Terminal-Phase Errors From Missing Distribution Compartments

Why Late-Phase Errors Deserve Attention

A model can describe the peak and early decline of a concentration–time profile reasonably well yet systematically underpredict concentrations later in the dosing interval. If this pattern recurs across individuals and is not explained by absorption, assay behavior, or data handling, it can indicate that the structural model is missing a slower distribution process. The important question is not simply whether a more complex model fits better, but whether the added distribution structure is supported by informative data.

The pattern is suggestive, not diagnostic by itself. Late residuals can also arise from an incorrect absorption description, a changing clearance process, or an unsuitable residual error model. Earlier material on residual patterns and on absorption-related misspecification provides relevant checks: interpret the timing and direction of residuals in context rather than assigning a mechanism from one plot.

How a Missing Compartment Shapes the Terminal Phase

For an illustrative intravenous bolus profile in a linear two-compartment model, the central concentration can be written as a sum of two exponential terms:

where \(A\) and \(B\) are positive concentration coefficients and \(\alpha\) and \(\beta\) are rate constants. The faster component, with rate \(\alpha\), reflects the early decline associated with distribution and elimination; the slower component, with rate \(\beta\), dominates later. These are profile-level terms: \(A\), \(B\), \(\alpha\), and \(\beta\) are not themselves the compartment volumes and clearances.

A one-compartment model instead assumes a single exponential decline, \(C(t)=C_0e^{-kt}\). When a two-exponential profile is forced into that form, one value of \(k\) cannot generally describe both early and late decline. The following limit makes the reason precise.

As \(t\) increases, \(e^{-(\alpha-\beta)t}\) tends to zero because \(\alpha>\beta\). Thus \(C(t)\) approaches \(B e^{-\beta t}\), and the late log-concentration slope approaches \(-\beta\). More exactly, differentiating the log concentration gives

The apparent decline rate therefore transitions from a faster value toward \(\beta\). A one-compartment model fitted mainly to early data may estimate a decline that is too fast for the terminal phase. Its late predictions will then be too low, producing positive observed-minus-predicted residuals at late times. If the one-compartment model is fitted to all observations, it may compromise across phases rather than matching the early slope exactly, but a systematic time-ordered residual pattern can remain.

Diagnosing the Pattern in Population Data

Start with residuals plotted against time after dose, preferably with observations and predictions on a scale that makes the residual definition clear. With observed-minus-predicted residuals, repeated positive values late in the profile indicate underprediction. With weighted residuals, inspect whether the pattern is systematic rather than a few large values. Look both across the population and within individuals: a pooled trend can conceal different individual patterns, while an isolated individual pattern may not support a population-wide structural change.

The timing matters. A distribution explanation is more plausible when residuals are reasonably centered earlier, then show a consistent late-phase departure, and when the data include enough late observations to characterize the decline. Consider whether those observations are above the quantification limit and whether the sampling schedule actually spans the transition. If only early samples are available, the slower phase may not be observed well enough to estimate its rate or magnitude.

Before adding a compartment, check the model’s prediction and residual calculations, dose and sampling times, units, assay limits, and the assumed absorption description where relevant. A late residual pattern is evidence of model failure, not a label for its cause. In particular, different structural problems can produce superficially similar underprediction.

Does the Peripheral Compartment Improve the Model?

A two-compartment model adds structural flexibility and parameters. Compare it with the simpler model using residual patterns, parameter estimates and uncertainty, and fit criteria computed on the same data under comparable estimation and error-model assumptions. The Akaike information criterion is \(AIC=-2\ell+2p\), and the Bayesian information criterion is \(BIC=-2\ell+p\log(n)\), where \(\ell\) is the maximized log likelihood, \(p\) is the number of estimated parameters, and \(n\) is the number of observations used in the criterion. Smaller values indicate a preferred balance of fit and complexity for that criterion; they do not by themselves establish the biological truth of a model.

The central-plus-peripheral model is useful when the late residual pattern is reduced and the new parameters are estimable with reasonable precision. In a conventional linear mammillary model, \(CL\) denotes clearance from the central compartment, \(Q\) intercompartmental clearance, \(V_c\) central volume, and \(V_p\) peripheral volume. A fitted peripheral compartment should not be treated as established merely because an optimizer returns numerical estimates.

Because the simpler model is recovered at a boundary of the larger model when exchange vanishes, conventional likelihood-ratio reference distributions may not apply automatically. In particular, parameters that describe a compartment can be undefined or uninformative under the null model. Do not interpret an ordinary likelihood-ratio p-value without checking the conditions for its reference distribution. Fit criteria and residual diagnostics are useful comparisons, but neither removes the need to evaluate uncertainty and identifiability.

Diagnostics, Alternatives, and Limitations

Warning signs of a poorly supported peripheral compartment include large standard errors, very high parameter correlations, estimates that change markedly with starting values or small data perturbations, and profiles that do not constrain a parameter to a useful range. Convergence by itself is not evidence of identifiability. Also inspect whether the improvement is driven by one individual or a small number of late observations.

The main alternative is to retain a one-compartment model if the apparent late pattern is not reproducible, is explained by data issues or another structural mechanism, or cannot support stable estimates for the added parameters. A more complex distribution model may be justified when the data show additional structure, but adding compartments beyond what the design can resolve risks trading interpretable parameters for an apparently improved fit. These choices are conditional on the sampling design, the population studied, and the purpose of the model.

A common mistake is to reason, “The two-compartment model has a lower AIC, so the peripheral compartment is proven.” The correction is that the criterion compares fit and complexity under its assumptions; it does not prove a mechanism or guarantee that each added parameter is identified. Another mistake is to call every positive late residual evidence of a missing compartment. Check for alternative causes, repeated patterns, adequate late sampling, and stable parameter estimates before making that interpretation.