The absorption and elimination of a drug in the body may be modelled with a mechanism

Chapter 22, Problem 22.32

(choose chapter or problem)

The absorption and elimination of a drug in the body may be modelled with a mechanism consisting of two consecutive reactions:

\(\begin{array}{cllll} A & \rightarrow & B & \rightarrow & C \\ \text { drug at site of } & & drug dispersed & & eliminated \\ administration & & in blood & & drug \end{array}\)

where the rate constants of absorption \((\mathrm{A} \rightarrow \mathrm{B})\) and elimination are, respectively, \(k_1\) and \(k_2\).

(a) Consider a case in which absorption is so fast that it may be regarded as instantaneous so that a dose of \(\mathrm{A}\) at an initial concentration \([\mathrm{A}]_0\) immediately leads to a drug concentration in blood of \([\mathrm{B}]_0\). Also, assume that elimination follows first-order kinetics. Show that, after the administration of n equal doses separated by a time interval \(\tau\), the peak concentration of drug B in the blood, \([\mathrm{P}]_m\), rises beyond the value of \([\mathrm{B}]_0\) and eventually reaches a constant, maximum peak value given by

\([\mathrm{P}]_{\infty}=[\mathrm{B}]_0\left(1-\mathrm{e}^{-k_2 \tau}\right)^{-1}\)

where \([\mathrm{P}]_n\) is the (peak) concentration of \(\mathrm{B}\) immediately after administration of the nth dose and \([\mathrm{P}]_{\infty}\) is the value at very large n. Also, write a mathematical expression for the residual concentration of \(\mathrm{B},[\mathrm{R}]_n\), which we define to be the concentration of drug \(\mathrm{B}\) immediately before the administration of the (n+1)th dose. \([\mathrm{R}]_n\) is always smaller than \([\mathrm{P}]_n\) on account of drug elimination during the period \(\tau\) between drug administrations. Show that \([\mathrm{P}]_{\infty}-[\mathrm{R}]_{\infty}=[\mathrm{B}]_0\).

(b) Consider a drug for which \(k_2=0.0289 \mathrm{~h}^{-1}\).  (i) Calculate the value of \(\tau\) required to achieve \([\mathrm{P}]_{\infty} /[\mathrm{B}]_0=10\). Prepare a graph that plots both \([\mathrm{P}]_n /[\mathrm{B}]_0\) and \([\mathrm{R}]_n /[\mathrm{B}]_0\) against n.  (ii) How many doses must be administered to achieve \([\mathrm{P}]_n\) value that is 75 per cent of the maximum value? What time has passed during the administration of these doses?  (iii) What actions can be taken to reduce the variation \([\mathrm{P}]_{\infty}-[\mathrm{R}]_{\infty}\) while maintaining the same value of \([\mathrm{P}]_{\infty}\)?

(c) Now consider the administration of a single dose \([\mathrm{A}]_0\) for which absorption follows first-order kinetics and elimination follows zero-order kinetics. Show that with the initial concentration \([\mathrm{B}]_0=0\), the concentration of drug in the blood is given by

\([\mathrm{B}]=[\mathrm{A}]_0\left(1-\mathrm{e}^{-k_1 t}\right)-k_2 t\)

Plot \([\mathrm{B}] /[\mathrm{A}]_0\) against t for the case \(k_1=10 \mathrm{~h}^{-1}, k_2=4.0 \times 10^{-3} \mathrm{mmol} \mathrm{dm}^{-3} \mathrm{~h}^{-1}\), and \([\mathrm{A}]_0=0.1 \mathrm{mmol} \mathrm{dm}^{-3}\). Comment on the shape of the curve.

(d) Using the model from part (c), set \(\mathrm{d}[\mathrm{B}] / \mathrm{d} t=0\) and show that the maximum value of [B] occurs at the time \(t_{\max }=\frac{1}{k_1} \ln \left(\frac{k_1[\mathrm{~A}]_0}{k_2}\right)\). Also, show that the maximum concentration of drug in blood is given by \([\mathrm{B}]_{\max }=[\mathrm{A}]_0-k_2 / k_1-k_2 t_{\max }\).