Modelling Drug Dosages to Reach Target Level

[Canuck156]

Although I don't want to be given a straight answer to this question, I would really appreciated some help to point me in the right direction.


The amount of a drug in the body follows the equation y=De^(-kt) , where D is the initial dosage, k=(ln2)/6.7 , and t is the time in days since the dose was taken.

A person suffering from a disease needs to have approximately 130 micrograms of the drug in their bloodstream. Their doctor prescribed the following pattern to follow for the medication:

Monday – 200 micrograms
Tuesday – 100 micrograms
Wednesday – 200 micrograms
Thursday – 100 micrograms
Friday – 200 micrograms
Saturday – 100 micrograms
Sunday – 100 micrograms

If these doses are taken, and the decay equation y=De^(-kt) is taken as the only factor influencing the amount of the drug in the system, the amount of drug will rise far past the required 130 micrograms.

However, it is known that the specified doses will result in the required level of the drug in the body. Therefore, it follows that there must be some other factor influencing the clearance of the drug.

The task is to modify the model (y=De^(-kt)) to include some other factor (you don’t have to state what the factor is) that will cause the level of the drug in the body to remain at approximately the 130 microgram level over an extended period of time. This can be done by any means, such as adding to, subtracting from, or multiplying by some factor, the given model.

Note: So far I have not even been able to find an equation to model the long term amount of drug in the body using the initial dosages, I have had to do this through a spreadsheet. Is it possible to find a simple equation to model this situation?

To anyone who can help, thanks a lot.

 

[HallsofIvy]

No, you won't be able to find what is normally thought of as a "simple" equation because the problem is inherently "piecewise". Taking Monday, at the time of taking the medicine, as 0, Tuesday, same time, as 1, etc. you have, for the first day
y= De^-kt with y(0)= D= 200 micrograms so y(t)= 200e^-kt for 0<= t<= 1.
At t=1, the residual medication is 200e^-k and now you add 100 micrograms: D= (100+ 200e^-k) so y(t)= (100+ 200e^-k)e^-kt for 1<= t<= 2.

That is, the formula is:
y = 200e^-kt for 0<= t<= 1
(100+ 200e^-kt for 1<= t<= 2
etc.

If you really WANT a formula in a single equation you could use Heaviside's step function H(x) which is defined to be 0 for x< 0, 1 for x<= 0 but that is just "hiding" the piecewise property.

By the way since k= -ln(2)/6.7, e^-kt is the same as (e^-ln(2))^t/6.7= (1/2)^t/6.7 which says that the medicine has a half-life of 6.7 days in the body.
As far as the "other factor" is concerned, it would be perfectly valid to add a "cutoff" to the formula. Using y= De^-kt , y will be 130 when -kt= ln(130/D) or
t= -ln(130/D)/k. Since k= -ln(2)/6.7, that is the same as k= (ln(130/D)/ln 2)(6.7)=
6.7 ln_{2[/sup](130/D). The only problem is determining when that will be less than 1 (so that you don't go over into the next day).
Using a spread sheet to do piecewise calculations is completely reasonable.}

 

[Canuck156]

Thanks, I've ended up using a spreadsheet, and it worked out really well.