Worked Analysis Of Competing Absorption Models

Why Compare Absorption Structures?

A concentration-time profile can often be described by more than one plausible absorption model. A first-order model represents absorption as a rate proportional to the amount remaining to be absorbed. A zero-order model represents input at a constant rate for a finite duration. These structures can produce similar profiles over some sampling schedules, yet imply different concentration predictions between observations and different fitted formulation effects. The question is not simply which curve follows the observed points most closely: it is whether the structural choice is supported by the data and whether it materially changes the bioequivalence quantity of interest.

This tutorial compares two candidate absorption structures fitted to the same hypothetical formulation-comparison data. The illustrative calculations focus on concentration predictions, residual fit, Gaussian-likelihood information criteria, and exposure ratios. Numerical optimizer output is used for the fit comparison; the examples do not represent a real study.

Two Candidate Input Structures

Consider a one-compartment disposition model with first-order elimination rate constant \(k\), in units of \(h^{-1}\). For formulation \(j\), let \(Q_j\) denote the concentration-scale quantity \(F_jD/V\), in mg/L, where \(F_j\) is bioavailability, \(D\) is dose, and \(V\) is apparent volume. These equations are used here to make the structural comparison explicit; they do not remove the need to check whether a one-compartment disposition model is adequate.

With first-order absorption at rate \(k_{a,j}\), concentration after an oral dose is

With zero-order input lasting \(T_j\) hours, the concentration is

Both models assume complete input of the scaled amount \(Q_j\) by the end of absorption and first-order elimination thereafter. In a formulation comparison with two formulations, a simple version uses a common \(k\), two formulation-specific scales \(Q_j\), and two formulation-specific absorption parameters: \(k_{a,j}\) for the first-order model or \(T_j\) for the zero-order model. Thus each candidate has five structural parameters. If the concentration errors are modeled as Gaussian with an unknown common variance, that variance is an additional estimated parameter for likelihood-based information criteria.

For either model, the complete-profile area under the concentration-time curve is \(Q_j/k\). The absorption shape changes the time course, but under these model assumptions it does not change the area for a fixed \(Q_j\) and \(k\). In practice, however, fitting a different shape can change the estimates of \(Q_j\), \(k\), or both, so the fitted exposure ratio can change.

Assumptions and Comparison Strategy

A direct likelihood comparison requires the candidate models to be fitted to the same response observations and to use comparable likelihood assumptions. For the calculations below, assume concentration residuals are independent, normally distributed with mean zero and a common unknown variance \(\sigma^2\), conditional on the model predictions. In an actual analysis, residual plots and the design must support the chosen error structure; the earlier material on residual error models and random-effects structures provides the relevant alternatives when these assumptions are not credible.

The models should also be identifiable from the sampling schedule. If few samples are collected around the absorption phase, \(k_a\) or \(T\) may be weakly determined, or may trade off against the scale and elimination parameters. A lower residual sum of squares alone does not establish that the fitted structure is scientifically adequate. Check convergence, parameter estimates and uncertainty, residual patterns over time and by formulation, and prediction at informative times. Use the same observations, handling of below-quantification values, and error model in both fits.

For Gaussian errors with estimated common variance, let \(SSE\) be the sum of squared concentration residuals and \(n\) the number of observations. The maximized likelihood estimates the variance as \(SSE/n\), not as \(SSE/(n-p)\). The Akaike information criterion is \(AIC=-2\ell+2p\), where \(p\) counts all estimated likelihood parameters, including the residual variance. With \(p=6\) here, it includes five structural parameters plus the common variance.

AIC compares relative information loss among the specified candidates; a smaller value is preferred, but it is not a test that proves a model true. AICc adds a small-sample correction, \(2p(p+1)/(n-p-1)\), when \(n>p+1\). The correction can be large when the number of observations is only modestly greater than the parameter count. If the same observations and same \(p\) are used for both candidates, the penalty terms cancel in their AIC difference, but the absolute AICc values may still warn that the comparison is based on a small sample.

Fit, Information Criteria, and Predictive Checks

The next hypothetical comparison summarizes output from fitting both structures to the same eight concentration observations. The residuals below are observed minus predicted concentrations, in mg/L. They are included so the sum-of-squares arithmetic can be checked directly. The fitted model predictions are treated as numerical optimizer output; the purpose is to compare the two fits, not to infer the parameter values from this compact diagnostic summary.

A predictive check asks whether the fitted model predicts observations it was not tuned to fit. For example, with sufficient data, one can hold out observations or subjects, fit each candidate to the remaining data, and compare prediction errors on the held-out data. The split must respect the design: in a crossover study, randomly holding out individual concentration records can leak information from the same participant into both sets. Holding out whole participants is often more defensible for assessing prediction to new participants. AIC and held-out prediction answer related but different questions: AIC estimates relative expected information loss under its likelihood assumptions, while validation measures performance on the chosen held-out sample.

How Structural Choice Changes the Exposure Ratio

For the simple models above, if the dose and disposition parameters are common across formulations, the formulation exposure ratio is \(Q_T/Q_R\), where \(T\) and \(R\) denote test and reference. The model-derived ratio can differ across structures because the fitted scales are estimated jointly with the absorption-shape parameters. This is especially plausible when sampling is sparse or when the assumed input shape is not a good description of the data. The following calculation treats the two sets of fitted scales as hypothetical outputs from fitting the same comparison dataset.

Diagnostics, Alternatives, and Limitations

Review residuals against time, predicted concentration, and formulation. A systematic early-time pattern can indicate that the input phase is misspecified; a late-time trend can indicate an inadequate disposition description. Unequal residual spread may contradict the assumed common variance. Also inspect parameter estimates, uncertainty, boundary behavior, and convergence from multiple starting values. A very large or unstable \(k_a\), an implausible absorption duration, or strong parameter correlation may indicate weak identifiability rather than a meaningful mechanistic finding.

When both models have similar fit, use predictive performance and scientific plausibility rather than selecting the one with the marginally smaller SSE. AIC or AICc is useful when the likelihood assumptions and observation set are aligned. Cross-validation or simulation-based predictive checks can provide complementary evidence. A more flexible structure may improve in-sample fit but may also estimate poorly and predict less reliably. Conversely, a simpler structure can be easier to interpret but fail to capture an important feature of the data. Model averaging or sensitivity analysis can communicate the consequences of structural uncertainty, but neither makes a misspecified likelihood correct.

The comparison is limited by the candidate set: the better of two structures may still be inadequate. These equations also assume the specified disposition form, complete absorption as represented, and no additional lag or transit process. If the data cannot distinguish the structures, report that uncertainty rather than treating the preferred criterion as proof. The earlier material on structural identifiability, bootstrap uncertainty, and sensitivity analysis gives additional tools for evaluating those limitations.