A drug is administered once every four hours. Let Dt be the amount the drug in blood system at t-th interval. The body eliminates a certain fraction p of the drug during each time interval. If the amount administered is Do find Dt and limit t->infinity Dt.
Basically I am looking for a equation for Dt. which can tell me at given time interval t how much of drug is still left in body.
On
Assume that the amount of the drug administered is a constant $A$ and that the amount of drug in the body at the start of the interval from $t$ to $t+1$ before the drug is administered is $D_t$. Then at the start of the interval from $t$ to $t+1$ after the drug is administered we have $D_t+A$ in the body and at the end of the interval we have: $$ D_{t+1}=(D_t+A)-p\times(D_t+A) $$
Now as $t\to \infty$ we should get $D_{t+1}\to D_t$ (that is the amount in the body at the end of an interval approaches that in the body at the end of the previous interval). Hence after a large number of intervals (that is for large $t$): $$ D_t\approx (D_t+A)-p\times(D_t+A) $$ Solving this gives: $$ D_t\approx A(1-p)/p $$ This approximation becomes better and better as the number of intervals increases, so that in the limit: $$ D_{\infty}= A(1-p)/p $$
The model I think is
$$D_t = t*D_0-p*D_0*(t-1)$$
The (t-1) depends on when the withdrawl happens before or after the drug has been administered into the body on any given time interval. If it is after, then the model is $$D_t = t*D_0-p*D_0*t$$