Compartmental pharmacokinetic modeling
Fit a simple concentration–time curve after a single intravenous bolus and estimate compartment parameters.
Concepts and reasoning
A model of how concentration changes
A compartmental pharmacokinetic model describes movement of drug between idealized, well-mixed spaces. Compartments are mathematical representations; they do not necessarily correspond to individual anatomical organs. The model links dose and time to concentration through parameters such as clearance and volume.
For a single intravenous bolus in one compartment, the model assumes immediate mixing and first-order elimination. Concentration declines exponentially. Volume connects the initial concentration to the dose, while clearance controls the volume of plasma effectively cleared per unit time.
A two-compartment model adds a peripheral space and exchange with the central space. An early distribution phase and a later terminal decline can then appear. The terminal slope is a combination of underlying rates; it is not generally equal to clearance divided by central volume.
Model structure and measurement noise
The structural model describes the expected concentration trajectory. The residual error model describes deviations of measured concentrations from it. Constant additive errors treat a one-unit error similarly at all concentration levels. Proportional or log-scale errors instead emphasize relative discrepancies. The choice changes the fitting objective and can change parameter estimates.
A visually close curve is not enough. Different parameter combinations can generate almost the same sampled trajectory, especially if early distribution or late elimination is sparsely measured. The Penn State early-phase course notes provide background on pharmacokinetic interpretation and study design.
Mathematical foundations
A one-compartment intravenous bolus
With dose D, volume V, and clearance CL,
The elimination rate k = CL/V has units of inverse time. The half-life and exposure are
For a two-compartment model, let A₁ and A₂ be drug amounts, Vc and Vp the central and peripheral volumes, and Q the intercompartmental clearance:
For an ideal central bolus, A₁(0) = D and A₂(0) = 0. The resulting concentration is a sum of two exponential terms. The macro-rates governing those terms depend jointly on CL, Q, Vc, and Vp.
Nonlinear least squares minimizes a sum of squared residuals under the chosen weighting. Local standard errors approximate the model near the fitted parameters; they can be misleading when the objective is flat, strongly curved, or close to a parameter boundary.
Worked example
Worked example: separating volume and clearance
Suppose an illustrative intravenous dose is 100 mg, V = 10 L, and CL = 1 L/h. Then C(0) = 10 mg/L and k = 0.1 per hour.
| Time (h) | Predicted concentration (mg/L) |
|---|---|
| 0 | 10.000 |
| 5 | 6.065 |
| 10 | 3.679 |
| 20 | 1.353 |
Half-life is log 2/0.1 = 6.931 hours, and AUC∞ is 100/1 = 100 mg·h/L.
Now double V to 20 L while keeping CL at 1 L/h. The initial concentration halves to 5 mg/L and the rate halves to 0.05 per hour, so the half-life doubles to 13.863 hours. AUC remains 100 because dose and clearance have not changed.
Alternatively, keep V = 10 L and double clearance to 2 L/h. The initial concentration stays 10, half-life halves to 3.466 hours, and AUC halves to 50. Volume and clearance therefore affect different features of the curve.
If all measurements are collected late, the data may estimate a decay rate reasonably while providing little information about the initial amplitude. In a two-compartment model, missing the early phase makes it particularly difficult to distinguish central volume and distribution parameters. More parameters cannot manufacture the missing early information.
Interpreting the results
Parameters describe a concentration model
In a one-compartment intravenous-bolus model, clearance and central volume determine the initial concentration and exponential decline. Clearance has volume-per-time units and volume scales dose to concentration. These are model parameters inferred from the curve, not direct anatomical measurements.
A two-compartment model adds exchange between central and peripheral compartments. The early and late decline phases depend jointly on clearance, volumes, and intercompartmental clearance; the late slope should not be interpreted as simply clearance divided by central volume.
Approximate parameter intervals quantify local fitting uncertainty under the residual model. Wide or unstable intervals can coexist with a visually close concentration fit because different parameter combinations predict similar curves. Report the dosing model, units, sampling range, and identifiability limitations with the estimates. A fitted curve is not by itself a validated dosing recommendation.
Assumptions and limitations
Match administration and error structure
A single intravenous bolus assumes immediate entry into the central compartment. Oral absorption, infusion duration, repeated dosing, time-varying clearance, or nonlinear elimination require different equations. Pooling observations from different individuals can confound between-person variation with the residual error of a single curve.
Ordinary nonlinear least squares with additive constant-variance errors gives each concentration residual the same error scale. If measurement error grows with concentration, a proportional or combined error model may be more appropriate. Fitting log concentrations would also change the objective and interpretation.
The sampled times must identify the desired compartments. Without early samples, central distribution may be weakly identified; without late samples, clearance and terminal behavior may be uncertain. Positivity constraints prevent negative parameters but do not guarantee biological plausibility or unique estimation.
Diagnostics and sensitivity analysis
Inspect identifiability as well as curve agreement
Compare observed and fitted concentrations and inspect residuals against fitted values and time. Systematic early or late departures suggest structural mismatch; increasing model complexity may fit noise if the sampling design cannot distinguish the added parameters.
Review the Jacobian condition number and parameter intervals. Strong parameter dependence can make small data changes produce large parameter changes while leaving the curve nearly unchanged. Compare defensible one- and two-compartment descriptions and seek additional informative sampling when necessary.
Check dose and unit conversions, administration route, and whether the earliest concentration is compatible with the assumed bolus. Confirm predictions in an independent or prospectively collected profile where feasible. Extrapolation outside the sampled interval and use for a different dosing regimen require validation beyond the fitted residual error.
References and other material
Read the methodological references and supporting tutorials below. The examples in this lesson are original teaching calculations, not findings from these sources.
- Penn State STAT 509: early phase studies — Free course notes introducing pharmacokinetic models and early development.