# Two-compartment model for depot intramuscular injections

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.

• You said you fit the model, right? What are you fitting it to? Can you plot x1, x2, x3 as a function of time? Commented May 10, 2023 at 19:07
• @BryanKrause Yes, I fit the model using the method of residuals. Since I cannot measure x1 and x3 directly, I used x2 only based on my measurements (blood tests). The curve does fit relatively well. It's just the number on a different scale. This is where I have this problem. Commented May 10, 2023 at 19:41
• Well, you have all the numbers you need to calculate x1 and x3. Can you plot them all (the predictions, that is)? Commented May 10, 2023 at 19:43
• I suspect there's either a problem with units, or something weird is happening with your model because you don't have enough information to distinguish between kd and ke. Commented May 10, 2023 at 20:01
• @Julia, it is difficult to comment the outcomes without your data and fitting protocol. I suggest you add data together with fits to the question. Commented May 11, 2023 at 13:07