Longitudinal Tumor Size and Survival Modeling for Exposure-Response Analysis of Drugs with Frequent Dose Reductions: Dose Justification of Abemaciclib in Patients with Metastatic Breast Cancer.

Dose omissions or dose reductions are commonplace in oncology clinical trials. Significant dose reductions and/or dose omissions mean that use of a single static summary exposure metric (such as the average steady‐state concentration, area under the curve, trough concentration or peak concentration) to reflect overall drug exposure in a clinical trial may not be appropriate. Indeed, this approach can lead to unexpected and biased exposure–response relationships where patients with longer survival (and therefore higher probability of dose reduction) have lower drug exposure as measured using common summary pharmacokinetic (PK) variables. In such cases, justification of the dose would be difficult since lower exposures (falsely) appear to lead to better survival.
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Similarly, using an early exposure metric before dose adjustments occur (e.g., cycle 1 or first dose PK) may have limited value in identifying the real exposure–response relationship since it does not reflect future changes in exposure due to changes in dose
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and does not take the dosing history into account. One might then be tempted to use cumulative (or time‐averaged) measures of exposure up until the event occurs. However, this approach also leads to biased exposure–response relationships because the drug exposure is somewhat driven by the response
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instead of the other way around. A longitudinal PK‐pharmacodynamic (PKPD) modeling approach is a more accurate representation of drug exposure in patients over time in a study. Since this approach captures the significant changes in dosing and drug concentrations in patients over time, this would result in better elucidation and characterization of the exposure–response relationship.
Progression‐free survival (PFS) is a common surrogate endpoint in oncology clinical trials. It is based on monitoring specific target tumor lesions over time according to the Response Evaluation Criteria in Solid Tumors (RECIST).
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While PFS is often the primary endpoint, information is lost when rich time‐course tumor size data is condensed into a single categorical outcome. When doses are adjusted or omitted (which is often the case with anticancer drugs), it is even more important to analyze the time course data of an efficacy measure that better tracks with these changes. At the same time, the categorical primary endpoint remains important and should be reported accordingly. Therefore, a simultaneous modeling approach that takes advantage of time course tumor size measurements and links them to survival would be advantageous. Indeed, the relationship between change in tumor size and overall survival through simultaneous modeling of the two measures has been previously reported and used for dose justification.
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Other work has also been reported to determine the relationship between various tumor size metrics and survival.
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Abemaciclib is an orally administered small molecule inhibitor of cyclin‐dependent kinases (CDKs) 4 and 6.
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Abemaciclib has been approved in the United States, Japan, Europe and other countries for the treatment of hormone receptor (HR)‐positive, human epidermal growth factor receptor 2 (HER2)‐negative breast cancer. Like many anticancer drugs, dose reductions and dose omissions of abemaciclib are permitted as needed for individual tolerability. Due to dose modifications, the use of standard, static PK metrics (such as Day 1 PK, cycle 1 PK, or average steady‐state concentration) to evaluate the exposure–response relationship for PFS is not recommended for reasons stated earlier. Indeed, empirical quartile Kaplan–Meier analysis yielded biased exposure–response as depicted in Figure

S4
. The figure uses steady‐state peak concentration, but the results were similar for area under the curve, trough concentrations, or when using PK exposure metrics on Day 1. Consequently, the objective of this study was to describe the exposure–response relationship of abemaciclib in patients with metastatic breast cancer by (i) characterizing the change in tumor size of target lesions over time and evaluating the impact of drug exposure, (ii) determining the relationship between change in tumor size and PFS and (iii) evaluating the impact of various patient factors on the exposure–response relationship.

METHODS
The MONARCH 2 clinical trial was registered at clinicaltrials.gov (NCT02107703). The study was conducted in accordance with the Helsinki Declaration and with approval from the corresponding institutional review boards.

Patients
The study participants were women with hormone receptor positive (HR+), human epidermal growth factor receptor 2 negative (HER2−) metastatic breast cancer who had progressed after endocrine therapy. A total of 663 patients were available for the analysis, of whom 477 had tumor size data (measurable disease) and were included in the tumor size modeling analysis. The remaining 186 patients had “non‐measurable” disease (bone metastases) and were included in the PFS analysis. Patients were randomized to receive either abemaciclib plus fulvestrant, or fulvestrant alone. The starting dose for abemaciclib in the study was initially 200 mg twice daily. Later during safety review, the starting dose was decreased to 150 mg twice daily, so that all newly enrolled patients would start at 150 mg twice daily, and all existing patients would also now receive 150 mg twice daily (unless modifications had already occurred). In both cases, the dose could be adjusted in 50 mg decrements due to adverse effects (such as diarrhea or neutropenia) or withheld altogether, as determined by the investigator. Further details about the MONARCH 2 clinical trial have been published.
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Data analysis
A nonlinear mixed effect modeling approach implemented in NONMEM 7.3.0 (ICON Plc. Gaithersburg, MD) was used to analyze the data. For exposure–response modeling of tumor size and PFS data, a multi‐step sequential modeling approach was used where the pharmacokinetic (PK) data were analyzed first. The population PK of abemaciclib which already included data from this study have been published elsewhere.
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Briefly, this was a semi‐mechanistic parent‐metabolite model that included parallel biphasic absorption and first pass effect (intestinal and hepatic metabolism). From that model, individual post hoc PK parameters with the dosing information were then used to predict concentration–time profiles of abemaciclib for exposure–response modeling. The exposure–response modeling was also a sequential process. First tumor size data were analyzed for those patients who had measurable disease. Then, using the model describing the change in tumor size, individual tumor size model post hoc parameters were used to predict the time course of change in tumor size and how it relates to PFS in patients with measurable disease. Patients without tumor size data (nonmeasurable disease) were also included in the PFS modeling dataset.

Exposure–response model for change in tumor size
Tumor size was defined as the sum of longest diameters of target lesions. To account for the delay in drug effect on tumor shrinkage, a series of three transit compartments was used in a model structure similar to that of Simeoni and colleagues.
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In this model, baseline tumor size was estimated and used to initialize the tumor size compartment, representing the viable proliferating tumor cells. A zero‐order growth rate for the tumor cells in this compartment was estimated. Unlike the Simeoni model which contained a switch from exponential tumor growth to zero‐order growth based on xenograft experiments, a zero‐order growth rate alone was used, as patients presenting in a clinical trial would have established tumors (linear phase of growth) as compared with xenograft inoculations which undergo exponential growth upon inoculation. The action of abemaciclib was to damage these “healthy proliferating” cells through a first‐order decline in the amount of cells that would then go through a series of three transit compartments until cell death as shown in Figure

S1
, adapted from Simeoni and colleagues.
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The set of equations is as follows:

where Healthy are the healthy proliferating tumor cells, EFFdrug is the drug effect (abemaciclib + fulvestrant), Damage N is the tumor size for each compartment containing cells damaged due to drug administration, and Tumor size is the model‐predicted tumor size (sum of diameters of target lesions) at a particular time.
Various pharmacodynamic models were tested for the abemaciclib concentration–effect relationship including linear, maximum effect (E
max) and power models. The stochastic approximation expectation–maximization (SAEM) estimation algorithm was used, along with a full covariance block for the random effects to optimize its efficiency, followed by importance sampling (IMP) for objective function evaluation. Upon obtaining a satisfactory tumor size model, the individual patient post hoc tumor parameters were then used in a sequential tumor size—PFS model.

Exposure–response model for progression‐free survival
A time to event modeling approach was used to fit the PFS data, and the Laplacian estimation algorithm in NONMEM was used. For the modeling, the post hoc PK parameters and post hoc tumor parameters were added to the time to event dataset to obtain drug concentrations and tumor size predictions for patients. Like the change in tumor size model, the observed dosing history was used for each patient, taking into account dose reductions and dose omissions. Naturally, the tumor size predictions were restricted to patients who had measurable disease (and thus tumor measurements were performed). For patients without measurable disease, tumor size was not used to inform the PFS model since tumor size measurements were not available to determine disease progression in those patients (clinical assessment was used). Various hazard functions were evaluated, including exponential, Gompertz, Weibull or log logistic functions. For patients with measurable disease, the design of the study was such that tumor assessments were performed approximately once every 2 months. This meant there would be a large time window during which progression could occur. Therefore, it was necessary to address this limitation through a modeling approach that included interval censoring. This was achieved through a model structure that calculated the likelihood of an event occurring between a time interval as the difference in the survival probability at the last known time when no progression was observed from the survival probability at the recorded time of the progression. This is illustrated in the equation below.

To explain in more detail, let us assume a patient in the dataset has an event at clinic visit number n, denoted by time, t

n
. The survival at that time (Surv
t,n
) was calculated from the cumulative hazard from zero up to time t
n as follows:A similar calculation was then done for the same patient for the previous clinic visit (n−1) where they did not have a disease progression event, denoted by time t

n−1. The difference between the two then gives the probability that the event occurred within the interval between t

n−1 and t

n.

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For patients with tumor size information, the absolute tumor size as well as the change from baseline (CFB) tumor value were separately tested as a predictors of the baseline hazard. Using the CFB approach has the advantage of increasing tumor size increasing the hazard and decreasing tumor size decreasing the hazard without influence from baseline. Outside of influencing the change in tumor size, drug effect was additionally tested as a predictor of the hazard, with separate effects being tested for those with tumor size information and those with no measurable disease. Linear, power, and E
max drug effect models were tested. In the modeling, patients with tumor size information had two drug effects, one coming through the tumor size model, and an additional drug effect representing the action of abemaciclib on PFS outside that mediated through the sum of longest diameters of target lesions. This would be reasonable since the definition of disease progression was not limited to tumor growth of target lesions, but also tumor growth of nontarget lesions and/or appearance of new lesions. Patients without measurable disease would thus have a single drug effect since they did not have tumor measurements during the course of the study. The PK—tumor size—PFS model was of the structure depicted in Figure

1
. Covariates evaluated in the modeling framework included Eastern Cooperative Oncology Group (ECOG) status (0 or 1), presence or absence of hepatic metastases, alanine aminotransferase (ALT) concentration, age, race, and patient population (measurable vs. nonmeasurable disease).
Per standard modeling practice, goodness‐of‐fit plots, visual predictive checks (VPCs) and bootstraps were used for model evaluation of the PK, tumor size, and PFS models.

RESULTS
Of the 663 patients in the MONARCH 2 PKPD analysis dataset, 223 were randomized to fulvestrant + placebo, while the remaining 440 received fulvestrant + abemaciclib.

Figure

S2
contains Kaplan–Meier plots for patients in the abemaciclib arm stratified by the number of dose reductions. Patients with longer survival experienced more dose reductions, thereby demonstrating the need to incorporate dynamic changes in dose/exposure using a time course modeling approach to correctly determine the exposure–response relationship. However, the figure should be interpreted with caution because of PK variability and the possibility that those with higher drug concentrations had a greater number of dose reductions.

Tumor size exposure–response model
The model that described the change in tumor size was comprised of linear growth followed by tumor shrinkage through a series of transit compartments. This tumor shrinkage was influenced in a time‐dependent manner by the abemaciclib plasma concentration through a linear model, as well as an effect of fulvestrant. Nonlinear drug effect models including E
max or power models did not result in improved model fit.where Efffulvestrant is the tumor shrinkage driven by fulvestrant, Effabemaciclib is the abemaciclib effect which depends on the plasma concentration at time t (conc
t
), and resist is an exponential decline in the drug effect which starts after a delay.
Parameter estimates for the tumor size exposure–response model are presented in Table

1
.
A positive relationship was observed between the abemaciclib plasma concentration and the tumor shrinkage, and is illustrated in Figure

2
using parameter estimates from the final model. The intercept on Figure

2
is 0.00647 week−1 representing the typical effect of fulvestrant alone. The abemaciclib exposure–response relationship may be viewed as shallow. For example, an abemaciclib plasma concentration of 200 ng/mL was associated with only twice the tumor shrinkage of 50 ng/mL.
The drug effect on tumor shrinkage was estimated to start decreasing after about 2 months. However, there was high variability (473%) in the start time (Table

1
). The ‘half‐life’ of this decrease in drug effect was estimated to be 2.51 weeks and 8.44 weeks for patients in the control arm and abemaciclib arm respectively with high variability (Table

1
). This gradual decline in drug effect over time is illustrated in Figure

S3
and the difference in decline between the abemaciclib and control arms is consistent with the expectation that combination therapy can protect against development of resistance and prolong drug effect. The high variability in the resistance parameters implies that while resistance occurs in some patients, many patients continued to experience tumor suppression throughout the follow‐up period, particularly those in the abemaciclib + fulvestrant group. Figure

S5
has sample individual plots that show significant variability in the tumor response over time, and this is reflected in the model parameters.
Prediction‐corrected VPCs (Figure

3
) demonstrated that the model could adequately predict the observed data. Because the tumor assessments were only performed until disease progression was observed (as per RECIST criteria, either a 20% increase in size from the nadir [or baseline], or appearance of a new lesions), it was necessary to account for this dropout effect in the simulations for the tumor size VPC. Model simulated values that were 20% greater than the nadir (or baseline) were therefore excluded, as were model simulated values that would have been obtained from patients who progressed for other reasons (appearance of new lesions, or progression of nontarget lesions). As a result, it was necessary to add the hazard for progression from the PFS model (described later) to simulate such events from the tumor size model. VPCs are presented (Figure

3
) for simulations conducted with and without incorporation of dropout. As expected, not accounting for dropout results in overprediction of tumor size at later time points.

Progression‐free survival exposure–response model
The dynamic PFS model included dosing history, the PK model (using individual post hoc PK parameters) and the change in tumor size model (using individual post hoc tumor model parameters) and may be viewed as PK‐PD‐PD model for patients who had measurable disease (had tumor measurements). One hundred and eighty‐six patients did not have measurable disease and thus did not have tumor measurements. These patients had bone‐only disease which could not be measured, although they were part of the study and progression was based on investigator clinical decision.
The hazard model that best described PFS was an exponential distribution of event times (constant hazard model). In patients with measurable disease, the hazard was influenced by the CFB tumor size as well as an independent linear drug effect which represents the action of abemaciclib on PFS outside of the sum of longest diameters of the target lesions. The same linear drug effect was used for patients who did not have measurable disease. The hazard model for PFS is shown below.where Haz is the hazard at time t; HBASE is the baseline hazard, CFB is the change in tumor size from the baseline tumor, THAZ is the effect of CFB on the hazard, SLOPE is the coefficient for the drug effect, CONC is the abemaciclib plasma concentration at time t, DECAY is the rate constant for the decline in drug effect with time on treatment.
CFB was more significant than absolute tumor size as a predictor of the hazard, meaning that the effect of abemaciclib on PFS was not affected by baseline tumor size. The baseline hazard was estimated separately for those with no measurable disease and it was 34% lower than that for those with measurable disease. Therefore, patients with no measurable disease had higher predicted survival than those with measurable disease. A mixture model was implemented for the parameter that linked the CFB with the hazard, THAZ. THAZ was estimated to have a stronger effect in 48% of patients with measurable disease, where a 3 mm change in tumor size resulted in a twofold change in the hazard. In the other 52% of patients with measurable disease, a 44 mm change in tumor size would be required to produce a twofold change in the hazard. This finding is in accord with the fact that of the 477 patients with measurable disease, 170 (36%) had either new lesions or had progression of nontarget lesions, meaning that for some patients, the reason for progression was outside progression of target lesions, although CFB in target lesions still played a role in the hazard.
A decrease in drug effect with time on treatment was noted, similar to that seen in the tumor size model. For the PFS model, in the typical individual with measurable disease the drug effect would be predicted to decrease to half of its initial effect after 12.2 weeks. This effect was not significant for patients without measurable disease, perhaps due to the subjective nature of the progression.
Covariates that were significant on the baseline hazard of progression were presence of hepatic metastases (doubled the hazard) and higher ECOG status (48% higher hazard for ECOG = 1 than ECOG status of 0). Parameter estimates for the final model are in Table

2
. The NONMEM model code for the simultaneous PK—tumor size—PFS model is available in the
supplementary material
. The VPC in Figure

4
shows that the model could adequately predict the observed PFS data. The model could therefore be used for simulations to visualize the dose exposure–response relationship for PFS and determine the impact of dose reductions.

Model application for dose justification and impact of dose reductions
The final PK/PD model which incorporated the time‐course of abemaciclib dosing and PK, tumor size information as represented by the sum of longest diameters of target lesions, and PFS was used to simulate more than 50,000 virtual patients for a range of dose levels to understand the dose exposure–response relationship following twice daily dosing. The simulations were done for a range of starting doses carried through to the end of the study with no dose reductions as these would be difficult to predict. The result is shown in Figure

5
which also shows good agreement between the model simulation and the observed data. The simulations show that when measured from the time of a baseline tumor size scan, a 150 mg twice daily starting dose carried through with no dose reductions would result in a median survival time of 19.6 months, while a dose of 140 mg twice daily (comparable to the 136 mg which was the average dose for the 150 mg twice daily post‐amendment population followed by dose reductions) resulted in a survival time of 18.7 months. Therefore, the overall impact of dose reductions as done in the clinical trial after a 150 mg twice daily starting dose was a small decrease in the survival time of 0.9 months (4.6% decrease).

DISCUSSION
There is no greater predictor of PFS than the sum of diameters of target lesions which is the metric used to determine whether or not disease progression has occurred in a patient. Therefore, simultaneous modeling of the change in tumor size and using it as a predictor of the ‘snapshot’ categorical PFS endpoint maximizes use of the available longitudinal data. Summary PK metrics, such as peak concentration, trough concentrations, or average steady‐state concentration have limited value for attempting to determine exposure–response relationships when the exposure varies significantly within an individual over time due to dose reductions or drug holidays. The dynamic PKPD model incorporating individual dosing history, fluctuating concentrations, change in tumor size, and PFS was developed to describe the relationship between dose, concentrations, and efficacy in the context of dose reductions. Tumor size measurements provide a more ‘real‐time’ assessment of the impact of dose reductions or dose omissions. The modeling approach presented here enabled elucidation of a positive linear relationship between abemaciclib concentration and tumor shrinkage and also PFS. Higher abemaciclib concentrations were predicted to be associated with larger reductions in tumor size and also longer PFS. The model enabled evaluation of pertinent covariates including the prognostic factors of ECOG status (ECOG = 1 had 48% higher hazard than ECOG = 0), presence of hepatic metastases (doubled the hazard), and measurable disease (34% lower hazard for patients without measurable disease) for patients in both the abemaciclib and placebo arms were taken into account. The reason for a lower hazard in patients without measurable disease is unclear, but could be due to the subjective nature of clinical observation. One limitation of the modeling is that the mixture model in patients with measurable disease represents unknown factors that result in CFB being less influential on the hazard for disease progression in some patients than in others. There may be some biological reasons such as differences in tumor activity (as opposed to tumor size alone) or other clinical reasons not captured in our dataset.
The average actual doses administered, taking dose reductions into account in the group of patients who started at a dose of 200 mg twice daily and 150 mg twice daily were 160 and 136 mg, respectively. Furthermore, due to PK variability, there was significant overlap in abemaciclib concentrations achieved in the two different starting dose groups. The efficacy observed regardless of the starting dose was similar in these two groups, due to in part to the relatively shallow concentration–response relationship and in part to PK variability. Consequently, the overall impact of dose reductions for individual tolerability as implemented in the clinical trial after a 150 mg starting dose was estimated to be a small decrease in survival time when considering the variability in response and patient population, as demonstrated through the simulation. The alternative to dose reductions would be complete cessation of treatment. This would deprive the patient of effective treatment, and this work suggests a negligible impact of dose reductions on efficacy. The model‐based predictions of negligible impact of dose reductions are consistent with actual findings from other studies of abemabiclib.
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Figure

5
shows that higher dose would result in greater PFS. However doses greater than 150 mg twice daily would be associated with greater adverse events (diarrhea in particular), hence a maximum starting dose of 150 mg in combination with fulvestrant is recommended
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and supported by the modeling and simulation. The negligible impact of dose reduction on efficacy is likely due to the initial doses resulting in drug exposure at the higher end of the exposure–response relationship. Doses in this dataset ranged from 50 to 200 mg twice daily and this could influence characterization of the exposure–response relationship. This fairly narrow dose range may be considered a limitation of the modeling analysis. Therefore, our findings related to the impact of dose reductions should be interpreted within the context of the studied dose range.
As mentioned in the introduction, we were driven to develop and implement the longitudinal modeling presented in this article because standard quartile‐based Kaplan–Meier analyses led to biased results. In general, multiple factors need to be considered before resorting to this more complex and computationally intensive modeling approach. These factors include: the PK profile of the drug in question (long half‐life vs. short half‐life); the duration of drug holidays in relation to the ‘forgiveness’ of the regimen to missed doses; the proportion of patients who undergo dose reduction; magnitude of dose reduction, among others. Yue et al. conducted a simulation study that provides insights into these aspects. Their study sought to determine the extent of dose reduction which would necessitate longitudinal PK/PD modeling instead of Kaplan–Meier quartile analyses.
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A key finding from their analyses was that the longitudinal approach is recommended when > 40% of patients have dose reductions, which was the case for abemaciclib.
While our model linked change in tumor size to PFS, the presented method can be readily applied to other relevant efficacy‐based endpoints in oncology, such as overall survival. Our longitudinal approach should also be considered for safety/tolerability endpoints because those endpoints also need to be informed by the appropriate drug exposure metrics and may be equally prone to the shortcomings of static PK analyses in studies with dose reductions. The relevance of linking tumor size to safety or tolerability endpoints can also be evaluated on a case‐by‐case basis.
In summary, the PKPD modeling and simulation analyses presented support the starting dose of 150 mg twice daily in combination with fulvestrant, with dose reductions allowed in 50 mg decrements by taking into consideration the balance of risk and benefit. We demonstrated the importance of implementing a simultaneous time course PK—tumor size—PFS modeling approach for determining the exposure–response relationship for a drug in the context of significant dose reductions. The concepts and modeling approach presented here should be applied in similar situations for modeling PFS or overall survival data.

Funding
Eli Lilly funded the study.

Conflict of interest
All authors are employees of Eli Lilly and Company.

Author contributions
E.C., S.C.C., and P.K.T. wrote the manuscript; E.C., S.C.C., and P.K.T. designed the research; E.C., S.C.C., and P.K.T. performed the research; E.C. analyzed the data.

Supporting information

Data S1.

Data S2.