Chapter 1.1, Problem 13P

Pharmacokinetics. Asimple model for the concentration $C\left(t\right)$ of a drug administered to a patient is based on the assumption that the rate of decrease of $C\left(t\right)$ is negatively proportional to the amount present in the system,

$\frac{dC}{dt}=-\text{}kC$,

where $k$ is rate constant that depends on the drug and value can be found experimentally.

Suppose that a dose administered at time $t=0$ is rapidly distributed throughout the body, resulting in an initial concentration $C_0$ of the drug in the patient. Find $C\left(t\right)$, assuming the initial condition is $C\left(0\right)=C_0$.

Consider the case where doses of $C_0$ of the drugare givenat equal time intervals $T$, that is, doses of $C_0$ are administered at times $t=0,T,2T,\mathrm{...}$. Denote by $C_n$ the concentration immediately after the $n\text{th}$ dose. Find an expression for the concentration $C_2$ immediately after the second dose.

Find an expression for the concentration $C_n$ immediately after the $n\text{th}$ dose. What is $\lim {}_{n\to \infty }{C}_n$?