I am trying to learn pharmacokinetics.
Let us consider an intramuscular depot injection of $5 \,\pu{mg}$ ($5\cdot10^9 \,\pu{pg}$) of a drug, whose half-time is 4 days. Let us say that the initial plasma concentration is $C(0) = 50 \,\pu{pg/ml}$. After 3 days a blood test was taken and showed the plasma concentration $C(3) = 170 \,\pu{pg/ml}$.
We consider a two-compartment model in which:
first-order absorption to the central compartment with rate $k_a$
first-order distribution from the central compartment to the peripheral compartment with rate $k_d$
first-order elimination from the central compartment with rate $k_e$
We plot the amount in the serum, by solving the system of differential equations:
$$ \left\{ \begin{array}{l} \frac{dx_1}{dt} = - k_a x_1 \\ \frac{dx_2}{dt} = k_a x_1 - k_d x_2 - k_e x_2 \\ \frac{dx_3}{dt} = k_e x_2 \end{array} \right. $$
Where $x_1$ is the amount in the depot, $x_2$ is the amount in the central compartment, and $x_3$ is the amount in the peripheral compartment.
We find that the amount of the drug in serum reaches, say $0.15 \,\pu{mg}$ ($1.5\cdot10^8 \,\pu{pg}$) on day 3. If we divide that by the amount of serum (e.g. $2.8 \,\pu{l} = 2800 \, \pu{ml}$), we obtain $$C(3) = \frac{1.5\cdot10^8 \,\pu{pg}}{2800 \, \pu{ml}} = 53571 \,\pu{pg/ml}$$ However the real concentration is just $170 \,\pu{pg/ml}$. Where does this all drug amount go? Why is there such a large mismatch?
I feel that I am missing something very basic.
Note: Here is an example of a two-compartment model.
