# Optimal design of time-to-event models with an emphasis on dropouts in disease progression studies

Objectives: Population optimal design is a tool to increase efficiency in drug development [1]. However, the population models used in drug development are becoming more and more advanced as the models incorporate additional variables like, for example, Time-To-Event (TTE), discrete type outcomes, etc. This increase in model complexity makes model-based optimal design more relevant but the methods to compute the optimal designs must be improved to be able to handle these types of models. From a maximum likelihood point of view the main issue with these more complex models is their increasing non-linearity. Recently a general method has been presented for optimizing experiments using these highly non-linear models by use of higher order approximations or exact calculations of the population likelihood [2].

Most applications of optimal design have focused on optimizing sample times or the population size of a study, however several examples of optimal design have been presented optimizing on other design variables such as doses, study length, infusion duration, dose orders, etc. [3,4,5]. In one previous study optimal design was used to optimize start and stop time of disease progression (DP) studies with fixed sample times [7]. However, the models used in this study did not include a dropout model which often will be an important part of a disease progression study [8,9].

The objective of this work is to show the importance and the development of a general method of optimal design for time-to-event models. In this work we apply these general methods to the optimization of disease progression studies using disease progression models with incorporated dropout model components.

Methods: A linear protective disease progression model was used with a linear natural history of the disease. All of the parameters in the model had either proportional or exponential between-subject variability terms. An additive residual error was used. The study was designed to have 200 individuals with both a pre-treatment and washout periods before and after the drug treatment. The sample times were fixed to be every integer time unit between 0-12. The optimization was done on the start and stop time of the treatment, i.e. tstart, tstop ~[0,12]. Two different models were used for the dropout, 1) a constant hazard corresponding to ~50% of the individuals having dropped out at time 12 and 2) a hazard depending on the increase of the typical individual disease progression status from baseline with different magnitudes of dropouts, i.e. very low (VL) ~99.95% survival at t=12, high (H) ~50% survival at t=12 and extremely high (EH) ~1% survival at t=12. For the constant hazard model a Fisher Information Matrix (FIM) was used with no correlation between the dropout parameter and the DP parameters: FIM=[FIMDP 0; 0 FIMDrop] where FIMDP=SUM(gi*FIM1,i), gi is the number of individuals that have dropped out from the study between time i and i+1 and FIM1,i is the FIM for one individual evaluated with the sample times until time i. With this setup the FIMDP could still be sufficiently approximated by a first order approximation and the FIMDrop could be calculated analytically for this simple model [10] or using the same technique as in Nyberg et al. [2] for a general model. Three different scenarios were investigated for the constant hazard; No dropout model (ND), dropout model included but FIMDrop was excluded from FIM, i.e. gi changes with time (D) and dropout included in FIM (ID).  For the DP dependent hazard model the FIM was defined as: FIM=[(FIMDP+FIMDP|Drop) (0 or cov(θDP, θDrop)); (0 or cov(θDP, θDrop)) (FIMDrop)] where FIMDP|Drop is the information the dropout model gives about the typical DP parameters, cov() is the covariance between the typical DP parameters and the dropout parameters or 0 if it is the covariance between the dropout parameters and a DP random effect. Again, the three different scenarios were investigated; (D), (ID) mentioned above and a scenario (DD) where FIMDrop was excluded from the FIM but information about the DP parameters was still available from the dropout model, i.e. FIMDP|Drop≠0. All methods and optimizations were implemented and performed in PopED 2.09 [5,6].

Results: The optimal designs were quite different for the scenarios investigated. For the constant hazard model the optimal design changed from tstart≈0 or 4 and tstop≈12 or 8 with |FIM|=1.6e+14 for the (ND) scenario to an optimal design at tstart≈4 and tstop≈12 for the (D) (|FIM|=2.8e+13) and (ID) (|FIM|=5.5e+15) scenarios. For the DP dependent hazard with high dropout (H) the optimal design was tstart≈4 and tstop≈12 with |FIM| = 2.7e+13 for the (D) scenario, tstart≈1 and tstop≈8 and |FIM|=1.8e+14 for the (DD) scenario and tstart≈1 and tstop≈9 with |FIM|=1.9e+17 for the (ID) scenario. The difference of including information about the DP parameters (DD) versus not including this information (D) could be viewed as an efficiency = (2.7e+13/1.8e+14)(1/7) ≈76%. This indicates that including the information from the dropout model (DD) will decrease the average uncertainty/parameter to ~76% of the uncertainty with the (D) scenario. If the magnitude of the dropout changed (VL or EH) the design changed accordingly and in some cases changing the design parameters slightly could give significantly worse results.

Conclusions: A general method for incorporating TTE models into optimal design calculations has been successfully implemented in PopED 2.09. The optimal designs change for DP models including dropout (sometimes dramatically) if dropout models are included in the optimization. This implementation is made with an emphasis on dropout models but the method is general and more complex TTE models can be used and population models with repeated time to event can be implemented. Furthermore, the assumption that the correlation between the DP random effects and the dropout model must be 0 or that the typical DP parameters are driving the dropout model could be relaxed; naturally, with a higher computational cost. Other extensions that could be interesting to investigate are to use other types of DP effects and models, e.g. slow onset of the drug effect. Again, these models should be straight forward to use within this framework.

References:

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