Supporting drug development as a Bayesian in due time?!
Sebastian Weber
Novartis Pharma AG, Basel
Objectives:
Bayesian approaches give the pharmacometrican additional control over models and can greatly facilitate model based drug development, e.g. partial pooling of few data-sets is a key strength. However, Bayesian approaches are often hampered by their enormous computational burden which can quickly become a problem as model evaluation times easily exceed days for ODE based models. I will present solutions to these practical problems at the example of Bayesian aggregation of average [1,2] data into a population non-linear mixed effect model. In will briefly introduce the Bayesian aggregation of average data approach and then describe generally applicable approaches which reduce the computation time from days to less than one hour. These include analytical shortcuts, approximations and a demonstration of the recently available parallelization technique in Stan [3].
Methods:
As generic example for a pharmacodynamic model an ODE based turn-over model is used. The data-set comprises about 1300 patients from three different studies which collected per patient monthly observations over a year of follow-up time. As the individual pharmacokinetic data was infeasible to measure, a simple one-compartment model was used with known typical patient parameter estimates as input for the turn-over pharmacodynamic model. The average data of an equivalent of 1200 patients was included in the model likelihood, using a nested simulation approach during MCMC integration with Stan.
Results:
The model was expressed using a single ODE for the turn-over equation while the one-compartmental pharmacometric model was solved analytically. Still, the model evaluation time on the individual patient data alone took 2.5 days when using the non-stiff Runge-Kutta 4/5 ODE solver in Stan. However, approximating the concentration time curve by a suitably chosen step function allows to analytically evaluate the turn-over equation in a stepwise manner. This lead to an 8x speedup in model evaluation to only 8h runtime while providing matching results. The accuracy of the approximation can be controlled by decreasing the time step size used in approximating the concentration time curve. A further approach is offered by the current development version of Stan which can perform within chain parallelization using the message passing interface (MPI). Applying MPI parallelization to the ODE based model resulted in a 62x speedup on 80 cores and a net runtime of just one 1h. For the analytical approximation model an 11x speedup on 15 cores was feasible which corresponds to only 45 minutes execution time.
Conclusions:
Excessive running times are an issue for wide spread adoption of any modeling technique. This has been a major drawback of Bayesian approaches for pharmacometricians. I demonstrate how analytical shortcuts or approximations can reduce or avoid the need for computationally intensive ODE solutions. While these techniques are generally applicable, they may require substantial investment of the modeler to work out problem specific analytical solutions. Thus, the availability of within-chain parallelization in Stan is a critical feature which enables reasonable running times for ODE based models in Stan.
References:
[1] Annals of Applied Statistics. 2017. Weber et al.
[2] PAGE Meeting, Alicante, Spain. 2014. Weber et al.
[3] Stan Development Team. 2018. http://mc-stan.org.
