19 Single and repeated oral doses
Developing a 2-compartment model
So far we have assumed that as soon as the drug is administered it is immediately present in the bloodstream since it is administered intravenously. However, many drugs are administered in other ways, such as orally through tablets. In this case the drug will first reach the gastrointestinal (GI) tract where it dissolves and is gradually absorbed into the bloodstream. In this case we need to include two compartments in our model – one for the amount of the drug in the GI tract (which we might denote by [latex]X_G[/latex]), and one for the amount of the drug in the bloodstream ([latex]X_B[/latex]). Notice that I have used the word amounts here, as the concentration would have little meaning in the GI tract.
Let’s initially assume that there is a one-off dose of the drug. The amount of the drug in the GI tract cannot be increased, and simply reduces as it is absorbed in to the bloodstream. Hence, we can describe the dynamics of this first compartment as,
[latex]\begin{equation} \frac{dX_G}{dt}=-aX_G. \end{equation}[/latex]
For the second compartment we can assume all the drug that leaves the GI tract arrives in the bloodstream, and this then decays as before as the body uses up the drug. Hence, the second equation will be,
[latex]\begin{equation} \frac{dX_B}{dt}=V\frac{dC}{dt}=aX_G-kX_B=aX_G-kVC. \end{equation}[/latex]
Even before deriving the solutions to these equations, we can get a good qualitative picture of what might happen. The amount of drug in the GI tract will decay exponentially down towards zero. Initially, the drug concentration in the bloodstream will be very low, suggesting little being lost due to decay, but the amount being absorbed in to the bloodstream would be relatively high, meaning the concentration will initially increase. As time goes on, the amount of drug left in the GI tract will decrease until a point is reached that the absorption of new drug is less than the decay of existing drug, and the bloodstream concentration will reduce. Eventually we would expect the bloodstream concentration to also approach zero.
Let’s show this formally. We can solve this pair of equations in turn. The first is in a fairly simple form and just yields,
[latex]\begin{equation} X_G(t)=X_G(0)e^{-at}. \end{equation}[/latex]
We can then substitute this in to the second equation, to give,
[latex]\begin{equation} \frac{dX_B}{dt}=V\frac{dC}{dt}=aX_G(0)e^{-at}-kVC. \end{equation}[/latex]
This can be re-arranged in to the form,
[latex]\begin{equation} \frac{dC}{dt}+kC=\frac{aX_G(0)}{V}e^{-at}. \end{equation}[/latex]
Written in this form, we can see it is possible to solve this equation by use of an integrating factor.
Exercises
[latex]\begin{equation} C(t)=\frac{aX_G(0)}{V(k-a)}(e^{-at}-e^{-kt}). \end{equation}[/latex]
Click for solution
The integrating factor here will be [latex]e^{kt}[/latex]. We multiply through every term by this to get,
[latex]\begin{align*} e^{kt}\frac{dC}{dt}+e