When mathematical modelling is applied to many different application areas, a common task is the estimation of states and parameters based on measurements. With this kind of inference making, uncertainties in the time when the measurements have been taken are often neglected, but especially in applications taken from the life sciences, this kind of errors can considerably influence the estimation results. As an example in the context of personalized medicine, the model-based assessment of the effectiveness of drugs is becoming to play an important role. Systems biology may help here by providing good pharmacokinetic and pharmacodynamic (PK/PD) models. Inference on these systems based on data gained from clinical studies with several patient groups becomes a major challenge. Particle filters are a promising approach to tackle these difficulties but are by itself not ready to handle uncertainties in measurement times.
In this article, we describe a variant of the standard particle filter (PF) algorithm which allows state and parameter estimation with the inclusion of measurement time uncertainties (MTU). The modified particle filter, which we call MTU-PF, also allows the application of an adaptive stepsize choice in the time-continuous case to avoid degeneracy problems. The modification is based on the model assumption of uncertain measurement times. While the assumption of randomness in the measurements themselves is common, the corresponding measurement times are generally taken as deterministic and exactly known. Especially in cases where the data are gained from measurements on blood or tissue samples, a relatively high uncertainty in the true measurement time seems to be a natural assumption. Our method is appropriate in cases where relatively few data are used from a relatively large number of groups or individuals, which introduce mixed effects in the model. This is a typical setting of clinical studies. We demonstrate the method on a small artificial example and apply it to a mixed effects model of plasma-leucine kinetics with data from a clinical study which included 34 patients.
MTU-PF with the standard PF and with an alternative Maximum Likelihood estimation method on the small artificial example clearly show that the MTU-PF obtains better estimations. Considering the application to the data from the clinical study, the MTU-PF shows a similar performance with respect to the quality of estimated parameters compared with the standard particle filter, but besides that, the MTU algorithm shows to be less prone to degeneration than the standard particle filter.
Thanks go to Michaela Höhne from the University of Kaiserslautern for her help in preparing the figures. Data was generated by Martin Adiels, Professor Marja-Riitta Taskinen, Helsinki University, and Professor Jan Borén, Göteborg University, and has been published elsewhere ([31,32]). We would like to thank the anonymous reviewers for their helpful suggestions and feedback. This work was partially supported by the Fraunhofer Internal Programs under Grant No. MAVO 819 620. The work was also supported by grants from the European Commission 7th Framework Programme (UNICELLSYS, grant No 201142), the Swedish Foundation for Strategic Research through the Gothenburg Mathematical Modelling Centre (GMMC), the Sahlgrenska Center for Cardiovascular and Metabolic Research (CMR), the Sigrid Juselius Foundation, and the Helsinki University Central Hospital Research Foundation, Helsinki, Finland.
Authors and Affiliations
- A. Krengel, Fraunhofer ITWM
- J. Hauth, Fraunhofer ITWM
- M.-R. Taskinen, Dept. of Medicine, University of Helsinki
- M. Adiels, Dept. of Mathematical Sciences and Dept. of Medicine, University of Gothenburg,
- M. Jirstrand, Dept. of Systems and Data Analysis, Fraunhofer-Chalmers Centre