Multistate model for pharmacometric analyses of overall survival in anticancer treatments
Sreenath M. Krishnan (1), Lena E. Friberg (1) and Mats O. Karlsson (1)
(1) Department of Pharmacy, Uppsala University, Uppsala, Sweden.
Objectives: Multistate models have advantages in analyzing time to event data in the presence of competing risks and time-varying covariates [1-3]. Beyer et al.  developed a multistate model for describing response, progression, and death following the initiation of treatment in patients with cancer. The model described the transition hazards of intermediate events using semi-parametric models with treatment arm as a binary covariate. Multistate models can also be formulated as fully parametric models with graded and time-varying covariates [5-6]. The aim of this study was to develop a multistate model operated by parametric hazard functions while allowing the investigation of predictors derived from the longitudinal tumor size model on the transition hazards.
Methods: Longitudinal tumor size data (sum of longest diameters, SLD) and survival times for 1000 subjects were simulated using a model developed by Bender et.al  for docetaxel treatment in HER2-negative metastatic breast cancer patients. The simulated profiles were assigned to have different states:
State 1- Stable disease (initial state for all subjects)
State 2 - Tumor decrease (>=30% decrease in SLD from baseline)
State 3 - Progressive disease (>=20% increase in SLD from tumor nadir)
State 4 - Second-line (50% of the subjects who had the progressive disease)
State 5- Death.
In the first step, the tumor growth inhibition model  was applied to the SLD data and the parameters were re-estimated. An approach similar to PPP&D  was then used for the joint modeling of the tumor and the multistate data. In that joint modeling, only individual SLD data up to t was used in the estimation of the hazards after time t.
Multistate model building: From the initial, stable disease, state patients could have events that led to a change of state. The subjects could have an event of death (state 5) from any state. Additionally, subjects with stable disease could have a response (i.e. tumor decrease) or progress. Patients with a response could progress and progressed patients could receive second-line treatment. A multistate model, where the transition intensities (λij) between each state were estimated, was developed to describe the observed data. The transition intensity for each transition was evaluated with different hazard distributions (constant hazard, Weibull, Gompertz). The investigated predictors on transition intensities included age, baseline tumor size, the relative change in SLD from baseline (rSLD), the relative change in SLD from previous measurement (dSLD), the relative change in SLD from tumor nadir and time to progression.
Results: In the final model, Weibull probability density functions described the transitions from stable disease to tumor decrease (λ12), stable disease to progressive disease (λ13) and progressive disease to death (λ35); while a constant hazard function explained the transition from tumor decrease to progressive disease (λ23). The hazard of death from the second-line was similar to be the same as to the hazard for death from the state of progression and the same function was used for the two transitions (λ45 = λ35). The estimated hazard of death from stable disease (λ15) and tumor decrease (λ25) was close to 0, hence the hazards were fixed to Gompertz-Makeham distribution to allow for a hazard no lower than the expected age-specific hazard . A mixture model with two sub-populations best described the transition from progressive disease to second-line (λ34), where pop-1 received second-line treatment within 6 weeks (range: 5-9 weeks) and pop-2 did not receive second-line treatment. A larger tumor baseline was associated with higher λ12 and λ35. The risk of disease progression from tumor decrease state (λ23) was lowered by a decrease in tumor size (smaller dSLD). The hazard of death after progression (λ35 and λ45) was higher for subjects who had higher rSLD values (higher tumor burden). No covariates were significant in predicting λ13.
Conclusion: The developed multistate model adequately described the transitions between different states and jointly the overall event and survival data. The tumor-multistate model allows for simultaneous estimation of transition intensities along with their tumor model derived covariates. The investigation of predictors and characterization of the time to develop response, the duration of response, the progression-free survival and the OS can be performed in a single multistate modeling exercise.
 Broët et. al., Breast Cancer Res. Treat (1999).
 Putter et. al., Biom J (2006).
 Proctor et. al., Pharm Stat (2016).
 Beyer et. al., Biom J (2019).
 Karlsson et. al., Clin Pharmacol Ther (2000).
 Moustafa et. al., PAGE 28 (2019) Abstr 9033 [www.page-meeting.org/?abstract=9033].
 Bender et al., PAGE 26 (2017) Abstr 7344 [www.page-meeting.org/?abstract=7344].
 Zhang et. al., J Pharmacokinet Pharmacodyn (2003).
 Missove et al., Theor Popul Biol (2013).