Bayesian modelling of time series data (BayModTS)-a FAIR workflow to process sparse and highly variable data

Abstract

Motivation: Systems biology aims to better understand living systems through mathematical modelling of experimental and clinical data. A pervasive challenge in quantitative dynamical modelling is the integration of time series measurements, which often have high variability and low sampling resolution. Approaches are required to utilize such information while consistently handling uncertainties.

Results: We present BayModTS (Bayesian modelling of time series data), a new FAIR (findable, accessible, interoperable, and reusable) workflow for processing and analysing sparse and highly variable time series data. BayModTS consistently transfers uncertainties from data to model predictions, including process knowledge via parameterized models. Further, credible differences in the dynamics of different conditions can be identified by filtering noise. To demonstrate the power and versatility of BayModTS, we applied it to three hepatic datasets gathered from three different species and with different measurement techniques: (i) blood perfusion measurements by magnetic resonance imaging in rat livers after portal vein ligation, (ii) pharmacokinetic time series of different drugs in normal and steatotic mice, and (iii) CT-based volumetric assessment of human liver remnants after clinical liver resection.

Availability and implementation: The BayModTS codebase is available on GitHub at https://github.com/Systems-Theory-in-Systems-Biology/BayModTS. The repository contains a Python script for the executable BayModTS workflow and a widely applicable SBML (systems biology markup language) model for retarded transient functions. In addition, all examples from the paper are included in the repository. Data and code of the application examples are stored on DaRUS: https://doi.org/10.18419/darus-3876. The raw MRI ROI voxel data were uploaded to DaRUS: https://doi.org/10.18419/darus-3878. The steatosis metabolite data are published on FairdomHub: 10.15490/fairdomhub.1.study.1070.1.

Fig. 1.

The FAIR BayModTS workflow reveals credible regions in the data space and allows statistical testing whether different datasets stem from the same data generating process. BayModTS applies a statistical Bayesian framework and a simulation model $s(t,\theta )$ to transform sparse and highly variable time series data into less noisy time courses with uncertainty estimates for model states. (A) Time series data and (B) retarded transient functions as a universal simulation model $s(t,\theta )$ are used as input together with parameter priors. (C) The Posterior distribution $p(\theta |D)$ is inferred using Markov Chain Monte Carlo sampling. (D) Simulated parameter posterior sample trajectories (Ensembles) are used to infer summary statistics that quantify model state uncertainties. We use 95% CI tubes. (E) Dynamics under different conditions can be visually compared via CI tubes.

Fig. 2.

BayModTS analysis calculates perfusion courses in ligated and nonligated liver lobes from MRI data after PVL. (A) Illustration of ligated (hypoperfused) and nonligated (hyperperfused) lobes of a rat liver subjected to PVL. (B) Exemplary MRI images for 0, 1, 2, 3, and 5 POD. The outlined area (orange) refers to the ROI annotation for RML. Perfusion is colour-coded, with yellow to red values indicating high values. (C) Boxplots of individual voxel values in the selected ROIs of $n=5$ animals per time point (raw data). (D) State-of-the-art visualization of MRI ROI data. Boxplots are based on per-animal averaged ROI voxel data. (E) BayModTS ensemble perfusion predictions with median (solid lines) and 95% CI tubes for all lobes. (F, G) Marginal distributions of the parameters $\sigma$ and $p_0$ per lobe, restricted to the 95% highest density interval.

Fig. 3.

Effect of severity and pattern of periportal steatosis on pharmacokinetics. (A) Mice were fed a normal diet for 2 weeks and a MCD and HF diet for 2 or 4 weeks. The MCD+HF diet induces hepatic microsteatosis after 2 weeks and macrosteatosis after 4 weeks. (B) Boxplots of plasma drug elimination time courses. The control group (red) consisted of n = 4 animals, while the 2 weeks (orange) and 4 weeks (yellow) groups consisted of n = 6 animals each. (C) Median ensemble prediction (dark lines) and 95% CI tubes of the BayModTS analysis with RTFs.