Compartment Models

• Compartment models describe the body as one or more well-mixed spaces the drug moves between, so that plasma concentrations over time can be described mathematically
  • The compartments are mathematical rather than anatomical, although they loosely correspond to groups of tissues with similar blood flow

One-Compartment Model

• The drug distributes instantly and evenly through a single volume ($V_D$), so after an IV bolus the only process is elimination:

$$\frac{dA}{dt}=-CL\cdot C=-k\cdot A\therefore C(t)=C_0{e}^{-kt},k=\frac{CL}{V_D}$$

• A plot of log concentration against time is a straight line with slope $-k$, and extrapolating it back to $t=0$ gives $C_0=\text{Dose}/V_D$

Two-Compartment Model

• The drug is given into a small central compartment ($V_1$: plasma and well-perfused organs such as the brain, heart, liver and kidney) and distributes to a peripheral compartment ($V_2$: poorly perfused tissue such as muscle and fat)
  • Elimination is assumed to happen from the central compartment
• Writing the amount of drug in each compartment as $X_1$ and $X_2$:

$$\begin{array}{rl}\frac{d{X}_1}{dt}& =-(k_{10}+k_{12})X_1+k_{21}{X}_2\\ \frac{d{X}_2}{dt}& =k_{12}{X}_1-k_{21}{X}_2\end{array}$$

• The solution after an IV bolus is the sum of two exponentials (biexponential):

$$C(t)=A{e}^{-\alpha t}+B{e}^{-\beta t},\alpha +\beta =k_{10}+k_{12}+k_{21},\alpha \beta =k_{10}{k}_{21}$$

- **Distribution phase ($\alpha$, fast):** plasma concentration falls steeply as drug moves out to the peripheral compartment, even though little has been eliminated yet
- **Elimination phase ($\beta$, slow):** once the compartments have equilibrated, plasma concentration falls at a rate set by clearance; the elimination (terminal) half-life is $0.693/\beta$

What equation describes plasma concentration after an IV bolus in a two-compartment model, and what does each phase represent?

$C(t)=A{e}^{-\alpha t}+B{e}^{-\beta t}$. The fast $\alpha$ phase is distribution from the central to the peripheral compartment; the slow $\beta$ phase is elimination after the compartments have equilibrated (terminal half-life $=0.693/\beta$).

Volumes in a Two-Compartment Model

• With two compartments, “the” volume of distribution depends on when it is measured, since the ratio of amount in the body to plasma concentration changes as the drug distributes:

Volume	Formula	Meaning
Central ($V_1$)	$\frac{\text{Dose}}{A+B}$	Volume at the moment of injection, before any distribution
Steady state ($V_{SS}$)	$V_1+V_2$	Amount in the body divided by plasma concentration at steady state
Area ($V_{\text{area}}$ or $V_{\beta }$)	$\frac{CL}{\beta }=\frac{\text{Dose}}{\beta \cdot \text{AUC}}$	Relates amount to concentration during the elimination phase
Extrapolated ($V_{\text{extrap}}$)	$\frac{\text{Dose}}{B}$	What back-extrapolating the terminal phase gives

• In order of size: $V_1<V_{SS}\le V_{\text{area}}<V_{\text{extrap}}$
• Back-extrapolating the terminal phase to $t=0$ gives its intercept ($B$) rather than the true initial concentration ($A+B$), which overestimates the volume:
• How much it overestimates depends on how fast distribution is compared with elimination. Integrating the biexponential for the AUC:

$$\begin{array}{rl}\text{AUC}& ={\int }_0^{\infty }\left(A{e}^{-\alpha t}+B{e}^{-\beta t}\right)dt=\frac{A}{\alpha }+\frac{B}{\beta }\\ \therefore V_{\text{area}}& =\frac{\text{Dose}}{\beta \cdot \text{AUC}}=\frac{\text{Dose}}{B+A\frac{\beta }{\alpha }}\end{array}$$

- When distribution is much faster than elimination ($\alpha\gg\beta$), the $A\beta/\alpha$ term is negligible and $V_{\text{extrap}}\approx V_{\text{area}}$, which is why the one-compartment back-extrapolation is usually good enough
- When distribution is slow or large (e.g. digoxin), the extrapolated volume overestimates

• The central volume determines the peak concentration straight after a bolus ($\text{Dose}/V_1$), whereas $V_{SS}$ is the one to use for a loading dose

Why does back-extrapolating the terminal phase overestimate the volume of distribution in a two-compartment model?

It gives the intercept of the elimination phase ($B$) rather than the true initial concentration ($A+B$), so $V_{\text{extrap}}=\text{Dose}/B$ is larger than $V_{\text{area}}=\text{Dose}/(B+A\beta /\alpha )$. The error is small when $\alpha \gg \beta$.