A fully hierarchical Bayesian approach to sequentially update population parameter uncertainty in MIPD
Franziska Thoma (1,2), Manfred Opper (1), Charlotte Kloft (3), Jana de Wiljes (1), Niklas Hartung (1), Wilhelm Huisinga (1)
(1) University of Potsdam, Institute of Mathematics/Institute of Biochemistry and Biology; Potsdam, Germany, (2) PharMetrX Graduate Research Training Program; Berlin/Potsdam, Germany, (3) Freie Universität Berlin, Institute of Pharmacy; Berlin, Germany
Introduction: In model-informed precision dosing (MIPD), PK/PD models are used to predict therapy outcome based on patient characteristics and data from therapeutic drug/biomarker monitoring [1]. In a Bayesian statistical context, a prior distribution accounts for the uncertainty on the individual (ind.) model parameters. The aim of MIPD is to assimilate the ind. patient data to reduce the uncertainty with time [4]. On the ind. level, this uncertainty is a manifestation of the variability on the population (pop.) level. As data of more patients becomes available, the question arises of how to update the ind.-level prior [1,2]. In a Bayesian context, this is achieved by accounting for uncertainty on the pop. level, resulting in a hierarchical Bayesian model. In [3], a two-stage sequential hierarchical approach was introduced to assimilate the patients data and to update the pop.-level prior. While the general approach was successful in a dense data scenario, it did not correct for pop. parameter bias in a sparse data scenario. We hypothesize that this is due to a parametric approximation that was used to represent the pop.-level uncertainty.
Objectives: Implement and test a fully hierarchical Bayesian approach not using a parametric approximation to sequentially update pop.-level uncertainty in clinical practice.
Methods: On the ind. level, particle filters have been shown to provide useful information on ind. uncertainty and thus yield more informed predictions and dosing recommendations [4]. Building on this approach, we implemented a hierarchical particle filter approach [5] that allows us to incorporate and update the uncertainty on the pop. level sequentially. In essence, this approach uses two nested particle filters that allow a sequential update of the parameter distributions both on the ind. and pop. level. In contrast to the approach proposed in [3], no parametric approximation is used on the population level. For comparison, a non-sequential hierarchical approach using Metropolis-Hastings (MH) was implemented.
The hierarchical particle filter was tested in a simple model system with two sampling scenarios: dense (ni = 10) or sparse (ni = 2) longitudinal data was available for a varying number of individuals. The results were evaluated by comparing the posterior predictive distribution to the data generating distribution using the Kullback-Leibler divergence (KLD).
Results: We have successfully implemented a fully hierarchical Bayesian approach that allows for a sequential update of the uncertainty on the pop. level from ind. observations. It was applied to a simple model system with sparse and dense sampling scenarios. Importantly, we were not only interested in the magnitude of uncertainty, but rather in its effect on future predictions. We quantify this by calculation of the divergence between posterior predictive and data-generating distribution. For this, we choose the KLD, which measures the amount of information loss when approximating a reference by some other distribution. Importantly, in using the posterior predictive distribution, we can evaluate the performance even in cases of poor identifiability on the pop. parameter level.
We showed that in all cases, the performances of the sequential and the referenced non-sequential MH-approach are comparable. However, as the sequential approach is aimed at updating models in clinical practice, it suits our context much better and complies with data privacy laws as no ind. patient information is stored beyond its immediate use to update the ind. prior.
As expected, when updating the pop. prior on many individuals (N = 1000) with dense longitudinal data, we can accurately estimate the data-generating pop. parameters and predictive performance is high (KLD statistics from 100 rep: Median (IQR) = 0.0019 (0.0008,0.0039)). However, both approaches still preform well when given sparse longitudinal data (0.0076 (0.0037,0.0188)). With data from only few individuals, lower performance is observed for both sampling scenarios.
Conclusions: We have shown that a hierarchical particle filter approach allows for sequential updating of the pop. uncertainty and performs well in an illustrative simple model system for both dense and sparse data scenarios. The latter is of particular interest as it best resembles the clinical situation. These results allow us to apply our approach to more complex models as well as investigate computationally more efficient particle filter approaches.
References:
[1] Keizer, Ron J., et al. "Model‐informed precision dosing at the bedside: scientific challenges and opportunities." CPT: pharmacometrics & systems pharmacology 7.12 (2018): 785-787.
[2] Polasek, Thomas M., Sepehr Shakib, and Amin Rostami-Hodjegan. "Precision dosing in clinical medicine: present and future." Expert review of clinical pharmacology 11.8 (2018): 743-746.
[3] Maier, Corinna, et al. "A continued learning approach for model‐informed precision dosing: Updating models in clinical practice." CPT: Pharmacometrics & Systems Pharmacology 11.2 (2022): 185-198.
[4] Maier, Corinna, et al. "Bayesian data assimilation to support informed decision making in individualized chemotherapy." CPT: pharmacometrics & systems pharmacology 9.3 (2020): 153-164.
[5] Chopin, Nicolas, Pierre E. Jacob, and Omiros Papaspiliopoulos. "SMC2: an efficient algorithm for sequential analysis of state space models." Journal of the Royal Statistical Society: Series B (Statistical Methodology) 75.3 (2013): 397-426.
