Impact of Event‐Related Treatment Modifications in Exposure‐Response Analyses of Repeated Time‐to‐Event Data

ABSTRACT

Exposure‐response (ER) analyses of repeated time‐to‐event (RTTE) data can be confounded when treatment modifications occur due to the event of interest. One particularly challenging scenario is when an event is preceded by an undocumented clinical worsening leading to treatment discontinuation shortly before the event. Event‐related treatment modifications introduce potential reversed causality in the ER analysis and bias, as events may predict exposure, rather than exposure predicting events. We systematically evaluated three analysis approaches in multiple simulated placebo‐controlled studies to assess their capacity to estimate unbiased ER relationships: (1) average exposure (2) time‐varying exposure (3) average exposure + percentile‐rank of the average amount of blinded study treatment received. The novel Approach 3 was designed to remove confounding from event‐related treatment modifications by leveraging information on blinded study treatment received (including in the placebo treatment arm). Approach 1 was biased in all scenarios with event‐related treatment discontinuations. Approach 2 generally resulted in unbiased estimates, except in the scenarios in which events were sometimes preceded by a treatment discontinuation. In these scenarios, only Approach 3 provided unbiased estimates with linear ER, though it showed slight bias with pronounced nonlinear relationships. In conclusion, treatment modifications associated with events warrant careful consideration in ER analyses of RTTE data. Using a ranked measure of the average amount of blinded study treatment received may be used to lower the risk of confounding due to treatment modifications.

Study Highlights

• What is the current knowledge on the topic?

  • ○
    Exposure‐response analyses of repeated time‐to‐event data can be confounded when treatment modifications occur due to events. While solutions exist for continuous outcomes in titration studies, no systematic evaluation has been performed for repeated time‐to‐event data where events may cause treatment discontinuation.
• What question did this study address?

  • ○
    We evaluated whether standard analysis approaches (average exposure, time‐varying exposure) provide unbiased exposure‐response estimates when event‐related treatment modifications occur, particularly when treatment discontinuation happens shortly before events due to unobserved clinical worsening.
• What does this study add to our knowledge?

  • ○
    We identified that time‐varying exposure models become biased when discontinuations occur before events and proposed a novel method using a ranked measure of the amount of blinded study treatment received as covariate that reduces confounding across all tested scenarios with event‐related treatment modifications.
• How might this change drug discovery, development, and/or therapeutics?

  • ○
    This methodology enables more accurate exposure‐response characterization in studies with event‐related treatment modifications, potentially improving dose selection decisions and regulatory submissions for drugs where events influence dosing patterns.

1. Introduction

Exposure‐response (ER) and pharmacokinetic‐pharmacodynamic analyses can support dose selection. Dose selection questions are inherently causal in nature: “how would the outcome of a patient (or a group of patients) change with dose level X versus dose level Y versus no treatment?” [1]. Accurate answers to these questions therefore rely on unbiased estimates of the causal and pharmacological relationship between drug exposure and response [2, 3].

ER analyses can be prone to confounding when analyzing studies in which treatment modifications occur, which may be due to a titration design, medical judgment, or logistical reasons. Recent literature showed that an unbiased ER of continuous outcome data can still be estimated from outcome‐driven titration studies provided that appropriate analysis techniques are used [4, 5, 6, 7]. However, no such investigations have been performed for repeated time‐to‐event (RTTE) data, where early events may cause treatment modifications such as treatment discontinuation. This could create a misleading pattern in the data: patients with lower average exposure have more events—not only because the drug may be less effective at lower exposures but because having events caused them to stop treatment, which lowered their average exposure.

One particularly challenging scenario is when the event of interest is preceded by (undocumented) clinical worsening leading to treatment discontinuation before the event. In such a case, the event is not caused by the treatment discontinuation, but clinical worsening is the common cause of the event and the treatment discontinuation. This methodological issue has been described for epidemiological studies on the effects of glucose‐lowering drugs on cardiovascular outcomes [8], although the issue is likely not limited to this setting and would also affect single time‐to‐event analyses.

Here, we simulated various RTTE datasets with varying influence of events on treatment discontinuation. We compared different analysis methodologies in their ability to provide unbiased estimates of the true ER relationship, including a new method that uses a ranked measure of the amount of received blinded study treatment as a covariate to avoid confounding from treatment modifications.

2. Methods

2.1. Simulation of Analysis Datasets

All simulated datasets consisted of a large four‐arm placebo‐controlled study in which patients were randomized to either the low or high dose level, and within each dose level also randomized to active treatment or placebo, resulting in four arms (10 mg active, 20 mg active, 10 mg placebo, 20 mg placebo). Patients are monitored for the occurrence of events for 5 years. To minimize the impact of random sampling on the estimated bias in these stochastic simulations, each dataset included 100,000 subjects [9].

Patients start on their randomized dose level (which can be placebo treatment), but may discontinue treatment during study follow‐up. Treatment discontinuation does not terminate study follow‐up (i.e., events are monitored after treatment discontinuation). The relation between the events and treatment discontinuation is the main differentiator of the three key scenarios (A, B or C):

• Scenario A. Random treatment discontinuation (constant rate of discontinuation of 10% per year)

• Scenario B. Event‐related treatment discontinuation, where each outcome event has a 25% probability of causing subsequent treatment discontinuation.

• Scenario C. As scenario B, but there is a conditional probability that if an event‐related discontinuation occurs, this discontinuation occurs the day before the event. This conditional probability of early discontinuation (CPED) was set to 0%, 25%, 50% or 100%. With CPED = 0%, the simulation is equivalent to Scenario B.

Within each key scenario, multiple subscenarios were simulated with different true ER relationships (no effect, linear ER relationship for log hazard, nonlinear sigmoid Emax relationship for hazard) and the value of CPED (Scenario C only). For simplicity, the simulations did not include covariates on the hazard. Inter‐individual variability was included on the hazard (known as frailty) and the drug clearance (used to calculate drug exposure). Additionally, several sensitivity analyses were performed, such as simulations in which the base hazard was increased (to increase the average number of events per subject, determining the shrinkage of the empirical Bayes estimates of the individual frailty term [10, 11]). A complete list of the simulation settings is provided in Table S1 and datasets and model codes in Data S1.

2.2. Calculation of Exposure‐ and Dose‐Related Analysis Variables

2.2.1. Pharmacological Dose Versus Blinded Study Treatment

The time‐varying received pharmacological dose level for patients at the time after first dose t is defined as DOSE(t). Based on this definition (further illustrated in Table S2), DOSE(t) is 0 for patients from the active treatment (verum) arm when t exceeds the time of treatment discontinuation (t _disc). Also, DOSE(t) is by definition always 0 for patients from the placebo arm and, consequently, does not contain any information on treatment modifications in this arm. Therefore, to uncouple the confounding association between treatment modifications and outcome, we define the time‐varying received blinded study treatment at time t as BlindedStudyTreatment(t), which does not distinguish and applies to both verum and placebo. For example, when a patient in the placebo arm receives a placebo treatment at time t at a dose level of 20 mg (i.e., a mock or sham version of 20 mg treatment), BlindedStudyTreatment(t) = 20 mg but the true (i.e., unblinded) treatment is placebo and therefore DOSE(t) = 0 mg. Any patient (including those in the placebo arm) has BlindedStudyTreatment(t) = 0 when t > t _disc, which signifies that no study treatment is given after treatment discontinuation. For patients in the active treatment arm, DOSE(t) = BlindedStudyTreatment(t). These two treatment variables have distinct roles in the ER analysis: DOSE(t), by informing drug exposure, drives the pharmacological effect, while BlindedStudyTreatment(t) is used to adjust for the confounding influence of event‐related treatment modifications.

2.2.2. Treatment History Variable for Blinded Study Treatment

BlindedStudyTreatment(t) is used to calculate BlindedStudyTreatment_rank, a subject‐level summary measure (percentile‐rank) of treatment history of blinded study treatment during the follow‐up period:

First, rank_i is defined as the rank of subject i when all subjects in a particular treatment arm g (e.g., placebo 20 mg arm) are sorted by their average (time‐averaged over total follow‐up duration) of BlindedStudyTreatment(t), with random ordering for ties. BlindedStudyTreatment_rank for each subject is then defined as:

$${\text{BlindedStudyTreatment}}_{\mathrm{rank}}=\left(\mathrm{rank}\_i-0.5\right)/n$$

where n is the number of subjects in treatment arm g. For example, a BlindedStudyTreatment_rank of 0.1 indicates that a subject's average BlindedStudyTreatment is the 10th percentile of the overall distribution of the average BlindedStudyTreatment within the treatment arm of that subject. The transformation to a percentile‐rank per treatment arm guarantees that BlindedStudyTreatment_rank is uniformly distributed between 0 and 1 in each treatment arm. This identical distribution between treatment groups was selected with the aim to avoid overadjustment bias [12, 13, 14].

2.2.3. Time Varying Exposure

Time‐varying drug exposure was expressed as area under the concentration‐time curve during the 24‐h dosing interval at steady‐state (AUC(t)):

$$\mathrm{AUC}\left(t\right)=\text{DOSE}\left(t\right)/\mathrm{ICL}$$

where DOSE(t) is the current dose at time t, and ICL is the individual drug clearance. In each simulation in which there was a true drug effect, AUC(t) was the driver of this effect.

2.2.4. Average Exposure

AUCavg represents the average drug exposure over a patient's entire follow‐up period. AUCavg is calculated by integrating the time‐varying AUC(t) values over the entire follow‐up period and dividing by the total follow‐up time (until study completion after 5 years or after occurrence of the 10th event). AUCavg is calculated across the entire follow‐up period because, in the simulated RTTE datasets, patient follow‐up does not terminate upon treatment discontinuation and AUCavg therefore reflects that patients who discontinue an effective treatment (resulting in a lower AUCavg) will have a smaller average treatment effect compared to patients who do not discontinue treatment.

2.3. RTTE Models

RTTE modeling of the simulated datasets was performed in line with prior publications on RTTE analyses in the pharmacometrics field [9, 15, 16] and briefly outlined below. With parametric RTTE models, the event data are used to estimate the hazard function h(t):

$$h\left(t\right)=h_0\left(t\right)\times \mathrm{EFF}\times \text{DEFF}\times e^{\mathit{\eta i}}$$

where h ₀(t) is the base hazard (in this simulation, all simulation and re‐estimation models use a time‐constant base hazard), EFF is the estimated treatment effect (estimated as a function of drug exposure AUC(t) or AUCavg), DEFF is the estimated covariate effect of BlindedStudyTreatment_rank (only with AUCavg+BlindedStudyTreatment_rank model), and ηi is the normally distributed frailty term for individual i. The probability density function for an event to occur at time t is defined as:

$$f\left(t\right)=h\left(t\right)\times S\left(t\right)=h\left(t\right)\times \exp \left(-{\int }_{t_{j-1}}^{t}h\left(t\right)\mathit{dt}\right)$$

where S(t) is the survival probability of not having an event within the interval between t _j−1 and t and is directly related to the cumulative hazard during this interval. Here t _j−1 reflects either the start of follow‐up (for the first event) or the time of the previous event (for subsequent events).

To each simulated dataset, three different RTTE models were fitted which used either AUC(t) or AUCavg as a driver of the pharmacological effect EFF: (1) AUC(t), (2) AUCavg, (3) AUCavg+BlindedStudyTreatment_rank included as a covariate. For all models, inter‐individual variability of the hazard (frailty) was estimated. The parameterisation of the ER relationship for EFF in the RTTE models (driven by AUC(t) or AUCavg) was equal to the simulation model of the respective dataset (i.e., linear, Emax or sigmoid Emax). When implemented, the effect of BlindedStudyTreatment_rank was modeled as a covariate influencing the hazard in all patients. Model development was performed to derive the functional form of the relationship between BlindedStudyTreatment_rank and the hazard.

2.4. Analysis of Modeling Results

For each model, the estimated ER relationship is compared to the “true” ER relationship from the simulation model. For simulations in which the true ER is linear, this is done by comparing the estimated slope of the ER (and its 95% confidence interval) with the true slope. For simulations with a nonlinear ER, this is done by plotting the 95% confidence interval of the estimated ER relationship against the true ER relationship.

3. Results

3.1. BlindedStudyTreatment_rank in the ER Model

For the simulated Scenarios B and C, the relationship between events and BlindedStudyTreatment_rank was generally well characterized with the following empirical function:

$$\begin{array}{c}\begin{array}{c}\text{DEFF}=\left(1+\frac{{{\text{DEMAX}\times \text{BlindedStudyTreatment}}_{\mathrm{rank}}}^{\text{DHILL}}}{{{\text{BlindedStudyTreatment}}_{\mathrm{rank}}}^{\text{DHILL}}+\mathrm{D}{50}^{\text{DHILL}}}\right)\\ \times \left(1+\text{DINT}\times e^{{\text{BlindedStudyTreatment}}_{\mathrm{rank}}\times \text{DSLOPE}}\right)\end{array}\hfill \end{array}$$ (1)

This empirical function uses a sigmoid Emax function to capture the strong difference in hazard between patients that discontinue versus those who do not. This sharp difference occurs because in the Scenarios B and C, a discontinuation guarantees that patients had at least one event; while not all events are associated with a discontinuation, all discontinuations are associated with an event occurring shortly before or after the discontinuation. The exponential function (defined by DINT and DSLOPE) characterizes that among those that discontinue, those with an early discontinuation (resulting in a lower BlindedStudyTreatment_rank) are more likely to have a higher hazard than those with a late discontinuation. The shape of DEFF versus BlindedStudyTreatment_rank for one of the simulated datasets of Scenario B is shown in Figure 1. In Scenario A, which has random treatment discontinuation, no (significant) relationship between BlindedStudyTreatment_rank and the hazard is expected, and the effect of BlindedStudyTreatment_rank was therefore implemented as a simple log‐linear relationship with hazard to avoid minimization issues.

FIGURE 1.

Illustration of BlindedStudyTreatment_rank as a covariate effect using a Scenario B simulation dataset. Left panel: Illustrates the relationship between the average BlindedStudyTreatment and its percentile‐rank BlindedStudyTreatment_rank across each study arm. Right panel: Estimated relationship between DEFF and BlindedStudyTreatment_rank from an RTTE model that uses AUC_avg + BlindedStudyTreatment_rank as predictors of the hazard (following Equation 1).

3.2. Simulations With Linear ER

3.2.1. Scenario A (Random Treatment Discontinuation)

For the base scenario A (random treatment discontinuation), two subscenarios were simulated, one with a true treatment effect on the hazard (with linear E‐R slope) and another effect without a true treatment effect (i.e., slope of ER = 0). All tested RTTE models (AUC(t), AUCavg and AUCavg + BlindedStudyTreatment_rank) resulted in an unbiased estimate of the slope of the ER relationship, with the true values (−0.25 or 0) falling within the 95% confidence interval of the parameter estimates (Figure 2).

FIGURE 2.

Comparison of the true exposure‐response (ER) slope (horizontal dashed line) versus the estimated ER slope and its 95% confidence interval (dot and error bars) for Scenario A, where treatment discontinuation occurs completely randomly. BST_rank = BlindedStudyTreatment_rank.

3.2.2. Scenario B (Event‐Related Treatment Discontinuation After Event)

For scenario B (each event has a 25% chance to cause a treatment discontinuation), two subscenarios were simulated, one with a true treatment effect on the hazard (with linear ER slope) and another effect without a true treatment effect (i.e., slope of ER = 0). Figure 3 shows that the slope estimates from the RTTE model with time‐varying AUC(t) are unbiased, while the slope estimates with AUCavg over‐estimate the treatment effect due to the confounding from the impact of events (via treatment discontinuation) on AUCavg. The addition of BlindedStudyTreatment_rank as a covariate to the ER model with AUCavg removes this bias.

FIGURE 3.

Comparison of the true exposure‐response (ER) slope (horizontal dashed line) versus the estimated ER slope and its 95% confidence interval (dot and error bars) for Scenario B, where events have a 25% probability of causing a treatment discontinuation after the event. BST_rank = BlindedStudyTreatment_rank.

3.2.3. Scenario C (Event‐Related Treatment Discontinuation, Sometimes Before the Event)

In scenario C, like Scenario B, each event has a 25% chance to cause a treatment discontinuation. The difference with Scenario B is that in Scenario C, when an event causes a treatment discontinuation, there is a conditional probability CPED that the discontinuation occurs 1 day before the event. Figure 4 compares the estimated linear slope of the ER compared to the true slope (0 or −0.25) for sub‐scenarios with different CPED (0%, 25%, 50% and 100%). The simulations with CPED = 0% are included to aid interpretation of Figure 4, but are essentially the same as Scenario B (apart from a different random seed). The RTTE model with AUC(t) is only unbiased in the CPED = 0% scenario, and with increasing values of CPED, the estimated slope becomes increasingly overestimated compared to the true value. The RTTE model with AUCavg is always biased, but independent of CPED. The RTTE model with AUCavg + BlindedStudyTreatment_rank is unbiased in all 8 simulations of Scenario C shown in Figure 4.

FIGURE 4.

Comparison of the true exposure‐response (ER) slope (horizontal dashed line) versus the estimated ER slope and its 95% confidence interval (dot and error bars) for Scenario C. In scenario C, like Scenario B, each event has a 25% chance to cause a treatment discontinuation. The difference with Scenario B is that in Scenario C, when an event causes a treatment discontinuation, there is a conditional probability (CPED) that the discontinuation occurs 1 day before the event instead of directly after the event. BST_rank = BlindedStudyTreatment_rank.

3.3. Simulations With Nonlinear ER

The simulations from Scenario C with a linear ER (for CPED = 25%) were then extended for a range of nonlinear ER functions (Emax and sigmoid Emax). Figure 5 shows a comparison of the true ER (solid line) with the 95% confidence interval of the estimated ER from the AUCavg + BlindedStudyTreatment_rank. For the Emax function (HILL = 1), the performance of the AUCavg + BlindedStudyTreatment_rank model was good, with the true ER generally falling within the 95% confidence interval. For the strongly nonlinear sigmoid Emax function (HILL = 5), a small bias between the estimate and true ER can be seen (Figure 5), as the estimated ER tends to be slightly “more linear” (less sigmoidal) than the true ER (reflected in an under‐estimation of the HILL parameter in these models). Similar results were seen when testing nonlinear ER functions for Scenario A for both the “AUCavg + BlindedStudyTreatment_rank” and “AUCavg” approaches (Figures S1 and S2). In line with the results with a linear ER shown in Figures 2 and 4, with a nonlinear ER, the RTTE model with AUC(t) is unbiased in Scenario A, but strongly over‐estimates the treatment effect in Scenario C (Figures S3 and S4).

FIGURE 5.

Comparison of the true ER relationship (solid line) and the estimated ER relationship (dashed line) and its 95% confidence interval (gray area) for the RTTE model using AUCavg + BlindedStudyTreatment_rank as predictors of the hazard in Scenario C, where events can cause treatment discontinuation and these discontinuations sometimes (in CPED = 25% of cases) occur 1 day before the event and directly after the event otherwise.

3.4. Additional Sensitivity Analyses and Models

The sensitivity analyses are described and plotted in the Supporting Information, and main findings summarized here. The use of BlindedStudyTreatment_rank was also tested as a covariate for an RTTE model that used AUC(t) as a driver, but this resulted in biased results (as shown Figures S6 and S7 and discussed in Supporting Information). Qualitatively (i.e., which models were biased or unbiased), similar results were seen for the alternative simulation for scenario B in which events in patients currently receiving 20 mg could lead to down‐titration to 10 mg, and events in patients receiving 10 mg could lead to treatment discontinuation (Figure S8). The same qualitative agreement was seen in the sensitivity analyses that investigated a higher base hazard (Figures S9 and S10).

4. Discussion

This work set out to clarify when commonly used ER approaches for RTTE data provide unbiased estimates and why they may fail in the presence of event‐related treatment modifications. We compared three analysis strategies—AUCavg, AUC(t), and using AUCavg with a new adjustment for the percentile rank of average received blinded study treatment (AUCavg + BlindedStudyTreatment_rank)—across scenarios where discontinuation is random (Scenario A), follows events (Scenario B), or sometimes precedes events (Scenario C).

When treatment discontinuation is random (Scenario A), all approaches were able to provide an unbiased estimate. In Scenario B, where events can cause treatment discontinuation, the time‐varying AUC(t) was unbiased. This finding for RTTE models is in line with recent literature on ER analysis in continuous outcome data in outcome‐driven titration studies, which showed that an unbiased ER can still be estimated from titration studies provided that the sources of variability (inter‐individual and/or intra‐individual variability) are adequately characterized by the model [4, 7]. This interpretation is further supported by a sensitivity analysis (Figure S5) that showed that, if the true interindividual variability of the hazard is high, a misspecified AUC(t) model without inter‐individual variability will be biased in Scenario B. In this study, we have only considered RTTE models with inter‐individual variability on the hazard (frailty) as this is often the only parameter for which IIV terms can be identified. The expansion to cases with IIV on ER parameters is a potential topic for future investigations.

In Scenario C, where the event‐related treatment discontinuations would sometimes occur before the event, the RTTE model with AUC(t) becomes increasingly biased as the frequency of discontinuations shortly before the event (i.e., CPED) increases because treatment discontinuations that occur before events create a misleading temporal relationship where low drug exposure appears to precede events, when in fact a precursor to the impending event caused the discontinuation. This temporal reversal increases the association between lower exposure and events, causing the model to overestimate the protective effect of the drug (steeper negative ER slope than the true value). The time‐varying AUC(t) model was remarkably sensitive to bias from this phenomenon; for example, in the “Effect, CPED = 25%” simulation (Figure 4), only ~5% of events were directly preceded by a treatment discontinuation, but this resulted in an over‐estimation of the effect slope by 50%.

Scenario C reflects the situation where (unobserved) clinical worsening of the patient may precede the event and this clinical worsening can cause a treatment discontinuation to occur shortly before the event (e.g., by subject or investigator decision, or due to logistical issues associated with the clinical worsening). Rather than a theoretical possibility, this methodological issue occurs in practice and has been described in the context of epidemiological studies on the effects of glucose‐lowering drugs on cardiovascular outcomes [8] and would also affect analyses of time‐to first event (TTE) data. A sign that this issue may be present is an over‐representation of events occurring shortly after treatment modifications. If this pattern is also observed in data from the placebo arm, it reinforces the hypothesis that (unobserved) clinical worsening caused the treatment modification rather than the alternative hypothesis that the treatment modification reduced or removed the beneficial drug effect, thus leading to clinical worsening and the event shortly afterwards.

The “AUCavg” models (i.e., without use of BlindedStudyTreatment_rank) were biased in Scenarios B and C, because the impact of events on the treatment discontinuation distorts the estimated ER relationship compared to the true ER relationship. When patients experience one or more events, they're more likely to stop treatment early, which reduces their average drug exposure (AUCavg). This creates a fundamental problem: patients with low AUCavg are not randomly selected but represent those who had more health problems. Therefore, comparing these patients to placebo patients or to patients who stayed on treatment longer is subject to confounding as apparent differences can be due to their underlying health status rather than the drug effect. In other words, low AUCavg becomes a marker for patients who had more events, creating a selection bias that makes direct comparisons misleading. The relationship between exposure and outcome becomes confounded because the average exposure itself is affected by the outcome.

In Scenario B and C, the adjustment for BlindedStudyTreatment_rank removes the bias seen with the AUCavg model. The rationale for the “AUC_avg + BlindedStudyTreatment_rank” approach is that event‐related treatment modifications result in a confounding association between outcome (events) and AUC_avg. This association is present when events cause treatment modifications, but also when an unobserved confounder causes both the treatment adaptation and the events, such as in Scenario C. According to the principles of causal inference, confounding can be avoided by adjusting for a variable on the confounding pathway (i.e., events cause treatment modifications which then affect AUC_avg). We used BlindedStudyTreatment_rank as a covariate to adjust for treatment modifications and their timing which applies to patients in both active and placebo treatment arms. Because the blinded study treatment in the placebo arm has no pharmacological effect, BlindedStudyTreatment_rank in the placebo arm specifically informs the confounding association between outcome and dose level, thus helping to remove this confounding from the estimated E‐R driven by AUCavg. The “AUC_avg + BlindedStudyTreatment_rank” was the only approach that provided unbiased ER estimates in Scenario C, as the AUCavg calculations are not significantly affected by whether discontinuation happens shortly before or after an event.

For situations that resemble Scenario C, the standard approaches (AUCavg and AUC(t)) may result in a strong bias of the estimated ER relationship, and the use of “AUCavg + BlindedStudyTreatment_rank” may be a promising approach. To avoid bias in the ER (driven by AUCavg) it is important that the RTTE model adequately characterizes the relationship between BlindedStudyTreatment_rank and the hazard. Due to its empirical nature, the parameterization of the BlindedStudyTreatment_rank needs to be investigated during model development on a case‐by‐case basis. One could consider starting RTTE model development based on placebo data only: to investigate the influence of covariates on the hazard and investigate (using appropriate RTTE model evaluation techniques [10, 16] that stratify by BlindedStudyTreatment_rank) a potential association of BlindedStudyTreatment_rank with the hazard that is not captured by baseline covariates. After this step, data from the active treatment could be introduced to the ER analysis. While the “AUCavg + BlindedStudyTreatment_rank” approach could in theory also be used in the absence of a placebo arm, we expect it will then be difficult to differentiate the effects of AUCavg and BlindedStudyTreatment_rank as these variables will be highly correlated then. We expect this method to be mainly applicable for analyses of studies with a randomized placebo‐controlled designs where the design of placebo arm(s) mimics the design for active treatment arm(s).

The “AUCavg + BlindedStudyTreatment_rank” method is not without limitations. With strongly nonlinear ER relationships, the method tends to underestimate the nonlinear property of the true ER relationship (Figure 5 and Figure S1). However, this is a limitation that applies in general to ER analyses of studies with time‐varying exposure that use averaged subject‐level exposure metrics and is thus also seen in base Scenario A with the “AUCavg” model without BlindedStudyTreatment_rank (Figure S2). Another limitation is that the use of AUCavg and BlindedStudyTreatment_rank as predictors in the RTTE model may complicate the use of this model for clinical trial simulations because BlindedStudyTreatment_rank is not known at baseline.

A general assumption (often made implicitly) in ER analyses, is that conditional on the covariates in the model, patients with a high and low exposure and placebo patients are comparable, such that the observed difference in outcome that is not explained by covariates can be attributed to the treatment effect. Similarly, the use of BlindedStudyTreatment_rank as a covariate in the RTTE model relies on the assumption that subjects with similar value of BlindedStudyTreatment_rank in both treatment and placebo groups are comparable. Violation of this assumption may occur if the active treatment induces discontinuation for reasons unrelated to underlying event risk (e.g., tolerability issues in otherwise healthier patients) and this could potentially result in a flattening or steepening of the estimated ER slope.

AUCavg (i.e., without using BlindedStudyTreatmentrank) should only be used when there is a negligible impact of events on treatment modifications. If events are not over‐represented after treatment modifications, the use of AUC(t) may be preferred [17], as this approach will also work with event‐related treatment modifications (in contrast with AUCavg) and because this approach is more straightforward in terms of model development and simulations than the AUCavg + BlindedStudyTreatment_rank approach. However, if events are over‐represented after treatment discontinuation (Scenario C, which may be an under‐diagnosed problem in pharmacometrics) the newly proposed AUCavg + BlindedStudyTreatment_rank may be the only approach that can provide unbiased ER estimates, provided that the empirical association between BlindedStudyTreatment_rank and the hazard is adequately modeled and the true ER is not strongly nonlinear. We investigated different treatment discontinuation and down‐titration scenarios. Other treatment modifications (up‐titration or temporary treatment interruption) were not evaluated, although similar results would be expected.

In conclusion, event‐related treatment modifications can strongly bias standard ER analyses which use AUCavg. When events are often preceded by treatment discontinuations, even time‐varying AUC(t) can result in strongly biased treatment effects. The proposed adjustment approach (AUCavg + BlindedStudyTreatment_rank) uses a ranked measure of the amount of blinded study treatment received as a covariate to avoid this bias and is a suitable alternative for situations where standard approaches (AUC_avg or AUC(t)) do not work. In future research, a wider applicability of BlindedStudyTreatment_rank to avoid confounding from treatment modifications in other ER analysis settings may be explored.

Author Contributions

S.C.G. and D.G., T.E. and N.S. wrote the manuscript; S.C.G., N.S., T.E. and D.G. designed the research; S.C.G. performed the research; S.C.G. analyzed the data.

Funding

This work was supported by Bayer AG, Wuppertal, Germany.

Conflicts of Interest

Dirk Garmann and Thomas Eissing are employees of Bayer AG and own shares or share options in the company. Sebastiaan Camiel Goulooze and Nelleke Snelder are employees of Leiden Experts on Advanced Pharmacokinetics and Pharmacodynamics (LAP&P Consultants BV), a contract research organization employed by Bayer AG.

Supporting information

Appendix S1: psp470231‐sup‐0001‐AppendixS1.docx.

Data S1: psp470231‐sup‐0002‐DataS1.zip.