Comparing Models by Dose-Time Extrapolation

Why Compare Models Outside the Observed Regime?

Two candidate pharmacokinetic SDEs can describe observed concentrations similarly yet imply very different predictions when the dose is increased, the dosing interval is changed, or follow-up continues beyond the last measurement. These differences matter because a model may be used precisely where data are sparse: to assess a new regimen, anticipate a late concentration, or examine variability at a dose not studied directly. A good fit within the observed range does not by itself establish that those predictions are reliable.

Dose-time extrapolation is a structured stress test, not a substitute for data at the target regimen. Specify the target quantity first, such as a concentration distribution at a stated time, a probability of exceeding a threshold, or an interval-specific peak or trough. Then compare candidate models under the same target scenarios, parameter information, and uncertainty calculation. The earlier material on calibration and model comparison motivates checking predictive distributions; here the emphasis is how those checks behave beyond the observed dose-time domain.

Keep three forms of extrapolation distinct. Dose extrapolation changes the administered dose while holding the schedule and prediction time fixed. Interval extrapolation changes the time between doses, which can alter accumulation and peak-trough patterns. Horizon extrapolation predicts farther beyond the observed follow-up. A scenario can involve more than one of these changes; if so, record each one. An extrapolation is only as interpretable as its target: comparing “predictions at a higher dose” is incomplete without a specified time and summary of the predictive distribution.

Predictions and Uncertainty Under Extrapolation

For a candidate model, a predictive distribution can include randomness in the latent SDE path and observation process, and uncertainty in estimated parameters. If random effects are part of the model, predictions may also average over the target population’s random effects or condition on a particular individual’s effects. State which prediction is intended. A population predictive distribution and a prediction conditional on estimated individual effects answer different questions.

When parameters are uncertain, a practical simulation procedure is to draw a parameter value from an appropriate uncertainty distribution, generate the relevant random effects if needed, simulate the SDE and observation process under the target regimen, and calculate the target quantity. Repeating this produces a predictive sample. The same parameter draws, initial conditions, dosing scenarios, and simulation conventions should be used across candidate models where they can be matched. Otherwise, differences in the reported spread may reflect different computational setups rather than model assumptions.

The following result makes that separation precise. Let \(\theta\) denote uncertain model parameters and \(Y\) a future quantity under a fixed target regimen. If \(E[Y^2]\) is finite, then

The first term is the average conditional predictive variance, which may include process and observation variability. The second is variation in the conditional mean as the uncertain parameter value changes. To prove the identity, write

Squaring and taking expectations, the cross term is zero because \(E[Y-E[Y\mid\theta]\mid\theta]=0\). The expected square of the first term is \(E[\operatorname{Var}(Y\mid\theta)]\), and the expected square of the second is \(\operatorname{Var}(E[Y\mid\theta])\). This proves the decomposition. \(\square\)

Dose Changes: What Scales and What Does Not?

A dose extrapolation asks whether candidate models imply plausible changes in both the level and variability of exposure when the dose changes. In a linear PK model, the mean concentration may scale proportionally with dose. In a multiplicative-noise model, absolute variability can scale with dose too, even if relative variability remains unchanged. Nonlinear dynamics, state-dependent diffusion, or dose-dependent parameters can break simple scaling. Thus, proportionality should be derived from the candidate model, not assumed from the fact that dose is doubled.

For the model in the previous example, the scaling claim can be proved directly. Let \(C_t^{(a)}\) be the solution starting at \(aC_0\), and \(C_t^{(1)}\) the solution starting at \(C_0\), driven by the same Brownian path. The explicit solution gives \(C_t^{(a)}=aC_t^{(1)}\) at every time. Consequently, its mean scales by \(a\), its variance by \(a^2\), and its coefficient of variation is unchanged. This is an exact property of this model, not a general SDE rule.

Interval Changes and Accumulation

Changing a dosing interval is not equivalent to multiplying a single-dose prediction. Residual concentration from earlier doses contributes to later concentrations, and stochastic evolution carries uncertainty between administrations. To see the calculation, take a constant bolus increment \(A\) and let the concentration evolve between doses according to the geometric Brownian model above. Over an interval \(\tau\), conditional on the concentration immediately after a dose, the mean is multiplied by \(e^{-k\tau}\) and the second moment by \(e^{(-2k+\sigma^2)\tau}\). At a bolus, the concentration jumps by \(A\); its mean increases by \(A\), and its second moment changes from \(E[C^2]\) to \(E[C^2]+2AE[C]+A^2\).

Comparison, Diagnostics, and Limits

No single comparison criterion answers every extrapolation question. Use observed-data fit and residual diagnostics to assess behavior where data exist, but do not treat them as proof of performance outside that range. Predictive calibration at observed regimens is useful evidence about the adequacy of predictive distributions there. It does not certify an unobserved high-dose or long-horizon prediction. Information criteria and held-out prediction can help compare candidates within the support of the data; when target-regimen outcomes are unavailable, they cannot supply an empirical score for that target. Extrapolation comparisons should instead make the model dependence visible, expose uncertainty, and guide which data would be most informative.

Several warning signs deserve attention. Candidate predictions may diverge sharply as dose or time increases; uncertainty bands may widen rapidly; simulated paths may approach implausible or inadmissible states; or the answer may depend strongly on an unverified parameter or diffusion specification. Check that stochastic simulations remain numerically stable under smaller time steps and that reported summaries are not artifacts of a few extreme paths. For positive concentrations, a model with additive noise may generate negative values; for multiplicative noise, positivity can hold while implausibly large values remain possible. These are model properties, not problems that can be fixed merely by plotting a different summary.

A common mistake is to call the candidate with the narrowest extrapolated interval “best.” Narrowness is not accuracy: it may result from omitted parameter uncertainty, an overly restrictive diffusion term, or fixing random effects at estimated values. Another mistake is to rank models by their agreement at extrapolated regimens when no outcomes there are available. The correction is to label such results as conditional projections, show the assumptions and uncertainty sources, compare a set of scientifically plausible scenarios, and seek target-regimen data when the decision warrants it. Extrapolation cannot identify which candidate is true when the candidates agree on observed data but differ only in unobserved regions.