Ségolène Siméon (1), Rémy Beaudouin (1), Katharina Brotzmann (2), Frédéric Y. Bois (3)
(1) INERIS, METO unit, Verneuil en Halatte, France. (2) University of Heidelberg, Aquatic Ecology and Toxicology, Centre for Organismal Studies, Germany. (3) CERTARA, Simcyp division, Sheffield, UK
Objectives: The forward Kolmogorov differential equations governing multistate models are usually solved with matrix exponentials. This imposes severe time-homogeneity limitations on the probability rates’ sub-models that can be used. We demonstrate here the application of numerical integration methods to solving general non-homogeneous forms of such models. Our application case study has also a substantive interest: the OCDE guideline on zebrafish embryo malformation testing recommends performing a separate dose-response analysis for each observation time and test outcome. However, outcomes at a given time are strongly conditioned by the whole set of past outcomes. That competing risk problem can be solved with time- and dose-nonlinear multi-state models.
Methods: We developed multi-state models for the time of occurrence of 55 developmental malformations and live events that are typically observed experimentally during the five first days of life of zebrafish, in the absence or presence of exposure to a chemical drug. The fish (60 or more per exposure group) were observed individually once per day (panel data) during continuous exposures to eight concentration levels of valproic acid. We performed statistical inference on the model parameters and structure using a Bayesian framework and numerical methods (MCMC). A mix of informative and non-informative priors were used. The likelihood of transitions was classically given directly by the model-computed transition probabilities over a day. Stiff differential equations solvers of the GNU MCSim package (LSODE or CVODE) were used. Hatching probability rate is clearly not constant in time and we compared three forms of its time-dependence. We also estimated the time- and water concentration-dependence of transition probability rates from a state to the others using flexible sub-models.
Results: Our results show that complex non-homogeneous multistate models can be quickly calibrated by numerical integration and MCMC sampling. We examined various sub-models (constant, piecewise constant, Hill, scaled Gaussian) of the hatching rate as function of time. A constant piecewise model with estimated discontinuity time was found to have the smallest Akaike information criterion and a fit R2 of 0.998. We also found that, due to inter-experiment variance in the staging of the embryos at start of the experiment, the hatching parameters were best modelled hierarchically across dose-levels. Proportional hazard sub-models were found to be sufficient to describe the dose dependencies of the transition rates between five states (normal, hatched normal, malformed, hatched malformed and dead). Using the complete model, complex dependencies between transitions leading to a range of cardiac malformation and death were observed and modelled successfully. The direct transition from normal to dead was not much dependent on dose; the malformation rate of normal or hatched embryos, and the death rate of malformed embryos, were strongly dose-dependent. Reversions from malformed to normal or hatched were dampened by increasing dose. Parameters were identified with 50% precision on average. Our model makes full use of the data and its results give a much finer picture of the mechanism of teratogenic effects of valproic acid in zebrafish, as a function of time and dose, than the OECD recommended approach.
Conclusions: Our models improve significantly the analysis of malformation effects in zebrafish embryos. Similar models could be used for other species and it would be interesting to see whether their predictions would have improved translatability to humans. We have previously developed a PBPK model of the zebrafish embryo and plan to use the organ concentrations it predicts instead of water concentration to condition instantaneous transition probability rates. It would also be possible to model in the same framework interaction experiments between several drugs. Finally, numerical solving of the models’ differential equations is fast and opens a vast range of new options for multistate modelling in general.
Reference: PAGE () Abstr 9289 [www.page-meeting.org/?abstract=9289]
Poster: Drug/Disease Modelling - Safety