Simulation-based Parameter Optimisation and Uncertainty Analysis Methods for Reaction-Diffusion-Advection Equations
Final Report Abstract
Reaction-diffusion-advection equations are used in many fields of engineering and natural sciences to model spatio-temporal processes, such as processes in bioreactors and fluid transport in the human body. As the parameters of these mathematical models are often unknown, they have to be determined from the available experimental data. In this project, we developed a novel simulation-based optimisation approach for reaction-diffusion-advection equations, which exploits the structure of the parameter optimisation problem. The approach used methods from control engineering and optimisation theory to formulate a coupled system of ordinary differential equations and partial differential equations. We were able to show that this system possesses the optimal points of the optimisation problem as stable steady states. This in turn allowed us to use adaptive numerical methods for solving optimisation problems for reaction-diffusion-advection equations. The optimisation approaches developed in the project was employed to determine the optimal parameter values and to perform uncertainty analysis using profile likelihoods. Instead of computing profile likelihoods only using repeated optimisation, we modified the coupled systems. The modified system evolved along the individual profiles and allowed profile calculation in reduced time. The method was used to improve our understanding of several important biological processes: (1) We developed a quantitative model for the guidance of dendritic cell to lymphoid vessels. This model described gradient formation in complex geometries. This enabled a more detailed study of an early step in the adaptive immune response, which is essential for fighting infectious diseases. (2) We developed a dynamical model for the integration of single cell sequencing data collected at different time points. These data are becoming available in a broad spectrum of applications, but the assessment of temporal dynamics has been difficult. The pseudodynamics framework allowed for the rigorous integration across time points, thereby enabling the prediction of the temporal evaluation. (3) We used a core finding of our study to develop more quantitative models of cancer signalling. Specifically, we used a hierarchical approach to parameterize a comprehensive model of cancer signalling using different large-scale data sets. The resulting model was the basis for drug response predictions and projects assessing the importance of genetic alterations. To render the approaches accessible to research groups and companies, we implemented them in open-source software tools. In combination with the scientific publications, this should provide the basis for a broad spectrum of future developments.
Publications
- Continuous analogue to iterative optimization for PDE-constrained inverse problems Inverse Problems in Science and Engineering, May 2018
R. Boiger, A. Fiedler, J. Hasenauer, and B. Kaltenbacher
(See online at https://doi.org/10.1080/17415977.2018.1494167) - Inferring population dynamics from single-cell RNA-sequencing time series data. Nature Biotechnology, 37, 461–468, April 2019
D. S. Fischer, A. K. Fiedler, E. M. Kernfeld, R. M. J. Genga, A. Bastidas-Ponce, M. Bakhti, H. Lickert, J. Hasenauer, R. Maehr, and F. J. Theis
(See online at https://doi.org/10.1038/s41587-019-0088-0) - Efficient parameterization of large-scale dynamic models based on relative measurements. Bioinformatics, 36(2):594–602, January 2020
L. Schmiester, Y. Schälte, F. Fröhlich, J. Hasenauer, and D. Weindl
(See online at https://doi.org/10.1093/bioinformatics/btz581) - AMICI: high-performance sensitivity analysis for large ordinary differential equation models. Bioinformatics, 37(20):3676–3677, October 2021
F. Fröhlich, D. Weindl, Y. Schälte, D. Pathirana, L. Paszkowski, G. T. Lines, P. Stapor, J. Hasenauer
(See online at https://doi.org/10.1093/bioinformatics/btab227)
