Why Sensitivity Analysis Matters
A predicted drug-drug interaction (DDI) ratio is not independent of how the study is observed and analyzed. A sparse sampling schedule can miss a peak or trough; an AUC method can represent the curve differently between measured concentrations; and shifting a victim dose can change its overlap with time-varying perpetrator effects. Nonadherence and missing samples can also make the analyzed exposure differ from the exposure under the intended regimen.
Sensitivity analysis asks whether the substantive conclusion holds across plausible alternatives to these choices. It is not a search for whichever calculation gives the preferred result. The alternatives should be scientifically defensible, their assumptions should be explicit, and the primary analysis should be specified before examining results. Here, an exposure ratio is defined as the AUC in the DDI condition divided by the AUC in the control condition over the same interval. Any numerical decision boundary used below is hypothetical and illustrative, not a regulatory threshold.
Sampling Schedules and AUC Estimation
For concentrations measured at times \(t_1,\ldots,t_m\), the linear trapezoidal estimate over a measured interval is the sum of areas formed by joining adjacent observations with straight lines. For one interval, this is the interval width multiplied by the average of its two endpoint concentrations. The estimate is exact if concentration changes linearly between those endpoints, but a sparse schedule may poorly represent a curved profile.
A log-trapezoidal interval instead assumes exponential change between positive endpoint concentrations. If \(C(t)=C_j e^{k(t-t_j)}\) on an interval, integrating that curve gives the log-trapezoidal expression below. It is not defined when an endpoint is zero, and it may be unsuitable when the concentration curve changes direction within the interval. A hybrid convention may use linear interpolation for rising concentrations and log interpolation for falling concentrations, but that convention too must be prespecified.
The derivation follows directly from integration. For \(C(t)=C_j e^{k(t-t_j)}\), the endpoint relation is \(C_{j+1}=C_j e^{k\Delta t}\), so \(k=\ln(C_{j+1}/C_j)/\Delta t\). Integrating the exponential over \(\Delta t\) gives \((C_{j+1}-C_j)/k\), which is the stated expression after substituting for \(k\). Thus, the formula encodes a specific interpolation assumption; it is not simply a more accurate version of the linear rule in every setting.
Worked Example: A Sparse Schedule Changes the AUC Ratio
Consider hypothetical concentrations in mg/L measured at 0, 2, 4, 8, and 12 hours. The control profile is 0, 4, 8, 4, and 1 mg/L; the DDI profile is 0, 8, 12, 8, and 2 mg/L. Compare the linear-trapezoidal AUC ratio using all five times with the ratio using only 0, 4, and 12 hours. Use the same 0–12-hour window in both conditions. For illustration only, suppose a prespecified internal criterion flags ratios above 1.65.
All five times. In the control condition, the AUC is \(2(0+4)/2+2(4+8)/2+4(8+4)/2+4(4+1)/2=4+12+24+10=50\) mg·h/L. In the DDI condition, it is \(2(0+8)/2+2(8+12)/2+4(12+8)/2+4(8+2)/2=8+20+40+20=88\) mg·h/L. The ratio is \(88/50=1.76\).
Only 0, 4, and 12 hours. The control AUC is \(4(0+8)/2+8(8+1)/2=16+36=52\) mg·h/L. The DDI AUC is \(4(0+12)/2+8(12+2)/2=24+56=80\) mg·h/L. The ratio is \(80/52=1.538\), or approximately 1.54. The dense schedule exceeds the illustrative 1.65 criterion, while the sparse schedule does not. The sparse schedule has not just reduced precision: under linear interpolation it has changed the estimated shape between observations and therefore the ratio itself.
This comparison supports evaluating sampling times against expected profile curvature, especially around absorption peaks and periods of rapid concentration decline. It does not establish that the denser estimate is necessarily the true AUC; denser sampling can still miss features, and the underlying measurement and model assumptions still matter.
Worked Example: Linear and Log-Trapezoidal AUCs
Suppose hypothetical concentrations at 0, 2, and 4 hours are 2, 8, and 4 mg/L in the control condition, and 2, 12, and 4 mg/L in the DDI condition. Compare linear and log-trapezoidal AUC ratios over 0–4 hours. For this example, all endpoints are positive and each interval is treated as monotonic. Use 1.30 as an illustrative, prespecified comparison boundary.
Linear method. Control AUC is \(2(2+8)/2+2(8+4)/2=10+12=22\) mg·h/L. DDI AUC is \(2(2+12)/2+2(12+4)/2=14+16=30\) mg·h/L. The ratio is \(30/22=1.364\).
Log method. For control, the two interval areas are \(2(8-2)/\ln(8/2)=12/\ln(4)=8.656\) mg·h/L and \(2(4-8)/\ln(4/8)=8/\ln(2)=11.54\) mg·h/L. The total is \(20.20\) mg·h/L. For DDI, the areas are \(2(12-2)/\ln(12/2)=20/\ln(6)=11.16\) mg·h/L and \(2(4-12)/\ln(4/12)=16/\ln(3)=14.56\) mg·h/L. The total is \(25.73\) mg·h/L. The ratio is \(25.73/20.20=1.274\).
The linear ratio exceeds 1.30, whereas the log ratio is below it. The difference arises from the assumed concentration path between observations, not from different measured endpoints. Review the prespecified AUC convention, curve shape, and intended estimand before interpreting either result as definitive.
Dose Timing, Adherence, and Missing Data
Dose timing is consequential when the interaction varies over time. For example, the perpetrator concentration or its effect on victim-drug clearance may change over a dosing interval. The timing of the victim dose then determines how its absorption and elimination overlap with that changing effect. A timing sensitivity analysis should compare plausible schedules using the same AUC window and should align the control condition to the corresponding clock time. Otherwise, a difference attributed to the interaction could instead reflect the timing of the control profile.
Adherence defines what exposure is being estimated. A result under recorded, imperfect adherence describes the regimen as actually taken; a prediction under full adherence describes a different exposure scenario. Excluding nonadherent participants can also change the target population and introduce bias if adherence is related to participant characteristics or exposure. The treatment of missed doses should therefore follow the study estimand and a prespecified rule, rather than an after-the-fact preference for a larger or smaller ratio.
Missing concentration observations raise a distinct issue: an AUC cannot be calculated from an unobserved concentration without an assumption or a model. Linear interpolation between surrounding samples, model-based prediction, and exclusion of a profile rely on different assumptions. A missing late sample may particularly affect a partial AUC that includes the late interval; extrapolating beyond the last observed concentration adds assumptions beyond interpolation. Record which observations are missing, how they are handled, and whether plausible alternatives change the conclusion.
Worked Example: Dose Timing Can Change a Predicted Ratio
In a hypothetical PBPK simulation, the victim drug is administered at either 08:00 or 20:00. The control AUC over the corresponding 24-hour interval is 100 mg·h/L at either time. With the perpetrator regimen, the predicted AUC is 165 mg·h/L for an 08:00 victim dose and 138 mg·h/L for a 20:00 victim dose. Suppose, only for this example, the prespecified comparison boundary is a ratio of 1.50.
For the 08:00 dose, \(R_{\mathrm{AUC}}=165/100=1.65\). For the 20:00 dose, \(R_{\mathrm{AUC}}=138/100=1.38\). The first exceeds the illustrative boundary, and the second does not. This does not show that one schedule is generally preferable. It shows that the conclusion depends on dose timing under the model assumptions. The simulation should use the intended perpetrator schedule, and each DDI AUC should be compared with a control AUC measured or simulated on the matching clock-time schedule.
Worked Example: Adherence Assumptions Change the Target Result
Suppose a small hypothetical simulation produces individual DDI-to-control AUC ratios of 1.8, 2.0, and 1.1. The third participant missed a perpetrator dose. For illustration, summarize these ratios with their geometric mean and compare with a prespecified boundary of 1.70. This calculation is descriptive and does not provide a reliable population estimate from only three participants.
If all three observed ratios are included, the geometric mean is \((1.8\times2.0\times1.1)^{1/3}=3.96^{1/3}=1.582\), approximately 1.58. If the missed-dose profile is excluded under a hypothetical full-adherence analysis, the result is \((1.8\times2.0)^{1/2}=\sqrt{3.6}=1.897\), approximately 1.90. The first is below 1.70 and the second is above it.
These are answers to different questions, not interchangeable corrections. The all-profile result reflects the ratios under the observed adherence pattern; the restricted result describes the adherent profiles in this example. A study intended to predict exposure under full adherence needs a defensible, prespecified adherence analysis and should report how missing or excluded observations affect the result.
How to Organize a Sensitivity Analysis
A useful analysis starts with one primary design and analysis specification, then varies the choices that are both uncertain and capable of affecting the interpretation. First vary one factor at a time to identify influential choices. Then evaluate a limited set of joint scenarios when factors plausibly act together—for example, sparse sampling combined with a different interpolation rule. One-factor-at-a-time results do not reveal all interactions between assumptions.
State the treatment conditions, AUC window, ratio direction, sampling schedule, AUC convention, dose timing, and adherence or missing-data rules.
Use clinically and computationally defensible schedules, integration methods, dosing times, and data-handling assumptions. Do not vary choices solely to obtain a desired result.
Keep the comparison window and ratio direction fixed unless changing them is itself the question. Apply the alternative consistently to the DDI and control conditions.
Report the estimates and identify whether any change in interpretation is driven by sampling, integration, timing, adherence, or missingness. Distinguish this scenario analysis from uncertainty intervals.
Sensitivity analysis and uncertainty analysis answer related but different questions. Sensitivity analysis shows the result under selected alternative assumptions. Uncertainty analysis quantifies uncertainty in model parameters, individuals, or trial outcomes; a set of scenario results is not automatically a confidence interval or a probability distribution. Earlier material on uncertainty propagation and confidence intervals provides the relevant tools for those separate questions.
Diagnostics, Alternatives, and Limitations
Warning signs include a ratio that changes substantially when one sample time is removed, a peak that lies between widely spaced samples, marked discrepancies between linear and log integration, or a conclusion that flips under a modest and plausible dose-time shift. For adherence and missing data, inspect dose records and the pattern of missing concentrations, including whether missingness is concentrated at particular times or conditions. Such findings do not by themselves identify the correct estimate; they identify where assumptions deserve scrutiny.
The main alternatives have different purposes. A denser sampling design can better characterize a curve but costs more and still depends on the AUC calculation. A prespecified interpolation convention makes the calculation reproducible but cannot recover unobserved profile features with certainty. A mechanistic PBPK simulation can evaluate timing and unobserved scenarios, but its predictions inherit model assumptions and should not be treated as measured data. Excluding nonadherent records may address a full-adherence question in some designs, but it is not a universal remedy for nonadherence. No single alternative dominates for every study.
Sensitivity analysis is limited by the alternatives selected: plausible but untested assumptions remain unexamined. Results may also be difficult to compare if the AUC windows, ratio directions, or control schedules differ. Report the primary specification alongside the sensitivity results, explain why each alternative is plausible, and state whether the interpretation is stable or assumption-dependent. When it is assumption-dependent, the responsible conclusion is that the available design or evidence does not support a single robust interpretation.
Common Pitfall
- Wrong reasoning: “The sparse-schedule ratio is smaller, so it is a conservative estimate and can replace the dense-schedule result.” A smaller estimate is not inherently conservative; it may reflect interpolation error or a missed feature, and its direction of bias is not guaranteed.
- Correction: Compare the estimates against the same prespecified estimand, investigate why they differ, and report the dependence on sampling and analysis assumptions. Do not select an estimate merely because it is smaller or supports a preferred interpretation.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Why should the DDI and control AUCs use the same time window when comparing a ratio?
- A log-trapezoidal interval has a zero concentration endpoint. Can the displayed log formula be applied directly? Explain.
- In the sampling example, what caused the sparse-schedule ratio to differ from the five-time ratio?
- Why is excluding a participant who missed a perpetrator dose not automatically a valid correction?
- What does it mean if plausible sampling schedules produce ratios on opposite sides of a prespecified comparison boundary?
Solutions. (1) Matching windows makes the numerator and denominator represent comparable exposure periods; different windows can change the ratio for reasons unrelated to the intended comparison. (2) No. The formula requires positive endpoints and contains a logarithm of their ratio; a zero endpoint requires another prespecified method or model. (3) The sparse schedule omitted the 2-hour observation and linearly interpolated across longer intervals, changing the estimated areas under the two profiles by different amounts. (4) Exclusion changes which observations contribute and may change the target population; it is defensible only if aligned with the estimand and prespecified handling rules. (5) The conclusion is sensitive to the sampling choice examined, so the boundary-based interpretation is not robust to those alternatives and the design or analysis assumptions need to be addressed.