Within subject variability in pharmacometric count data modeling analysis: dynamic inter-occasion variability and stochastic differential equations
Chenhui Deng, Elodie L. Plan and Mats O. Karlsson
Pharmacometrics Research Group, Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden
Objectives: Parameter variation in studies can be characterized as within subject variability (WSV) in pharmacometric models. WSV has previously been successfully modeled using inter-occasion variability (IOV) [1], but also stochastic differential equations (SDEs) [2]. Studies with count data as endpoint are often long-term with frequent recordings and WSV can be expected to be important. However, the traditional IOV implementation method (IOV) necessitates occasions to be predefined and, to the best of our knowledge, SDEs have so far only been applied to continuous models. Hence, this study aims to develop and evaluate approaches addressing WSV in count models based on IOV and SDEs.
Methods: Base models were derived from a Poisson distribution, where λ accounts for mean count. A dynamic IOV (dIOV) approach was applied to depict the change of IOV over time on λ or other parameter; it used a sum of sequential functions involving the parameters: occasion length, amplitude and shape factor. The implementation of SDEs [2] was adapted to estimate parameter variability on λ. The models were fitted to 2 published pharmacodynamic data sets, seizure counts [3] and Likert pain scores [4]. In addition, IOV with occasion length fixed to the estimate of dIOV was conducted. Likelihood ratio test and graphical evaluation were used to quantify the potential improvement of model-fit in comparison to the published models. Simulations were used to explore further the capabilities of the two approaches.
Results: When WSV was defined as IOV, the OFV drops compared to a model without WSV were only 84 (df=2) and 227 (df=2), for the seizure count and Likert pain score data sets, respectively. The corresponding decreases for dIOV models were 201 (df=4) and 1022 (df=6), where in the latter data set IOV and dIOV were introduced on both λ and the inflated probability of change in score. When including SDE in the models, the OFV dropped by 159 (df=2) and 1421 (df=1), respectively. Simulations confirmed the systematic gains in introducing WSV as dIOV or SDE compared to IOV and enabled model misspecification detection when present.
Conclusions: The proposed approaches in this study offer strategies to characterize WSV in count data models. The developed dIOV method can capture the change of IOV over time in rich long-term studies and improve model-fit. The adapted SDE approach can be applied to quantify parameter variability and detect model misspecification.
References:
[1] M.O. Karlsson and L.B. Sheiner, JPKB, 21 (6), 1993 [2] C.W. Tornøe, et al., Pharmaceutical Research, 22 (8), 2005 [3] I.F. Trocóniz, et al., J PK&PD, 36, 2009 [4] E.L. Plan, et al., CPT, 91(5), 2012
Acknowledgement: This work was supported by the DDMoRe project.