20 A two compartment bolus model
A bloodstream-tissue model
For our final chapter (well done for making it!) we will return to the scenario of intravenous bolus doses. In our earlier models we assumed the body acted as a single compartment for the drug, basically considering the drug concentration in the bloodstream. When we looked at the orally administered drug models we assumed there was now a second compartment where the drug had to enter the body (the GI tract) before it could move on to the bloodstream. In that case things simplified because the dynamics of the first compartment were very simple – exponentially decreasing to zero – and only concentrate on what happened in the bloodstream.
Now we will consider a more complicated case where there are two compartments, but the drug can now move in and out of both of those compartments. A biological example might be the bloodstream as our main compartment, and tissue as the second compartment. We will assume the drug is again administered by an intravenous bolus and so appears immediately in the bloodstream. The concentration of the drug in the bloodstream will again be given by [latex]C[/latex], and the concentration in the tissues will be given by [latex]D[/latex]. While in the bloodstream the drug is used up and is eliminated. The drug can also pass in to tissues at some rate [latex]a[/latex] and can pass back from tissues to the bloodstream at rate [latex]b[/latex]. We will assume the drug is not eliminated while in the tissue. Our mathematical model will thus look like,
[latex]\begin{align} \frac{dC}{dt}&=-kC-aC+bD\\ \frac{dD}{dt}&=aC-bD. \end{align}[/latex]
Solving a system of linear equations
Since both equations explicitly depend on both variables, we cannot handle this case in quite the same way as the previous pharmacokinetic models. We also cannot solve the system one equation at a time – i.e. apply an integrating factor to solve one, then substitue that solution into the second. However, we have a well established toolbox for dealing with a system of linear ordinary differential equations. I have not provided any explicit background review material for these methods, but will try to take the explanations quite slowly.
First note that we can write this out in matrix form, that is,
[latex]\begin{equation*} \frac{d}{dt}\left(\begin{array}{c}C\\D\end{array}\right)=\left(\begin{array}{cc}-k-a &b\\a & -b\end{array}\right)\left(\begin{array}{c}C\\D\end{array}\right). \end{equation*}[/latex]
For a system like this we expect to see solutions of the form,
[latex]\begin{equation} \left(\begin{array}{c}C(t)\\D(t)\end{array}\right)=\left(\begin{array}{c}u_1\\u_2\end{array}\right)e^{\lambda_1 t}+\left(\begin{array}{c}v_1\\v_2\end{array}\right)e^{\lambda_2 t}. \end{equation}[/latex]
This has become an eigenvalue/eigenvector problem since, substituting these desired solutions into our matrix equation we have,
[latex]\begin{equation} \lambda_1\left(\begin{array}{c}u_1\\u_2\end{array}\right)e^{\lambd