From Interim Measurements to a Regimen Decision
An interim adaptation is a sequence of linked calculations, not a dose change justified by one striking concentration or biomarker value. The trial team first checks that the interim data can be interpreted under the prespecified models. It then updates the population pharmacokinetic (PK) model, connects exposure to a pharmacodynamic (PD) target, predicts the consequences of candidate regimens, and applies a rule fixed before examining the interim results.
This worked example uses hypothetical data from a once-daily drug. The endpoint is the percentage reduction in a disease biomarker at a fixed assessment time; larger reductions are better. Concentrations are trough concentrations at steady state, measured in mg/L. The example is deliberately small enough to calculate by hand. In an operational trial, the models, data checks, prediction code, and decision rule would be prespecified and independently verified.
The calculations below use a simplified population model. They assume the interim participants and future participants are drawn from the same target population; concentration measurements are available at the specified steady-state time; and the PK model is adequate over the candidate dose range. The PD model is assumed linear over the observed concentration range. The safety threshold is also prespecified. These assumptions are substantive: if they are not credible, the numerical probabilities do not become reliable merely because the arithmetic is correct.
Worked Example: Updating the Population PK Model
Worked Example: Updating the Population PK Model
Suppose 12 hypothetical participants receive 100 mg once daily. Their observed trough concentrations have a geometric mean of 6.40 mg/L, so the sample mean of the log concentrations is \( \bar{x}=\ln(6.40)=1.8563 \). For this illustration, assume log concentration at 100 mg follows a normal population model with unknown typical log concentration \( \theta \) and known individual log-scale standard deviation \( \sigma=0.12 \):
Assume conditional independence across participants and a prior \( \theta\sim N(\ln 8,0.20^2) \). This prior is centered on a typical concentration of 8 mg/L at 100 mg. For a normal likelihood with known variance and a normal prior, the posterior variance and mean are:
Thus the posterior standard deviation is \( \sqrt{0.001165}=0.0341 \), and the posterior median typical concentration at 100 mg is \( \exp(1.8628)=6.44 \) mg/L. The data shift the typical concentration below the prior value of 8 mg/L. The shift is modestly uncertain, rather than a known fixed correction. The assumed common log-scale variability and the independence of participants are part of this calculation; clustering or materially different variability would require a different likelihood.
For a dose \(D\), assume dose-proportional exposure across the candidate range: the typical concentration is multiplied by \(D/100\). Conditional on \( \theta \), a future participant’s log concentration has standard deviation 0.12. Integrating over posterior uncertainty in \( \theta \) gives the predictive log-scale standard deviation \( \sqrt{0.12^2+0.0341^2}=0.1248 \). Therefore, in this example:
This predictive distribution includes both variation among future participants and uncertainty about the population typical concentration. Using only the posterior mean would omit the first component; treating the interim estimate as known would omit the second.
Linking Exposure to the Biomarker Target
The interim PK observations alone do not establish whether a regimen is likely to produce the desired biomarker response. For that step, suppose six hypothetical participants have paired trough concentration and biomarker-reduction measurements at the same prespecified assessment time. A simple local exposure–response model is \(Y_i=a+bC_i+e_i\), where \(Y_i\) is reduction in percentage points and \(e_i\) is a mean-zero residual. The linear model is used only over the observed concentration range, and it assumes measurements are comparable in timing and definition.
Worked Example: Estimating a Biomarker Target Concentration
The six hypothetical paired observations are \( (C,Y)=(5,17),(6,20),(7,24),(8,27),(9,31),(10,34) \), with concentration in mg/L and response in percentage points. Their means are \( \bar{x}=7.5 \) mg/L and \( \bar{y}=25.5 \) percentage points. The least-squares slope and intercept are:
The fitted relationship is \( \hat{Y}=-0.4286+3.457C \). Setting the fitted reduction to the target of 30 percentage points gives:
The target concentration lies within the observed range of 5 to 10 mg/L, so this calculation does not extrapolate beyond these data. It is still a model-based threshold, not a guarantee that every participant above 8.80 mg/L will reach the biomarker target. The probability calculation must account for the distribution of future concentrations. For simplicity, assume the fitted PD relationship and its residual uncertainty were evaluated and accepted in advance; this illustration propagates PK predictive uncertainty and uses 8.80 mg/L as the concentration threshold.
Predictive Comparison and the Prespecified Rule
Consider candidate doses of 100, 125, 150, 175, and 200 mg once daily. A regimen must satisfy both parts of the prespecified rule: the probability that a future participant’s trough is at least 8.80 mg/L must be at least 0.70, and the probability that it exceeds the safety threshold of 12.0 mg/L must be no more than 0.10. The safety threshold and cutoffs are hypothetical design choices, not universal standards.
For each candidate, the predictive distribution is lognormal. If \(m_D=6.44(D/100)\) is the posterior median concentration, then:
where \( \Phi \) is the standard normal cumulative distribution function. The strict or non-strict inequality at a concentration threshold makes no numerical difference for a continuous distribution.
Worked Example: Applying the Adaptation Rule
At 150 mg, the posterior median concentration is \(6.44(150/100)=9.66\) mg/L. The target-attainment probability is:
The predicted probability of exceeding 12.0 mg/L is:
Thus 150 mg passes both requirements: 0.772 is at least 0.70, and 0.041 is below 0.10. Repeating the same calculations for the other candidate doses gives the rounded comparison below. The adaptation rule selects the lowest candidate dose that meets both requirements.
| Dose (mg) | Predictive median (mg/L) | Target attainment | Above safety threshold | Rule result |
|---|---|---|---|---|
| 100 | 6.44 | 0.023 | <0.001 | Fails target attainment |
| 125 | 8.05 | 0.238 | 0.001 | Fails target attainment |
| 150 | 9.66 | 0.772 | 0.041 | Qualifies |
| 175 | 11.27 | 0.976 | 0.307 | Fails safety requirement |
| 200 | 12.88 | 0.999 | 0.712 | Fails safety requirement |
Under the stipulated rule, the regimen changes from 100 mg to 150 mg for the next eligible participants. This is a decision about predicted outcomes under the model and the stated thresholds. It is not a claim that 150 mg has been proven clinically superior or that each participant will attain the target.
Why the Calculation Works, and What to Check
The PK update follows from the normal-normal model. The sample mean of 12 independent log concentrations has distribution \( \bar{X}\mid\theta\sim N(\theta,0.12^2/12) \). Multiplying its likelihood by the normal prior gives a normal posterior. The posterior precision is the sum of prior precision and data precision; the posterior mean is the corresponding precision-weighted average. For a future participant, the log concentration can be written as \( \theta+\varepsilon_{\mathrm{new}} \). Posterior uncertainty in \( \theta \) and new-participant variation are independent under the model, so their variances add. This gives the predictive variance \(0.0341^2+0.12^2\), rather than either variance alone.
Before acting on a calculation like this, the team should inspect whether the assumptions and data support it. Important checks include:
- Data comparability: confirm dose, sampling time, steady-state status, units, assay version, and biomarker assessment window. A shifted sampling time can mimic a dose-related concentration change.
- Distributional adequacy: compare observed log concentrations with predictive distributions and inspect outliers or patterns by site, dose, and relevant covariates. A few unexpectedly high or low values may signal data errors or model misspecification, not a population shift.
- Dose proportionality: check whether proportional scaling is defensible over the proposed doses. Saturable absorption, changing clearance, or dose-dependent bioavailability can invalidate the \(D/100\) scaling.
- Exposure–response adequacy: check that the biomarker timing and response definition are consistent and that the concentration range supports the fitted relationship. A linear fit near its boundary should not be used to justify distant extrapolation.
- Predictive calibration: compare interim observations with predictions made before the interim, where available. Systematic underprediction or overprediction is evidence to investigate before adaptation.
A key limitation of this compact example is that it treats the PD target concentration as fixed after fitting the interim response relationship. A full analysis may propagate uncertainty in the slope, intercept, residual variation, and PK parameters jointly. If the response model has substantial uncertainty, the target-attainment probability should integrate over that uncertainty; otherwise the rule may look more decisive than the evidence warrants. The introductory material on predictive diagnostics and model uncertainty provides the broader rationale for such checks.
Alternatives, Failure Modes, and Interpretation
A point-estimate approach might multiply the estimated typical concentration by each dose and choose the dose whose predicted mean response reaches 30%. That approach ignores between-participant variability and parameter uncertainty: a regimen can have a satisfactory mean while leaving many participants below target or exposing an appreciable fraction above the safety threshold. The posterior predictive comparison addresses those two risks directly, conditional on its model assumptions.
A different alternative is to choose the regimen maximizing a prespecified utility that trades off efficacy and safety continuously. That can be useful when no clear threshold rule is appropriate, but it requires defensible utility weights and can be harder to communicate. The threshold rule used here is transparent and auditable; its limitation is that candidates just above and just below a cutoff may be treated differently despite similar predicted probabilities. In either approach, predictions inform an adaptation; they do not replace clinical judgment, safety oversight, or later evaluation of clinical outcomes.
A common mistake is to read the 0.772 target-attainment probability as a 77.2% probability that the model is correct, or as a 77.2% chance that the next group as a whole will attain the target. The proper interpretation is narrower: conditional on the interim data, the specified models, and the 150 mg regimen, the predictive probability that one future participant’s trough is at least 8.80 mg/L is 0.772. Likewise, the safety probability is a predicted individual-level probability under the model, not a guarantee about the cohort.
Another failure mode is changing the target, safety limit, or candidate set after seeing which dose appears attractive. That converts a prespecified decision into a result-dependent rule and makes the stated operating properties inapplicable. If diagnostics reveal serious model failure, the appropriate response is the prespecified fallback—such as pausing the adaptation for review—not silently substituting a more favorable model.
Confirm timing, units, dose records, and model-relevant measurements before interpreting parameter updates.
Combine the interim likelihood with the prespecified prior, then assess predictive fit and model assumptions.
Propagate relevant population and individual uncertainty to the target and safety probabilities.
Adapt only if a candidate satisfies every prespecified condition; otherwise use the stated fallback.
Check Your Understanding
Answer each question using the assumptions and numbers in the example. Brief solutions follow the questions.
- Why is the predictive log-scale variance larger than the posterior variance of the population parameter alone?
- At 150 mg, what are the two decision probabilities and why does this dose qualify?
- Suppose a candidate regimen has target-attainment probability 0.82 but probability above the safety threshold 0.14. Does it qualify under this rule?
- What does the 0.772 target-attainment probability mean, and what does it not mean?
- If no candidate regimen meets both probability cutoffs, what should determine the next action?
Solutions. 1. The predictive variance includes both uncertainty in the population typical concentration and new-participant log-scale variability: \(0.0341^2+0.12^2=0.1248^2\). 2. The target-attainment probability is 0.772 and the probability above 12.0 mg/L is 0.041; these meet the minimum 0.70 and maximum 0.10 requirements. 3. No. Although 0.82 meets the target criterion, 0.14 exceeds the allowed safety probability of 0.10, and both conditions are required. 4. Conditional on the interim data, model, and 150 mg regimen, it is the predictive probability that one future participant’s trough is at least 8.80 mg/L. It is not the probability that the model is true or a guarantee that a particular participant will respond. 5. The prespecified fallback action should determine what happens, for example continuing the current regimen or pausing for review; the rule should not be rewritten after seeing the results.