Pharmacokinetics studies the evolution of a drug after its administration in the body, by measuring its plasma concentration, that is to say its concentration in the plasma. In this exercise we study the evolution of plasma concentration in a patient of the same dose of drug, considering different modes of administration.

\textbf{Part A: administration by intravenous route}

We denote $f ( t )$ the plasma concentration, expressed in microgram per litre ( $\mu \mathrm { g } . \mathrm { L } ^ { - 1 }$ ), of the drug, after $t$ hours following administration by intravenous route. The mathematical model is: $f ( t ) = 20 \mathrm { e } ^ { - 0,1 t }$, with $t \in [ 0 ; + \infty [$. The initial plasma concentration of the drug is therefore $f ( 0 ) = 20 \mu \mathrm {~g} . \mathrm { L } ^ { - 1 }$.

\begin{enumerate}
  \item The half-life of the drug is the duration (in hours) after which the plasma concentration of the drug is equal to half the initial concentration. Determine this half-life, denoted $t _ { 0,5 }$.
  \item It is estimated that the drug is eliminated as soon as the plasma concentration is less than $0.2 \mu \mathrm {~g} . \mathrm { L } ^ { - 1 }$. Determine the time from which the drug is eliminated. The result will be given rounded to the nearest tenth.
  \item In pharmacokinetics, we call AUC (or ``area under the curve''), in $\mu \mathrm { g } . \mathrm { L } ^ { - 1 }$, the number $\lim _ { x \rightarrow + \infty } \int _ { 0 } ^ { x } f ( t ) \mathrm { d } t$. Verify that for this model, the AUC is equal to $200 \mu \mathrm {~g} . \mathrm { L } ^ { - 1 }$.
\end{enumerate}

\textbf{Part B: administration by oral route}

We denote $g ( t )$ the plasma concentration of the drug, expressed in microgram per litre ( $\mu g.L^{-1}$ ), after $t$ hours following ingestion by oral route. The mathematical model is: $g ( t ) = 20 \left( \mathrm { e } ^ { - 0,1 t } - \mathrm { e } ^ { - t } \right)$, with $t \in [ 0 ; + \infty [$. In this case, the effect of the drug is delayed, since the initial plasma concentration is equal to: $g ( 0 ) = 0 \mu g . \mathrm { L } ^ { - 1 }$.

\begin{enumerate}
  \item Prove that, for all $t$ in the interval $[ 0 ; + \infty [$, we have:\\
$g ^ { \prime } ( t ) = 20 \mathrm { e } ^ { - t } \left( 1 - 0,1 \mathrm { e } ^ { 0,9 t } \right)$.
  \item Study the variations of the function $g$ on the interval $[ 0 ; + \infty [$. (The limit at $+ \infty$ is not required.) Deduce the duration after which the plasma concentration of the drug is maximum. The result will be given to the nearest minute.
\end{enumerate}

\textbf{Part C: repeated administration by intravenous route}

We decide to inject at regular time intervals the same dose of drug by intravenous route. The time interval (in hours) between two injections is chosen equal to the half-life of the drug, that is to say the number $t _ { 0,5 }$ which was calculated in A - 1. Each new injection causes an increase in plasma concentration of $20 \mu \mathrm {~g} . \mathrm { L } ^ { - 1 }$. We denote $u _ { n }$ the plasma concentration of the drug immediately after the $n$-th injection. Thus, $u _ { 1 } = 20$ and, for all integer $n$ greater than or equal to 1, we have: $u _ { n + 1 } = 0,5 u _ { n } + 20$.

\begin{enumerate}
  \item Prove by induction that, for all integer $n \geqslant 1 : u _ { n } = 40 - 40 \times 0,5 ^ { n }$.
  \item Determine the limit of the sequence $( u _ { n } )$ as $n$ tends to $+ \infty$.
  \item We consider that equilibrium is reached as soon as the plasma concentration exceeds $38 \mu \mathrm {~g} . \mathrm { L } ^ { - 1 }$. Determine the minimum number of injections necessary to reach this equilibrium.
\end{enumerate}