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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.

Deepak
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    How much of the drug is administered every 4 hours? Also, do you know how to solve a linear difference equation? – John Douma Oct 11 '15 at 05:21
  • @JohnDouma the amount is not given and we don't have to find the amount of drug at any given interval. Just the formula for any interval. I do understand basic linear difference equations. – Deepak Oct 11 '15 at 05:27
  • Assume the amount of the drug administered is a constant $a$. Then $D_{t+1}=D_{t} + a - pD_{t}= (1-p)D_t+a$. You can now solve this as a linear difference equation and get an expression for $D_t$. – John Douma Oct 11 '15 at 05:30

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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: $$ D_{\infty}= A(1-p)/p $$

Conrad Turner
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  • The handwavy argument can be made fully rigorous (avoiding to write things like $D_t\to D_{t+1}$, which is absurd), noting that the recursion formula is equivalent to $$D_{t+1}-D_\infty=(1-p)(D_t-D_\infty),$$ hence $$D_t=D_\infty+(1-p)^t(D_0-D_\infty)\to D_\infty,$$ for every $D_0$, since $|1-p|<1$. – Did Oct 19 '15 at 20:49
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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 $$D_t = t*D_0-p*D_0*t$$

Satish Ramanathan
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