easy to convert into an Excel spread sheet, I would appreciate even more.

Not sure if this is what you are looking for:

Let A denote the amount of medicine, D the intial dose, h the half-life period and t the elapsed time in hours. Then the reduction of the medicine is modeled by:



With your example h = 15.

Hi Earboth,

Thanks for your reply! So A(t) is the dosage the patient has to receive (the "replenishing dosage") after the elapsed time to be up to the initial dosage again, right? And I guess "e" is the "elapsed" time.

Based on these assumptions I transfered the formula to Excel, but the results don't look right. After an elapsed time of 8 hours I get 0.463604 for the replenishing dosage, and after 15 hours (exactly the half time) 0.153037. Did I misinterpret the formula? - Ah, und ich spreche auch Deutsch...

Ok, after a little tinkering I came to the following conclusions (hope they are right):

- "e" is the constant 2.71828183, not "elapsed time"
- If I add "D-..." after the equal sign, the result of this formula will be equal to whatever the body has eliminated in the meantime. The way it was, the result told me what still was there, but I need to know how much I have to replenish.

So now the spread sheet formula is: =(C5-(C5*(2.71828183^((-1/C6)*(LN(2))*C7)))), where C5 is the initial dosage, C6 the half time and C7 the elapsed time.

I checked with "elapsed times" of 15 and 30 hours and the result was right. Hope that means the whole fomula is written correctly now.

Originally Posted by Uwe

Ok, after a little tinkering I came to the following conclusions (hope they are right):

- "e" is the constant 2.71828183, not "elapsed time"
- If I add "D-..." after the equal sign, the result of this formula will be equal to whatever the body has eliminated in the meantime. The way it was, the result told me what still was there, but I need to know how much I have to replenish.

So now the spread sheet formula is: =(C5-(C5*(2.71828183^((-1/C6)*(LN(2))*C7)))), where C5 is the initial dosage, C6 the half time and C7 the elapsed time.

I checked with "elapsed times" of 15 and 30 hours and the result was right. Hope that means the whole fomula is written correctly now.

Your formula is OK if you only want to calculate exactly one value. If you want to get a sequence of values to observe at which point the medicine has to be replenished, you must be very careful to use absolute cell refering(?) at the appropriate place. I've attached a screen-shot. Examine the formula I used!

You mean cell references like $c$5, right? Yeah, and being able to plot a curve is nice, too.

Something else I added to the spread sheet was a function that calculates the difference between what you should give the patient and what you actually gave him (like for example giving him 3mg instead of 3.2571). That will tell the physician when he (occassionally) has to add or subtract a little to/from the dosage.

Thanks again for all your help, Earboth (and whatever your real name is). Tonight I'll see this doctor and discuss with him how to make the spread sheet a little more attractive and easier to use for a physician. But those are just tiny details. The main point was the formula you gave me. I would have never thought that I would get such a clear reply and so fast!