Why a Dose Change Does Not Reset Concentration
A proposed dose change alters future drug input; it does not instantly remove drug already absorbed from earlier doses. An individual forecast must therefore combine the contribution of the proposed regimen with residual contributions from the actual dosing history. This distinction matters especially soon after a change, when concentrations can reflect both the previous and the new dose.
The forecast is conditional on an individual’s posterior distribution for pharmacokinetic parameters. For each plausible parameter value, the dose history is propagated through the concentration model. The resulting set of profiles describes parameter uncertainty; if the aim is to predict a future measured concentration rather than the underlying concentration, residual measurement variability must also be included.
Concentration Contributions Under a Linear Oral Model
For a precise calculation, consider a one-compartment model with first-order oral absorption and elimination. Let \(D_j\) be the amount of dose \(j\), given at time \(t_j\); let \(F\) be bioavailability, \(V\) the apparent volume of distribution, \(k_a\) the absorption rate constant, and \(k\) the elimination rate constant. Assume \(k_a>k>0\), consistent units for dose and volume, and instantaneous dose administration into the absorption process. The concentration contribution from that dose at time \(t\geq t_j\) is
Under linear, time-invariant PK, the total concentration is the sum of the contributions from all doses. For a forecast after time zero, the sum must include both observed historical doses and proposed future doses that have occurred by the forecast time. A dose scheduled after the forecast time contributes nothing yet.
where \(\theta\) denotes the individual PK parameters and \(\mathcal{D}\) is the complete dosing schedule. This expression assumes the same parameter values apply across the forecast period, doses are recorded correctly, and concentration contributions add linearly. It does not automatically accommodate saturable elimination, time-varying clearance, or a formulation change that alters absorption.
In Bayesian forecasting, the individual posterior \(p(\theta\mid \text{data})\) represents uncertainty about the parameters after considering the observed concentration data and the model. For each posterior draw \(\theta^{(m)}\), evaluate the same dose-history equation. The resulting collection of profiles reflects parameter uncertainty. A predictive distribution for a future measured concentration additionally includes the specified residual error model.
Worked Example: The First Concentration After a Dose Increase
Worked Example: The First Concentration After a Dose Increase
Suppose, hypothetically, an individual has been taking 100 mg orally every 12 hours. Doses occurred at times \(-24,-12,\ldots\) hours, and a proposed 150 mg dose is given at time 0. Forecast concentration at hour 4. For this calculation, use fixed individual parameters \(k=0.120\ \mathrm{h}^{-1}\), \(k_a=1.20\ \mathrm{h}^{-1}\), and \(F k_a/[V(k_a-k)]=0.0111111\ \mathrm{L}^{-1}\). These values are illustrative, not data from a real study.
At hour 4, the new dose has age 4 hours. Its contribution is
The preceding 100 mg doses have ages 16, 28, 40, and so on hours. Their combined residual contribution is a geometric sum:
Thus the forecast is \(1.01759+0.21348=1.23107\ \mathrm{mg/L}\), approximately \(1.231\ \mathrm{mg/L}\). The old doses account for about \(0.213\ \mathrm{mg/L}\) at this time. Omitting them would underpredict the model-based concentration, even though the proposed dose is larger than the former dose.
Posterior Uncertainty Changes the Profile, Not the Dose History
The preceding calculation used one fixed parameter set to make the arithmetic visible. An individual posterior generally contains multiple plausible parameter values. A practical forecast evaluates the full dose-history calculation for each posterior draw, keeping the proposed schedule fixed unless uncertainty about the schedule itself is explicitly represented. If parameter draws change the profile’s shape as well as its scale, each draw needs its own calculation; multiplying one profile by a common factor would then be inappropriate.
Worked Example: Summarizing Forecasts Across Posterior Draws
Continue the hypothetical example, and suppose three equally weighted posterior draws have identical absorption and elimination rates but differ in a parameter combination that scales concentration. Relative to the forecast in the first example, let the scale factors be 0.900, 1.000, and 1.200. These three values are a small illustration of draw-by-draw propagation, not a reliable substitute for a large posterior sample.
The hour-4 forecast for each draw is its scale factor times \(1.23107\ \mathrm{mg/L}\):
The equal-weight arithmetic mean is
The mean forecast is not the forecast from the mean parameter values in general: the concentration equation is nonlinear in some parameters, so evaluating it at average parameter values need not equal averaging the draw-specific concentrations. Here the median of these three forecasts is \(1.23107\ \mathrm{mg/L}\). With a real posterior sample, summaries should be computed from all draws, and any interval should be described as a posterior interval for the model-based forecast, subject to the model and dosing assumptions.
Transient Behavior and the New Steady-State Profile
The new regimen’s steady-state profile is not the same as the first profile immediately after a dose change. Under repeated oral doses of amount \(D\) every \(\tau\) hours, the dose ages at steady state, at a time \(s\) hours after a dose with \(0\leq s<\tau\), are \(s,s+\tau,s+2\tau,\ldots\). Summing their individual contributions gives a pair of geometric series.
This is a profile over one interval of the repeated regimen. The formula follows because \(e^{-k(s+n\tau)}=e^{-ks}(e^{-k\tau})^n\), and the sum over \(n=0,1,\ldots\) is \(e^{-ks}/(1-e^{-k\tau})\); the absorption terms sum in the same way. Both series converge because \(0<e^{-k\tau}<1\) and \(0<e^{-k_a\tau}<1\).
Worked Example: Comparing the Transient Profile With Steady State
Use the hypothetical individual and proposed 150 mg every-12-hour regimen from the first example. Immediately before the next dose, 12 hours after the first 150 mg dose, the contribution from that dose is
The earlier 100 mg doses, with ages 24, 36, and so on hours, contribute
The transient concentration just before the next dose is therefore approximately \(0.39488+0.08174=0.47662\ \mathrm{mg/L}\). In contrast, the steady-state concentration just before a 150 mg dose on the new regimen is
The transient trough is lower because the earlier doses were 100 mg, not 150 mg. In this example the difference narrows as the new regimen replaces the old dose contributions. The time course depends on the individual elimination and absorption rates and on the prior schedule; it should not be assumed that steady state is reached after a fixed number of doses for every individual.
Assumptions, Diagnostics, and Alternatives
The superposition calculation is appropriate when the model is linear in dose and parameters are treated as stable over the forecast horizon. Before interpreting a profile, check that the dose times, amounts, route, and formulation are represented correctly; that units for time and rates agree; and that posterior draws are used consistently with the individual’s observed data. A plausible-looking curve does not validate those assumptions.
- Check dose-time bookkeeping. Confirm which doses occurred before the forecast time and which are only planned for later. A dose given at time 0 must be included at time 4, while one planned for time 8 must not be.
- Check the transition. Immediately after a change, compare the calculated profile with the contributions from old and new doses separately. A discontinuity in concentration at the dose time may indicate an implementation error for this oral absorption model.
- Check posterior plausibility. Extreme profiles may arise from posterior draws with weakly identified parameters, prior-data conflict, or a coding error. Inspect draw-specific parameter values and profiles rather than silently deleting inconvenient draws.
- Check model adequacy. If concentrations show systematic time patterns not captured by the model, or the drug has nonlinear or time-varying PK, simple dose superposition may be unsuitable. A mechanistic state model or another adequately justified PK model may be needed.
A single point estimate of the individual parameters produces one conditional profile and is simpler to communicate, but it suppresses parameter uncertainty. Posterior draw propagation retains that uncertainty and can represent correlations among parameters, provided the joint draws are preserved. Neither approach repairs an incorrect dose history or a misspecified PK model. A model that accounts for changing parameters or nonlinear dose dependence may be more appropriate in some settings, but it requires those features to be supported and estimated rather than assumed away.
Check Your Understanding
Work through the questions, then compare with the solutions below.
- Why must doses given before a dose change remain in a transient forecast?
- Under the one-compartment oral model, what is the age of a dose given at time 3 hours when forecasting at time 9 hours?
- What mathematical condition allows the steady-state contribution from repeated doses to be written as a geometric series?
- When may concentration forecasts from posterior draws not be summarized by simply scaling one reference profile?
Solutions. (1) Earlier doses may still contribute drug to the concentration at the forecast time, so their residual contributions remain part of the model sum. (2) Its age is \(9-3=6\) hours. (3) For positive \(k\), \(k_a\), and interval \(\tau\), the series ratios \(e^{-k\tau}\) and \(e^{-k_a\tau}\) lie strictly between 0 and 1, so the geometric sums converge. (4) If posterior draws change parameters that alter profile shape, such as \(k\) or \(k_a\), rather than only a common concentration scale, each draw requires its own full dose-history calculation.