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# Solved problem on Pharmacokinetics

Problem 1: Plasma half life of a drug is 30 minutes. It follows the $1^{st}$ order metabolism kinetics. Find out the elimination rate constant of the drug.

Solution 1: We know that \begin{equation*} t_{1/2}=\frac{.693}{k} \end{equation*} putting the value of $t_{1/2}$ in the equation, \begin{equation*} 30 min=\frac{.693}{k} \end{equation*} therefore, \begin{equation*} k=\frac{.693}{30 min} \end{equation*} so, the value of elimination rate constant is 0.023/minutes.

Problem 2: 500 mg of drug A is given to a patient. Calculate the amount of drug to be present in plasma after 1 hour, taking the value of elimination rate constant as 0.5/minutes.

Solution 2: We know that, \begin{equation*} \ln{A_0}-ln{A_t}=k\times t \end{equation*} here, $A_0$=500 mg; k=0.5/minutes; t=60 minutes; therefore, \begin{equation*} \ln{500 mg}-ln{A_t}=0.5/min\times 60 min \end{equation*} \begin{equation*} 6.215 mg-ln{A_t}=30 \end{equation*} \begin{equation*} ln{A_t}=-23.785 mg \end{equation*} \begin{equation*} A_t=e^{-23.785} mg \end{equation*} \begin{equation*} A_t=4.68 mg \end{equation*} so, 4.68 mg drug A will be present in plasma after one hour of administration.

Problem 3: 100 mg of the drug A is administered to a patient. The drug follows zero order kinetic of elimination. Find out the elimination rate constant of the drug. Find out the elimination rate constant of the drug [$t_{1/2}$=20 minutes].

Solution 3:For zero order kinetics, the correlation of A and $k_0$ are as follows: \begin{equation*} t_{1/2}=\frac{A_0}{2\times k_0} \end{equation*} putting the given values in the equation, we obtain that \begin{equation*} 20 min=\frac{100 mg}{2\times k_0} \end{equation*} therefore, \begin{equation*} k_0=\frac{100}{2\times20} mg/min \end{equation*} \begin{equation*} k_0=\frac{100}{2\times20} mg/min \end{equation*} \begin{equation*} k_0=2.5 mg/min \end{equation*} The elimination rate constant of the drug is 2.5 mg/minutes.