Visual Predictive Checks for Repeated Dosing

Why Repeated Dosing Needs an Explicit Time Axis

A concentration measured two hours after a dose is not necessarily comparable with a concentration measured two hours after another dose. The patient may have received different previous doses, may not yet have reached a repeating exposure pattern, or may have a different interval between doses. A VPC for repeated dosing therefore needs to specify which occasion is being assessed and how time within that occasion is represented.

The central practical choice is whether to summarize concentrations by dose occasion, by elapsed time after the most recent dose, or by both. Occasion-specific summaries can reveal changes across successive cycles, including accumulation. Elapsed-time alignment makes samples at comparable positions within an interval easier to compare. Neither alignment is automatically right: the choice follows the predictive-check target defined for the analysis.

Constructing Occasion-Aligned Summaries

First define the occasions in a way that matches the regimen and the question. For a regular regimen, occasion 1 might begin at the first dose and end immediately before the second; occasion 2 begins at the second dose, and so on. The exact treatment of an observation at a dose time must be specified consistently. A sample taken just before a dose is generally a pre-dose or trough sample for the interval that is ending, whereas a sample immediately after the dose belongs to the interval that is starting. The data convention and the model's event ordering must agree.

Within each occasion, define bins on \(\tau\), then calculate a statistic such as the median or selected quantiles of concentration in each bin. To assess accumulation, retain the occasion index as a separate grouping variable: compare the same time-after-dose bin across occasions. To assess within-interval shape, summarize across the selected occasions at matched \(\tau\), provided those occasions are intended to be comparable. If the target is both accumulation and interval shape, present both views rather than collapsing the two dimensions into one.

For each simulated replicate, apply the same occasion assignment, bin boundaries, and summary calculation as for the observed data. The reference envelope for a statistic is formed from its replicate-specific values. This is important: the comparison is between the observed statistic and the distribution of that statistic under the model, not between the observed statistic and an unstructured pool of all simulated concentrations. As in other VPCs, replicate data should preserve the relevant design, including individual dosing histories, sample times, and the model's variability.

Reference Envelopes and a Short Justification

Let \(T_b\) denote a chosen statistic, such as the median concentration in bin \(b\), calculated from the observed data. For replicate \(r\), calculate the same statistic \(T_b^{(r)}\). The reference envelope at a specified central level is obtained from the empirical distribution of \(T_b^{(1)},\ldots,T_b^{(R)}\), where \(R\) is the number of replicates. For example, a central 90% envelope uses the 5th and 95th percentiles of those replicate-specific statistics. It describes the range of values that the statistic can take under the fitted model and the simulated design; it is not a confidence interval for a model parameter.

The reason to calculate the statistic within each replicate is straightforward. A predictive check asks how the model would reproduce the feature being checked if the experiment were repeated under the model. Each replicate is one such simulated experiment, so it must undergo the same grouping and summarization as the observed experiment.

Here \(C_{ij}^{(r)}\) is a simulated concentration, \(g\) is the selected summary, and \(Q_u\) is the empirical \(u\)-quantile. The observed statistic is calculated using the corresponding observed concentrations and the same bin definition. With a small number of replicates, envelope endpoints are imprecise; replicate count and Monte Carlo precision should therefore be adequate for the intended comparison.

Accumulation and Within-Interval Predictions

Accumulation is a change in exposure across successive doses or occasions as residual drug from earlier doses contributes to later concentrations. It is not identified merely because later-occasion concentrations are higher: dose changes, changing intervals, time-varying covariates, adherence differences, or selection of different samples can also produce that pattern. A VPC can assess whether the model reproduces the observed occasion-specific distributions under the recorded dosing history; it cannot by itself establish the cause of a trend.

A useful display has occasion on one dimension and time after dose on the other, with separate summaries or panels for each occasion. At matched \(\tau\), compare observed and simulated medians and lower and upper quantiles. A systematic difference between observed and simulated summaries that grows with occasion number is consistent with a mismatch in the model's predicted accumulation pattern. A discrepancy that recurs at the same \(\tau\) across occasions instead points toward a within-interval prediction issue, such as a mismatch in peak timing or decline. These are diagnostic clues, not unique explanations.

When dosing intervals differ, \(\tau\) in hours remains directly interpretable, but a bin at 4 hours after dose may represent different parts of two intervals. If the target is position within each interval, one can additionally use the phase \(\tau/\Delta_{ik}\), where \(\Delta_{ik}\) is the interval duration. This rescales time to a fraction of the interval; it should be used only when that relative position is scientifically meaningful. It does not make pharmacokinetics identical across intervals or remove the need to account for each individual's actual regimen in simulation.

Diagnostics, Alternatives, and Limitations

Before interpreting a repeated-dosing VPC, check that dose times, dose amounts, interval lengths, and sample times used in the simulations match the intended design. Verify that observations are assigned to the correct interval, particularly around dose boundaries and for samples taken before a dose. Inspect counts by occasion and time bin: a visually dramatic quantile based on very few observations can be unstable. Also check whether some individuals contribute many samples or many occasions; repeated observations from one individual are dependent, so they must not be treated as if they were independent subjects when describing the simulation design.

The main alternatives answer different questions. A conventional elapsed-time VPC that pools all occasions summarizes performance across the pooled observations, but can conceal a poor fit in early or late cycles. An occasion-stratified VPC retains the cycle pattern, but divides the data into smaller groups and can become noisy. A phase-scaled VPC can compare relative position in intervals of different lengths, but can obscure differences in absolute elapsed time. Prediction correction can address certain systematic prediction differences under its stated setup, but it does not replace correct occasion assignment or a clear accumulation target.

A VPC is a graphical predictive diagnostic, not a parameter-estimation procedure or a causal analysis. Envelopes can overlap even when a structured discrepancy exists, especially with sparse bins; conversely, isolated points outside an envelope need not imply a consequential model failure. The check is limited by the chosen summaries, binning, sample size, simulation count, and the fidelity of the recorded regimen. Use multiple views that answer prespecified questions, and interpret patterns in the context of the model and design rather than treating a single crossing of an envelope as a pass-or-fail rule.