Exposure‐Response Analysis for Time‐to‐Event Data in the Presence of Adaptive Dosing: Efficient Approaches and Pitfalls

ABSTRACT

Analyzing exposure‐response (E‐R) relationships for time‐to‐event (TTE) endpoints presents challenges due to the inherent time‐dependent nature of the data. Some authors address these difficulties by using a fixed timepoint approach, where exposure is assessed at a predetermined time rather than dynamically over time. (e.g., initial exposure or last exposure). The aim of the current work is to compare the use of time‐static and time‐varying metrics to assess the E‐R relationship through simulations. PK exposures were simulated from a one‐compartment model and TTE data from a parametric proportional hazard model, involving the weekly average PK concentration as a time‐varying covariate. Several scenarios were considered to handle the type of dosing (fixed or adaptive), the accumulation of the drug (low or strong), the type of event (efficacy, safety or independent), and the timing of the event onset (early or late). Wald tests on the exposure effect parameter were performed to assess the significance of the E‐R relationship. For each simulation scenario, the type‐I error and the power of the Wald tests were reported, revealing that no time‐static metric consistently produced reliable results across all conditions. In order to ensure adequate statistical properties, we recommend using time‐varying exposure, which shows good performance across all scenarios.

Study Highlights.

• What is the current knowledge on the topic?

  • ○
    In exposure‐response analysis, the use of time‐static metrics relying on patients' follow‐up is not recommended as it may lead to false conclusions. Current findings promote the use of time‐static metrics based on initial exposure.
• What question did this study address?

  • ○
    A detailed assessment of time‐static metrics in various settings depending on the nature of the exposure‐response (E‐R) relationship, the dosing design (fixed or adaptive), the accumulation of the drug (low or strong) and the event onset (early or late) is lacking.
• What does this study add to our knowledge?

  • ○
    As already pointed out in the literature, time‐static exposure metrics derived at the end of a patient's follow‐up should be avoided. In the case of fixed dosing design, metrics based on initial exposure typically yield correct results. However, in the case of adaptive dosing designs, those metrics can weaken the power to detect significant E‐R relationships. Overall, approaches based on time‐varying exposure metrics are preferable to adequately explore E‐R relationships.
• How might this change drug discovery, development, and/or therapeutics?

  • ○
    Modelers should prefer the use of time‐varying metrics when assessing exposure‐response relationships, in particular when the dosing design is adaptive. Those analyses can be efficiently implemented in a prespecified R‐based workflow.

1. Introduction

Exposure‐response (E‐R) analysis is crucial for understanding the relationship between drug exposure and the response to a drug. The modeling of the E‐R relationship allows for quantitatively describing how a variation in exposure (e.g., with different dose levels) may influence the magnitude of the response, and thus inform decision‐making and optimize interventions. Adaptive dosing is increasingly used in clinical studies and clinical practice for its flexibility and efficiency [1]. These designs allow for dose modifications, delays, etc., often based on safety criteria, which give more flexibility to the clinicians [2] but also introduce complexities into the E‐R analysis [3].

Commonly, pre‐screening methods such as graphical explorations are used to first explore any potential E‐R relationships [4]. When the response is a time‐to‐event (TTE) outcome, boxplots of the observed exposure in responders and non‐responders [5], or Kaplan–Meier (KM) plots stratified by quantiles of exposure [6] are usually derived. Such approaches require the use of a time‐static metric often based on the initial exposure [7] or on the last exposure [8]. Exposure metrics based on initial exposure may intuitively lack power because they are insensitive to the individual dosing histories. The use of exposure metrics based on the last exposure (or the average exposure until the event) may be even more problematic because of the unequal observed duration times. Current recommendations [7, 9, 10] suggest the use of time‐static metrics relying on initial exposure, as some studies showed their better performance characteristics compared to metrics relying on last exposure, which can be misleading because exposure duration is inherently linked to the outcome; patients who do not experience an event remain in the study longer and thus have more time to accumulate drug or receive more doses, creating a spurious exposure‐response relationship.

Some studies have described how the exposure‐response relationship can be altered by dose titrations in the context of continuous outcomes [11, 12, 13]. Recent conference posters have also evaluated the behavior of these metrics in the context of time‐to‐event (TTE) [14] data and adaptive dosing designs [15]. Moreover, most studies evaluating time‐static exposure metrics for E‐R analysis typically focused on a pre‐defined context, and the literature lacks a detailed assessment in numerous situations relying on the type of event, the speed of the event onset and the accumulation of the drug.

The objective of this work is to compare and assess the efficient approaches to accurately describe exposure‐response relationships across an extensive clinical trial simulation. The use of time‐static metrics based on first exposure, exposure at steady state based on first dose or exposure before the event is assessed and compared with the use of the true time‐varying exposure. For adaptive dosing scenarios, more complex scenarios involving correlation between the dose reduction process and the TTE process are considered through joint modeling. The impact of mis‐specifying the correlation structure is also evaluated.

2. Methods

2.1. Simulation Scenarios

2.1.1. Simulation of PK Exposures

Drug exposures were simulated from a one‐compartment pharmacokinetic model, parametrized in terms of volume and clearance. Each simulated subject $i=1,\dots ,N$ was randomly assigned to one of three intravenous dose levels (1:1:1 allocation ratio), which was administered weekly over a 30‐week period (week 0 to week 29).

The plasma concentration, at any time $t$ (in weeks), can be derived by the following equation:

$$C_i\left(t\right)=\sum _{k=1}^{K_t}\frac{D_{i,k}}{V_i}{e}^{-\frac{C{l}_i}{V_i}\left(t-t_k\right)}$$ (1)

where $K_t$ is the number of doses administrated before time $t$, $D_{i,k}$ the ${\mathit{k}}^{\mathit{th}}$ dose administrated for subject $i$, and $t_k$ the time (in weeks) when the ${\mathit{k}}^{\mathit{th}}$ dose was administrated. $V_i$ and $C{l}_i$ are the individual volume and clearance parameters, function of the vector of fixed effect parameters $\left(V,\mathit{Cl}\right)$ and random effects parameters (${\eta }_{V_i},{\eta }_{C{l}_i}$) of variance–covariance matrix $\Omega =\mathrm{diag}\left({\omega }_V,{\omega }_{\mathit{Cl}}\right)$.

We denote $\mathit{CA}{V}_i\left(t\right)$ the average plasma concentration for individual $i$ during the week preceding any time $t$ in dosing cycle $k$:

$$\begin{array}{c}\mathit{CA}{V}_i\left(t\right)=\frac{1}{\tau }{\int }_{t_k-1}^{t_k}{C}_i\left(x\right)\mathit{dx}\end{array}$$ (2)

As an example, $\mathit{CA}{V}_i\left(1\right)$ represents the average concentration observed between week 0 and week 1. τ is the inter‐dose interval equal to 1.

2.1.2. Simulation of Time‐to‐Event Data

Time‐to‐event data was generated according to a proportional hazard model involving the weekly individual average plasma concentration ($\mathit{CA}{V}_i\left(t\right)$) as a time‐varying covariate with a link coefficient $\beta$ (see Equation 2). Gompertz function was considered for the baseline hazard, with rate and shape parameter $g$ and $r$:

$$h_0\left(t\right)=g\times \mathit{exp}\left(r\times t\right)$$

$$\begin{array}{c}{h}_i\left(t\right)=h_0\left(t\right)\exp \left(\beta \times \mathit{CA}{V}_i\left(t\right)\right)\end{array}$$ (3)

Individual hazards were simulated starting from week 1 until week 30. Patients who did not experience an event before week 30 were censored at week 30, and no other censoring process was considered in the simulations.

2.1.3. Description of the Various Scenarios

Several scenarios were considered regarding the generation of PK exposure and TTE data, denoted in the following as the main scenarios (see Figure S1). Regarding PK exposure, two scenarios of dosing design as well as two scenarios of drug accumulation, were considered. Regarding TTE data, three scenarios for the E‐R relationship as well as two scenarios of the event onset were considered.

For the dosing design scenarios, the first one assumes a fixed dosing regimen, where each subject receives the same initially assigned dose every week throughout the entire follow‐up period. The second one assumes an adaptive dosing regimen, meaning that subjects can experience dose reductions over the follow‐up period, according to the following logistic process:

$$\begin{array}{c}\mathit{\text{logit}}\left(P\left(D{R}_i\left(t_k\right)\right)\right)=\mathit{log}\left(\frac{p}{1-p}\right)+\alpha \times \mathit{CA}{V}_i\left(t\right)\end{array}$$ (4)

Of note, the dosing strategy is exposure‐driven, with an additional stochastic process mimicking the investigator's decision, differing from a per‐protocol schedule. The probability of dose reduction at time $t_k$ (when the ${\mathit{k}}^{\mathit{th}}$ dose is administrated) depends on a probability $p$ and on the individual average exposure. Parameter $\alpha$ is defined as a positive number meaning that higher exposure leads to higher probability of experiencing a dose reduction, similar to what might be expected in the presence of dose limiting adverse events. The binary observations for dose reduction (yes/no) are generated using a uniform distribution, meaning that a dose reduction is applied at the next occasion if $\mathit{\text{logit}}\left(P\left(D{R}_i\left(t_k\right)\right)\right)>$ $u\left(u~U\left[0,1\right]\right)$. Note that dose reductions are decided on a weekly basis and can occur several times for a same patient. If a dose reduction occurs, the dose is reduced by a fixed amount $\gamma$, potentially down to a minimum dose (see Section 2.1.4).

For the drug accumulation scenarios, the first one assumes low accumulation, while the second assumes strong accumulation (adjusting on volume parameter ($V$)).

For the E‐R relationship, three cases were handled in the simulations considering either no E‐R relationship ($\beta =0$), positive E‐R relationship ($\beta >0$), or negative E‐R relationship ($\beta <0$).

For the event onset scenario, the first one assumes early occurrence ($r<0$) and the second one late occurrence ($r>0$) in the time of the study.

Additional simulations were performed to handle correlation between parameters $\alpha$ (effect of exposure on the probability of dose reduction) and $\beta$ (effect of the exposure on the hazard of event). To handle it, individual ${\alpha }_i$ and ${\beta }_i$ were drawn from a multivariate normal distribution with mean $\left(\begin{array}{c}\alpha \\ \beta \end{array}\right)$ and variance–covariance matrix $\left(\begin{array}{cc}{\omega }_{\alpha }^2& {\omega }_{\alpha ,\beta }\\ {\omega }_{\alpha ,\beta }& {\omega }_{\beta }^2\end{array}\right)$. Such correlation can be seen as a result of pharmacodynamic sensitivity: the variability in the sensitivity of drug targets (e.g., receptors or enzymes) can make some patients more responsive to a given drug exposure. Enhanced sensitivity can lead to greater efficacy but also increases the likelihood of adverse events, necessitating dose reductions. Those additional simulations were performed for all E‐R relationship cases, and obviously under the adaptive dosing design. They are denoted as the additional scenarios in this article.

2.1.4. Simulation Settings

$K=100$ datasets involving $N=300$ patients were simulated for each of the scenarios mentioned above. The initial doses considered were $100$, $200$ or $300$ mg. Parameter $\gamma$ (representing the reduction of the dose in mg when there is a dose reduction) was set to 50 mg. The minimum dose that could be given to a patient was set to 50 mg. The parameters used to simulate data were defined to be consistent over all the scenarios investigated and to obtain at least one third of the patients having an event, and around one half of the patients having at least one dose reduction (in the adaptive dosing scenarios). They are summarized in Table S1. For the additional scenarios, a high correlation of 0.9 was assumed between $\alpha$ and $\beta$. The random effect variances (${\omega }_{\alpha }^2$ and ${\omega }_{\beta }^2$) were twice lower in case of low accumulation scenarios compared to strong ones, to ensure the relative impact of IIV was comparable across the different exposure ranges produced by each scenario. The $\omega$ values are given in Table S2.

PK exposure was simulated using mrgsolve (version 1.0.6) [16] package available in R, and TTE data using R.

2.2. Exposure‐Response Analysis

The analysis of E‐R relationships across all the main simulated scenarios was based on graphical explorations and TTE modeling using time‐static or time‐varying metrics. Three time‐static metrics were considered:

• The average exposure during the first week, denoted as $\mathit{CA}{V}_{w_1}$

• The average exposure during the week before the event or the end of the study, denoted as $\mathit{CA}{V}_{w_{\mathit{TE}}}$

• The average exposure at steady state based on the starting dose, denoted as $\mathit{CA}{V}_{\mathit{SS}}$

One time‐varying exposure was considered: the weekly average exposure (i.e., true exposure used to simulate TTE data). Note that the true individual exposure metrics were derived from the simulations and not re‐estimated.

Firstly, for each simulation scenario and each time‐static exposure metric, KM plots stratified by exposure tertiles and log‐rank tests were performed to evaluate the differences between survival curves across the tertile exposure groups. Also, exposure distribution among patients with and without events were derived, and differences of mean exposures were evaluated by a t‐test. We defined $D_S$ as the difference in estimated survival probability at week 30 between the first and third tertile groups, and $D_M$ as the difference in mean exposure between subjects who experienced the event and those who did not, to indicate the direction of the exposure‐response (E‐R) relationship. The tests were performed in R, using the survdiff [17] (for log‐rank test) and t‐test [18] (for t‐test) functions. Across all the main scenarios and exposure metrics, we reported the type‐I error (when absence of E‐R relationship) and the power (when positive or negative E‐R relationship) of the log‐rank tests and t‐tests, assuming a type I error rate of 0.05.

Secondly, for each scenario and each exposure metric, parametric TTE models were estimated including the average concentration as a covariate (time‐static or time‐varying). Gompertz function was specified for the baseline hazard. The Wald test on $\beta$ was performed and associated p‐value was derived. Estimation of $\theta =\left\{g,r,\beta \right\}$ was performed using the flexsurvreg function of the flexsurv [19] package available on R software. In practice, it is common to use semi‐parametric approaches, where the baseline hazard is not specified. That is why estimation was also performed using Cox model (involving the average concentration as a time‐static or time‐varying covariate) with the coxph function of the R survival package. Similarly, across all the main scenarios and exposure metrics we reported the type‐I error and the power of the Wald test on $\beta$.

For each of the additional scenarios, parametric TTE model as defined in Equation (2) was estimated simultaneously with a logistic regression process to model the dose reduction mechanism (Equation 3) as well as the correlation between the two processes. The vector of parameter to estimate was composed of the fixed effects $\mu$ and the variance–covariance matrix of the random effects $\Omega ,$ defined as: $\mu =\left\{g,r,\beta ,p,\alpha \right\}$, $\mathrm{\Omega }=\left(\begin{array}{cc}{\omega }_{\beta }^2& {\omega }_{\beta ,\alpha }\\ {\omega }_{\beta ,\alpha }& {\omega }_{\alpha }^2\end{array}\right)$. Estimation was performed in R with the SAEM algorithm [20], using the code of the saemix package [21], already extended for multi‐response estimation [22]. The initial estimates for the population parameters have been set to the estimated parameter values when fitting the two submodels separately, for the first simulated data set of each scenario. Initial parameters for random effect variances and covariance were set to 100% of the initial parameter value. To allow for better stability and ensure convergence, the algorithm settings involved 10 chains and the number of iterations was put respectively to 1000 and 300 for the exploratory and smoothing phases. Estimation of the survival part independently from the logistic process (i.e., mis‐specifying the correlation structure) was also performed using the flexsurv package. The impact of mis‐specifying the correlation between the dose reduction process and the TTE process was assessed by reporting the type‐I error and the power of the Wald test on $\beta$ for both true and mis‐specified models. Relative estimation errors (REE) on $\beta$ were also assessed in both cases.

3. Results

This section is organized as follows. First, preliminary explorations are presented using graphical analysis and statistical tests on the first simulated dataset of each main scenario, followed by a global summary across all simulations. Second, the TTE modeling analysis is presented similarly, starting with results from the first simulated dataset before providing a global summary. Finally, results regarding additional scenarios are presented. Additional details for interpretation are provided in Appendix S5.

3.1. Exposure‐Response Explorations

3.1.1. Illustration on First Simulated Data Sets

To focus on the essential principles, this section presents the results relative to the first data set of two main scenarios: (i) positive E‐R relationship (efficacy event in the following), early events and strong accumulation, and (ii) positive E‐R relationship, late events and low accumulation of the drug. Results for the other scenarios can be found in Appendix S3 (Table S2 in Appendix S3 is a reader's guide indicating each scenario's location). In each case, results with fixed and adaptive dosing design are shown.

3.1.2. Positive E‐R Relationship, Early Events and Strong Accumulation

Figure 1 presents the weekly average concentration for the first 10 subjects, as well as a KM plot of the event times of all the subjects, for the first simulated data set belonging to the scenario with a positive E‐R relationship, early events and strong drug accumulation.

FIGURE 1.

Individual weekly average concentration (CAV) trajectories for the 10 first subjects (top) and Kaplan–Meier curves including risk table (bottom) for the first simulated data set of the following scenario: positive E‐R relationship, early events, strong accumulation and fixed (left) or adaptive (right) dosing design.

Figure 2 shows the KM plots with log‐rank tests p‐values for the first simulated data set of the current scenario, stratified by tertiles of exposure and based on the three time‐static exposure metrics. Corresponding boxplots with t‐tests p‐values can be found in Figure S14 (in Appendix S3b). Under adaptive dosing design, the E‐R relation appears weakened for metrics relying on initial exposure, particularly for $\mathit{CA}{V}_{\mathit{ss}}$. More importantly, the exposure‐response relationship based on the average exposure the week before the event $\left(\mathit{CA}{V_w}_{\mathit{TE}}\right)$ is reversed: the probability of efficacy is higher in patients having lower exposure the week before the event ($D_s<0,p=3.8\times {10}^{-19}$ and $p=5.2\times {10}^{-32}$ for fixed and adaptive dosing design respectively). Same conclusion is also obtained looking at boxplots (Figure S14): the exposure is lower in patients who did experience the efficacy event than in patients who did not ($D_M<0,p=6.4\times {10}^{-9}$ and $p=1.8\times {10}^{-17}$ for fixed and adaptive dosing design respectively). Patients who stayed longer in the study accumulated the drug exposure (even if they also had dose reductions), while patients who left early the study did not have the opportunity to accumulate. Note that on the contrary, if the relationship between the exposure and the response is negative (higher exposure leading to lower probability of event), the opposite effect is observed: the E‐R relation is not reversed but instead exaggerated (see Figures S23 and S24 of the Appendix S3c) as accumulation will benefit for patients with the longest follow‐up.

FIGURE 2.

Kaplan–Meier plots using each of the three time‐static exposure metrics and stratified by exposure tertiles. Ds is the difference of estimated survival probability at week 30 between the first and the third tertile group, and p is the p‐value of the log‐rank test. The scenario for generating data is the following one: positive E‐R relationship, early events and strong accumulation of the drug.

3.1.3. Positive E‐R Relationship, Late Events and Low Accumulation

Figure 3 presents the weekly average concentration for the first 10 subjects, as well as the KM plot of the event times of all the subjects, for the first simulated data set belonging to the scenario with a positive E‐R relationship, late events and low accumulation.

FIGURE 3.

Individual weekly average concentration (CAV) trajectories for the 10 first subjects (top) and Kaplan–Meier curves including risk table (bottom) for the first simulated data set of the following scenario: positive E‐R relationship, late events, low accumulation and fixed (left) or adaptive (right) dosing design. For the adaptive dosing scenario (right), dose reductions decisions are represented by black crosses.

Figure 4 shows the KM plots and log‐rank test p‐values relative to first simulated data set of the current scenario, stratified by tertiles of exposure and based on the three time‐static exposure metrics. Corresponding boxplots with t‐test p‐values can be found in Figure S21 (Appendix S3b). Under adaptive dosing design, the E‐R relationship appears even more weakened than in the presence of strong accumulation (Figure 2). Time‐static metrics relying on initial exposure showed non‐significant difference in event probability between the exposure tertile groups ($p=0.35$ and $p=0.13$ for $\mathit{CA}{V}_{w1}\text{and}\mathit{CA}{V}_{\mathit{ss}}$ under adaptive dosing design). Similar conclusion is showed with the boxplots, and while the difference in exposure means when considering $\mathit{CA}{V}_{\mathit{ss}}$ is significant for adaptive dosing design ($D_M>0,p=0.04$), it appears very small (Figure S21). In patients who initiated treatment with either high or low doses, subsequent dose reductions and the low accumulation result in a convergence of drug exposure levels across individuals, irrespective of their initial dosing regimen. Note that a similar trend was observed in the presence of negative relationship between exposure and response (see Figures S32 and S33 of the Appendix S3c). In contrast to the previous scenario with early events and strong accumulation, the relationship between $\mathit{CA}{V_w}_{\mathit{TE}}$ and the outcome is not reversed. Patients experiencing an early efficacy event drop out before accumulating high drug levels, creating an artificial relationship between lower exposure and better efficacy. This effect is mitigated in low accumulation/late event scenarios, where exposure profiles are less divergent and have more time to converge due to longer follow‐up times (events come later).

FIGURE 4.

Kaplan–Meier plots using each of the three time‐static exposure metrics and stratified by exposure tertiles. Ds is the difference of estimated survival probability at week 30 between the first and the third tertile group, and p is the p‐value of the log‐rank test. The scenario for generating data is the following one: positive E‐R relationship, late events and low accumulation of the drug.

3.1.4. Summary on All Simulated Data Sets

The heatmap in Figure 5 illustrates the type‐I error and the power of the log‐rank test across the main scenarios and exposure metrics. Overall, the use of CAV before the week of event ($\mathit{CA}{V_w}_{\mathit{TE}}$) should be avoided, due to extremely high type‐I errors (100% in some cases) introduced by unequal patient follow‐up durations. Note that under the negative E‐R relationship ($\beta <0$) and strong accumulation scenario, $\mathit{CA}{V_w}_{\mathit{TE}}$ leads to good conclusions in 100% of cases. However, even though the conclusion is correct and moves in the right direction, the relationship was found overstated (see Figures S23 and S24 as an illustration on one data set). In cases of low drug accumulation and adaptive dosing design, exposure metrics based on initial exposure may lead to incorrect conclusions, because dose reductions cause patients' long‐term exposures to become similar, regardless of their starting dose which obscures the true E‐R relationship.

FIGURE 5.

Heatmap showing the type‐I error (when β = 0) or power (when β ≠ 0) of the log‐rank test, across each scenario and time‐static metric.

Similar results were obtained when considering the t‐test (see the heatmap in Appendix S6, Figure S34).

3.2. Exposure‐Response Modeling for the Main Scenarios

3.2.1. Illustration on First Simulated Data Sets

In accordance with the exploration section, this section presents the results relative to the first data set of the two main scenarios: (i) positive E‐R relationship, early events and strong accumulation; and (ii) positive E‐R relationship, late events and low accumulation of the drug.

In summary, the TTE modeling confirmed the trends observed in the initial explorations: the use of $\mathit{CA}{V}_{w_{\mathit{TE}}}$ showed poor results for scenario when early events and strong drug accumulation, while the use of exposure based on initial data made the E‐R relationship weak and/or non‐significant in case of late events and low drug accumulation.

Results for all the scenarios can be found in Appendix S4: scenarios when no E‐R relationship in S4a, when a positive E‐R relationship in S4b and when a negative E‐R relationship in S4c.

3.2.2. Positive E‐R Relationship, Early Events and Strong Accumulation

Table S7 shows the parameter estimates for the first simulated data set of the current scenario, depending on the exposure metric considered in the modeling and both dosing designs. The use of $\mathit{CA}{V}_{w_{\mathit{TE}}}$ as a time‐static metric led to a misleading conclusion, as it reported a significantly negative $\beta$ estimate (−0.46 with 95% CI [−0.57, −0.34] and −0.87 with 95% CI [−1.03, −0.71] for fixed and adaptive design respectively). The true time‐varying metric supported reliable estimation of the underlying parameters for both fixed and adaptive dosing design.

3.2.3. Positive E‐R Relationship, Late Events and Low Accumulation

Table S10 shows the parameter estimates for the first simulated data set of the current scenario, depending on the exposure metric considered in the modeling. The use of time‐static metrics based on initial exposure showed a weak link between exposure and probability of event (when dosing design is adaptive, $\widehat{\beta }\left[95\%\mathrm{CI}\right]=0.11\left[-0.003,0.21\right]$ for $\mathit{CA}{V}_{w_1}$, and 0.04 [0.01, 0.07] for $\mathit{CA}{V}_{\mathit{SS}}$). Similar to the earlier case, the true time‐varying metric yielded good parameter estimates.

3.2.4. Summary on All Simulated Data Sets

The heatmap in Figure 6 presents the type‐I error and the power of the Wald tests on parameter $\beta$, across the main scenarios and exposure metrics. Overall, only the time‐varying metric provides reliable results whatever the scenario considered. Type‐I error (assessed when no relationship between exposure and response is simulated) is controlled for time‐varying modeling and for time‐static metrics based on initial exposure. The use of CAV before the week of event should be avoided. In case of low accumulation of the drug and adaptive dosing design, exposure metrics relying on initial exposure can lack power (power between 70% and 80%). As it is hard to know in practice in which scenario we are, we recommend using time‐varying exposure in order to avoid any potential mis‐leading conclusion.

FIGURE 6.

Heatmap showing the type‐I error (when β = 0) or power (when β ≠ 0) of the Wald test on β across each metric (time‐static or time‐varying) and each scenario.

The heatmap using the Cox model with time‐varying exposure is shown in Figure S35 of the Appendix S7. The performances appear similarly good as the parametric approach. This finding is notable because Cox proportional hazard models are quick to fit and do not require any assumptions about the shape of the baseline risk. As such, they could serve as an initial step to screen and gain a better understanding of the potential E‐R relationship.

3.3. Additional Simulation Scenarios

The heatmap in Figure 7 presents the type‐I error and power of the Wald tests on parameter $\beta$, across the additional scenarios where correlation was generated between the dose reduction process and the hazard of events. Type‐I error is controlled for all the scenarios when true model is fitted, and for almost all scenarios when the mis‐specified model is fitted (except when strong accumulation and late events where type‐I error = 10%). Power is very high whatever the scenario for both true and mis‐specified models.

FIGURE 7.

Heatmap showing the type‐I error (when β = 0) or power (when β ≠ 0) of the Wald test on β across each of the additional scenarios.

REEs on $\beta$ parameter are reported in Figure S36, for both true and mis‐specified models. Ignoring the correlation between the dose reduction process and the event process has limited impact on parameter estimation, as the distribution of REEs remains often centered around zero with moderate uncertainty. A moderate underestimation is observed in the presence of a positive E‐R relationship. The true model performs well when the ER relationship is absent or positive, although it tends to slightly overestimate in the case of a negative E‐R relationship.

4. Discussion

This work was performed to evaluate several graphical and modeling approaches for characterizing exposure‐response relationship under different scenarios. We showed that the use of time‐static metrics can be problematic in many cases. Firstly, the use of exposure metrics relying on last patient's observations can lead to strong mis‐leading conclusions, especially when the event occurred early and the accumulation is strong. However, in practice, the degree of strong accumulation considered in this scenario is likely to be less pronounced. An extreme case was presented to explicitly highlight the potential bias, but in the absence of such strong accumulation, the magnitude of the resulting mis‐leading conclusions would likely be more limited. We emphasize that any other time‐static exposure metrics computed during patient's follow‐up should be avoided, due to same limitations (e.g., the maximal concentration during the follow‐up ($C_{\mathit{max}}$) or the average concentration during the whole follow‐up ($\mathit{CA}{V}_{\mathit{TE}}$)).

Previous publications have pointed out the issue with exposure metrics dependent on the follow‐up time (e.g., CAV _TE ) and instead recommended using initial exposure [7, 9, 10]. However, we showed in this work that in some cases, the use of metrics relying on initial exposure can lead to a weaker power to detect the E‐R relationship. The heatmap we provided can be used as a decision tree for modelers. However, we emphasize that it is hard in practise to know in which situation we are. The determination of whether the event occurred early or late can be subjective. Also, the nature of the event is known (efficacy or negative event), but it is unclear whether the E‐R relationship coefficient ($\beta$) is different from zero. That is why we strongly recommend using time‐varying exposure to avoid wrong conclusions in the assessment of the E‐R relationship whenever exposure can be anticipated to change over time in some form of systematic manner (e.g., dose titration, accumulation, auto‐induction and auto‐inhibition). The use of time varying exposure metrics provides a more robust foundation for quantitative risk–benefit assessments and increase the reliability and predictive scope of model‐based simulations.

E‐R analysis for TTE endpoints with time‐varying exposure metrics can be conducted using pharmacometrics software such as Monolix or NONMEM. However, some R packages also allow the estimation of TTE models with time‐varying covariates, in a full parametric setting (flexsurvreg function of the flexsurv package [19], phreg function of the eha package [23]) or in a semi‐parametric setting (coxph function of the survival package [17]). The use of a Cox proportional hazards modeling approach may be especially appealing for screening purposes allowing for an efficient, fully pre‐specified analysis without the need for stepwise model building. However, parametric modeling approaches may still offer advantages especially when it comes to the development of a model that can be used for simulations.

When the dose reduction and event time processes are correlated, beyond what can be explained with drug exposure, mis‐specifying such correlation appears to have a limited impact on the E‐R analysis. A sensitivity analysis varying the level of IIV would be of interest. Overall, if a biological link is strongly suspected, a joint model is the most rigorous approach. However, this requires advanced tools beyond standard packages, which significantly increases run times.

The simulation framework we proposed in this article was for one rich sample size and did not account for dose delays, or treatment interruptions which are often observed in adaptive designs. Those settings can also affect E‐R analysis, and thus additional investigations considering additional scenarios and sample sizes could be performed in the future. Although this analysis does not model the full complexity of intercurrent events found in clinical practice, the main conclusions of this work still hold under the assumption that these events are independent of exposure.

To avoid introducing additional bias in our assessment, we made the simplifying assumption that true PK exposure was known. The choice of a weekly average concentration aligns with the weekly dosing regimen, although a more granular metric could also have been explored. A more complex alternative would have been to jointly simulate PK and time‐to‐event (TTE) data, and then re‐estimate the exposure metrics.

Finally, this project is closely related to causal inference and could also be presented using specific Directed Acyclic Graphs (DAGs), but that is outside the scope of this article.

Author Contributions

A.L.‐M., F.L.L., R.A., F.M., and M.B. wrote the manuscript; A.L.‐M, M.B. and F.M. designed the research; A.L.‐M, M.B and F.M. performed the research; A.L.‐M. analyzed the data; A.L.‐M, M.B. and F.M. contributed new reagents/analytical tools.

Conflicts of Interest

The authors declared no competing interests for this work. France Mentré is an expert advisor for Pharmetheus. As Editor‐in‐Chief of CPT: Pharmacometrics and Systems Pharmacology, France Mentré was not involved in the review or decision process for this paper.

Supporting information

Data S1: psp470149‐sup‐0001‐DataS1.docx.

Lavalley‐Morelle A., Le Louedec F., Anziano R., Mentré F., and Bergstrand M., “Exposure‐Response Analysis for Time‐to‐Event Data in the Presence of Adaptive Dosing: Efficient Approaches and Pitfalls,” CPT: Pharmacometrics & Systems Pharmacology 15, no. 1 (2026): e70149, 10.1002/psp4.70149.