Two-Compartment Model & Population Pharmacokinetics

Two-Compartment Model & Population Pharmacokinetics

1. Two-Compartment Model — Mass Balance

Central compartment (volume $V_c$, contains plasma + highly perfused organs) and peripheral compartment (volume $V_p$, muscle, fat, connective tissue). Intercompartmental transfer rate constants $k_{12}$ (central→peripheral) and $k_{21}$ (peripheral→central). Elimination occurs only from the central compartment with rate constant $k_{10} = k_e$.

$$\frac{dC_c}{dt} = -(k_{12}+k_{10})C_c + k_{21}C_p \cdot \frac{V_p}{V_c}$$

$$\frac{dC_p}{dt} = k_{12}\frac{V_c}{V_p}C_c - k_{21}C_p$$

After IV bolus of dose $D$, the biexponential solution:

$$C_c(t) = Ae^{-\alpha t} + Be^{-\beta t}$$

2. Macro Rate Constants α and β

The macro rate constants are the eigenvalues of the transfer matrix:

$$\alpha + \beta = k_{12} + k_{21} + k_{10}$$

$$\alpha \cdot \beta = k_{21} \cdot k_{10}$$

Solving: $\alpha = \frac{(k_{12}+k_{21}+k_{10}) + \sqrt{(k_{12}+k_{21}+k_{10})^2 - 4k_{21}k_{10}}}{2}$. The coefficients $A$ and $B$ are obtained from initial conditions:

$$A = \frac{D}{V_c} \cdot \frac{\alpha - k_{21}}{\alpha - \beta}, \qquad B = \frac{D}{V_c} \cdot \frac{k_{21} - \beta}{\alpha - \beta}$$

AUC$_\infty = A/\alpha + B/\beta$. Total clearance: $CL = k_{10} V_c = D/\text{AUC}$.

3. Population Pharmacokinetics — NLME Models

Individual patient PK parameters (e.g., $CL_i$, $V_{d,i}$) vary due to age, weight, renal function, and genetic factors. The NONMEM (nonlinear mixed-effects) model:

$$CL_i = \theta_{CL} \cdot (CRCL_i/80)^{\theta_1} \cdot e^{\eta_{CL,i}}$$

where $\theta_{CL}$ is the population mean clearance (fixed effect), $(CRCL/80)^{\theta_1}$ is the creatinine clearance covariate, and $\eta_{CL,i} \sim N(0,\omega^2)$ is the between-subject random effect (BSV). Residual variability $\varepsilon_{ij} \sim N(0,\sigma^2)$ captures assay error and model misspecification.

4. Bayesian Dose Individualisation

Given a population PK model (prior) and one or two measured drug concentrations (likelihood), Bayesian estimation updates the patient-specific parameter estimates. The MAP (maximum a posteriori) estimate minimises:

$$\text{OFV} = \sum_j \frac{(C_{obs,j} - C_{pred,j})^2}{\sigma^2} + \frac{\hat{\eta}^2}{\omega^2}$$

MAP Bayesian estimation with one vancomycin trough sample reduces the prediction error from ~50% (population prior) to ~15% (individual posterior), enabling precise AUC-guided dosing per the ASHP/IDSA 2020 vancomycin consensus guidelines.