Aligning VPCs Across Doses and Regimens

Why Dose and Regimen Alignment Matters

A visual predictive check (VPC) compares observed data summaries with the corresponding summaries from replicate data simulated under a fitted model. For a concentration VPC, those summaries are commonly quantiles within time bins. A time axis that ignores how and when doses were given can combine observations that do not represent the same pharmacological situation. A resulting trend may then reflect a mixture of dose histories rather than a clear assessment of predictive performance.

The choice of alignment is part of defining the predictive target. Elapsed time from the start of a study may be useful when the question concerns the whole study course. Time since a dose may be more useful for a single-dose profile. For repeated or changing doses, a regimen-specific panel may be needed because the same time since the latest dose can follow different prior exposure.

Dose alignment does not mean that every VPC should use time since the last dose. It means that the display should match the scientific question and preserve important features of the data-generating design. In particular, simulations used for the VPC should reproduce the observation times, dose amounts, dose times, and other design features relevant to that target. If they do not, discrepancies between observed and simulated summaries may be caused by a mismatch in the comparison rather than by the model.

Choosing an Axis and a Comparison

For a single dose, time since administration is usually a direct way to compare the absorption and disposition profile across subjects. If sampling times are recorded from study start and dose times differ, calculate time since dose from the event records rather than treating study time as though it were dose-relative. If doses differ in amount, separate dose panels or dose strata can show whether the model reproduces each level. A pooled display is appropriate only when the pooled population is itself the intended target and the design has been represented correctly in simulation.

For repeated dosing, the time since the most recent dose is helpful but may not be sufficient. Concentration after a dose can depend on earlier doses, the dosing interval, missed or delayed doses, and whether the subject has reached the relevant treatment phase. Thus, observations at the same \(s_{ik}\) can be aligned on the horizontal axis but still belong to different predictive conditions. Use panels or strata for regimen features that materially change the expected profile, such as dosing interval or dose level. When doses change over time, consider whether the target is the current-dose response, the entire dose history, or a particular treatment phase; these are not interchangeable.

A normalized regimen phase can sometimes help label a position within an interval. If a subject receives doses every \(\tau_i\) time units, define \(u_{ik}=s_{ik}/\tau_i\). This puts observations between dose events on a zero-to-one phase scale when the interval is regular. It does not make different intervals pharmacologically equivalent: two observations at the same \(u_{ik}\) can occur different numbers of hours after dose and after different accumulated histories. Use this scale only when relative phase is meaningful, and retain interval or regimen information in the panels or strata.

What the VPC Summaries Compare

Before interpreting a dose-aligned VPC, specify the summary and the population to which it refers. Suppose observations are assigned to a time bin \(b\), and the target summary is the \(p\)th quantile of concentration \(Y\). The observed summary is \(q_{p,b}^{\mathrm{obs}}\), calculated from observed concentrations in that bin and stratum. For each simulated replicate \(m\), calculate the same quantile, \(q_{p,b}^{(m)}\), using the corresponding simulated records and design.

A simulation-based prediction interval for the quantile is formed from the distribution of these replicate summaries. For a central 95% interval, its endpoints are the 2.5th and 97.5th percentiles of the \(M\) simulated quantiles. This interval describes variation in the quantile summary across replicate datasets generated under the model and design; it is not a confidence interval for a fixed model parameter. The observed quantile is compared with this interval, alongside the other quantile and time-bin patterns.

The calculation is meaningful only when each replicate uses the same inclusion rules, binning rule, alignment, and stratum definition as the observed data. If subjects are assigned to dose strata using their observed dose level, the simulated records must preserve that assignment. If a stratum is too sparse to estimate its quantiles stably, a jagged or extreme observed curve may be sampling noise rather than evidence of model failure.

Why Pooling Can Hide or Invent Patterns

A pooled quantile is a valid summary of a pooled population if that is the target. It can nevertheless be hard to interpret when subjects follow very different dosing conditions. The pooled median need not resemble the median profile of any actual dose or regimen group. The issue follows directly from how a quantile is computed: it depends on the combined ordering of all values, not on an average of the subgroup quantiles.

A useful way to choose the display is to start with the predictive question, then ask what features need to match within a comparison. The following sequence makes that choice explicit.

Diagnostics, Alternatives, and Limitations

A dose-aligned display is going wrong when its construction hides a relevant mismatch or creates groups too small to interpret. Check dose and sample-time records, especially around dose changes and repeated doses. Verify that observations at a plotted time truly have the intended relationship to a dose event. Inspect the number of subjects and observations in each bin and stratum; unstable counts can produce irregular quantile curves and wide simulation intervals. Also check that simulated data preserve the observed design. A simulation that applies one nominal dose history to subjects who actually received several histories does not provide the intended comparison.

There are several legitimate alternatives, depending on the question. A pooled VPC summarizes overall predictive performance under the combined design but can conceal subgroup-specific misfit. A VPC stratified by dose or regimen is easier to interpret within each condition but may be noisy when groups are small. A study-time VPC retains the full course and can reveal calendar-time or treatment-phase patterns, but it may overlay observations at different times after dose. A time-since-dose VPC makes post-dose timing easier to compare, while requiring regimen history to be retained when accumulation or interval differences matter. These displays answer related, not identical, questions.

Do not create many narrow strata solely because dose records differ in minor ways. Excessive splitting reduces precision and can make a check unreadable. Conversely, merging markedly different schedules just to obtain smoother curves can conceal failure. Combine groups only when the combined predictive target is meaningful and the model simulations reproduce their composition. The right choice balances interpretability, data support, and relevance to the intended predictive target.

Finally, dose-normalizing concentrations is not a general substitute for alignment or stratification. Such normalization assumes a suitable dose-proportional relationship; it can be misleading when pharmacokinetics are nonlinear or when dose history differs. Likewise, a shared normalized interval phase does not erase differences in actual elapsed time or accumulated exposure. These transformations should not be used to make unlike profiles appear comparable without a defensible model and scientific rationale.