Section: Research Program
Research axis 3: Model and parameter identification combining stochastic and deterministic approaches in nonlocal and multi-scale models
Marie Doumic, Dirk Drasdo, Aurora Armiento, Thibault Bourgeron, Rebecca Chisholm, Tommaso Lorenzi
Mamba developed and addressed model and parameter identification methods and strategies in a number of mathematical and computational model applications including growth and fragmentation processes emerging in bacterial growth and protein misfolding, in liver regeneration , TRAIL treatment of HeLa cells , growth of multicellular spheroids , blood detoxification after drug-induced liver damage , .
This naturally led to increasingly combine methods from various fields: image analysis, statistics, probability, numerical analysis, PDEs, ODEs, agent-based modelling methods, involving inverse methods as well as direct model and model parameter identification in biological and biomedical applications. Model types comprise agent-based simulations for which Mamba is among the leading international groups, and Pharmocokinetic (PK) simulations that have recently combined in integrated models (PhD theses Géraldine Cellière, Noémie Boissier). The challenges related with the methodological variablity has led to very fruitful collaborations with internationally renowned specialists of these fields, e.g. for bacterial growth and protein misfolding with Marc Hoffmann (Paris Dauphine) and Patricia Reynaud-Bouret (University of Nice) in statistics, with Philippe Robert (Inria RAP) in probability, with Tom Banks (Raleigh, USA) and Philippe Moireau (Inria M3DISIM) in inverse problems and data assimilation, and with numerous experimentalists.
Direct parameter identification is a great challenge particularly in living systems in which part of parameters at a certain level are under control of processes at smaller scales.
Estimation methods for growing and dividing populations
In this domain, all originated in two papers in collaboration with J.P. Zubelli in 2007 , , whose central idea was to used the asymptotic steady distribution of the individuals to estimate the division rate. A series of papers improved and extended these first results while keeping the deterministic viewpoint, lastly . The last developments now tackle the still more involved problem of estimating not only the division rate but also the fragmentation kernel (i.e., how the sizes of the offspring are related to the size of the dividing individual) . In parallel, in a long-run collaboration with statisticians, we studied the Piecewise Deterministic Markov Process (PDMP) underlying the equation, and estimated the division rate directly on sample observations of the process, thus making a bridge between the PDE and the PDMP approach in , a work which inspired also very recently other groups in statistics and probability ,  and was the basis for Adélaïde Olivier's Ph.D thesis ,  and of some of her more recent works  (see also axis 5).
Model identification for growing multicellular spheroids
For multicellular spheroids growing under different conditions, first an agent-based model on an unstructured lattice for one condition has been developed and then stepwise extended for each additional condition which could not be captured by the present model state   (axis 5). The multicellular dynamics has been mimicked by a master equation, intracellular processes by ODEs, and extracellular molecular transport processes by partial differential equations. The model development was based almost completely on bright field image sequences, whereby image segmentation parameter identification was performed by investigation of sensitivity and specificity of the segmentation with a biologist expert serving as gold standard. The Akaike and Bayesian Information Criteria were used to evaluate whether parameters introduced due to the model extension led to a significant increase of the information. It turned out that the final model could predict the outcome of growth conditions not considered in model development.
A similar stepwise strategy has been recently performed to identify the pressure and strain constraints of growing multicellular cell populations subject to mechanical stress. Here, a novel deformable cell model has been used that permits to display cell shape changes explicitly .
Data assimilation and stochastic modelling for protein aggregation
Estimating reaction rates and size distributions of protein polymers is an important step for understanding the mechanisms of protein misfolding and aggregation (see also axis 5). In , we settled a framework problem when the experimental measurements consist in the time-dynamics of a moment of the population.
To model the intrinsic variability among experimental curves in aggregation kinetics - an important and poorly understood phenomenon - Sarah Eugène's Ph.D, co-supervised by P. Robert , was devoted to the stochastic modelling and analysis of protein aggregation, compared both with the deterministic approach traditionnally developed in Mamba  and with experiments.
Model identification in liver regeneration
Based on successful model predictions for models addressing different aspects of liver regeneration, we extracted a general workflow on how modelling can inform liver disease pathogenesis . Liver has a complex micro-architecture ensuring its function (axis 5). Hence many clinical questions require quantitative characterisation of micro-architecture in normal liver and during degeneration or regeneration processes which has been performed using the software TiQuant (  see software) or other tools we generated. Disease specific and personal information can be used to build a list of hypotheses on the question of interest, which then can be systematically implemented in mathematical models and in simulation runs tested against the data. As confocal micrographs only display part of lobules, statistically representative lobules were constructed to permit definition of boundary conditions for flow and transport.
In order to compare data and model results quantitatively, quantitative measures characterising the processes under study have to be defined, and measured in both experiment and model. Models in a context where micro-architecture is important were based on agent-based models representing each hepatocyte as well as blood vessels explicitly. They were parameterised by measurable parameters as for those physiologically relevant ranges can be identified, and systematic simulated parameter sensitivity analyses can be performed. Movement of each cell was then mimicked by an equation of motion, describing the change of position as a function of all forces on that cell including active migration. If the best model disagreed with the data, the underlying hypotheses were considered incomplete or wrong and were modified or complemented. Models quantitatively reproducing data were either used to predict so far unknown situations, or the key mechanisms were directly challenged by our experimental partners. Along this line, two unrecognised mechanisms could be identified (axis 5).
Statistical methods decide on subsequently validated mechanism of ammonia detoxification
To identify the mechanisms involved in ammonia detoxification , 8 candidate models representing the combination of three possible mechanisms were developed (axis 5). First, the ability of each model to capture the experimental data was assessed by statistically testing the null hypothesis that the data have been generated by the model, leading to exclusion of one of the 8 models. The 7 remaining models were compared among each other by the likelihood ratio. The by far best models were those containing a particular ammonia sink mechanism, later validated experimentally (axis 5). For each of the statistical tests, the corresponding test statistics has been calculated empirically and turned out to be not chi2-distributed in opposition to the usual assumption stressing the importance of calculating the empirical distribution, especially when some parameters are unidentifiable.
Philippe Robert, Inria Rap, for the stochastic process modelling 
Marc Hoffmann, Université Paris-Dauphine, for the statistical approach to growth and division processes , M. Escobedo, Bilbao and M. Tournus, Marseille, for the deterministic approach.