Section: New Results

Proliferation dynamics and its control

Cell division dynamics in structured cell populations

Participants : José Luís Avila Alonso [DISCO project-team, Inria Saclay IdF] , Annabelle Ballesta, Frédérique Billy, Frédéric Bonnans [Commands project-team, Inria Saclay IdF] , Catherine Bonnet [DISCO project-team, Inria Saclay IdF] , Jean Clairambault, Luna Dimitrio, Marie Doumic-Jauffret, Xavier Dupuis [Commands project-team] , Olivier Fercoq [MaxPlus project-team, Inria Saclay IdF] , Stéphane Gaubert [MaxPlus project-team, Inria Saclay IdF] , Germain Gillet [IBCP, Université Cl. Bernard Lyon 1] , Philippe Gonzalo [IBCP, Université Cl. Bernard Lyon 1] , Pierre Hirsch [INSERM Paris (Team18 of UMR 872) Cordeliers Research Centre and St. Antoine Hospital, Paris] , Thomas Lepoutre [now in DRACULA project-team, Inria Rhône-Alpes, Lyon] , Jonathan Lopez [IBCP, Université Cl. Bernard Lyon 1] , Pierre Magal [University Bordeaux II] , Anna Marciniak-Czochra [Institute of Applied Mathematics, Universität Heidelberg] , Jean-Pierre Marie [INSERM Paris (Team18 of UMR 872) Cordeliers Research Centre and St. Antoine Hospital, Paris] , Roberto Natalini [IAC-CNR, Università Sapienza, Rome] , Silviu Niculescu [DISCO project-team, Inria Saclay IdF] , Hitay Özbay [Bilkent University, Ankara, Turkey] , Benoît Perthame, Ruoping Tang [INSERM Paris (Team18 of UMR 872) Cordeliers Research Centre and St. Antoine Hospital, Paris] , Vitaly Volpert [CNRS Lyon, UMR5208, Camille Jordan Institute, Lyon] , Jorge Zubelli [IMPA, Rio de Janeiro] .

1. Transition kernels in a McKendrick model of the cell division cycle. This theme has continued to be developed with identification of model parameters by FUCCI imaging in collaboration with G. van der Horst's team in Amsterdam and with F. Delaunay's team in Nice, within the C5Sys European network, coordinated by F. Lévi (Villejuif) [10] , [11] , [12] , [39] , [42] , [43] . Main young researchers on this theme, F. Billy has concluded her 2-year Inria postdoc at Bang, leaving for an industrial company in November 2012, and O. Fercoq (team MaxPlus, Saclay) has defended his PhD thesis at École Polytechnique in September 2012, only to leave for a postdoc position dedicated to optimisation theory in Edinburgh.

2. Modelling haematopoiesis with applications to AML. This theme has been active through a collaboration with Inria teams Commands (F. Bonnans, X. Dupuis) and Disco (JL Avila, C. Bonnet), and J.-P. Marie's team at St Antoine Hospital leukaemic tumour bank, where A. Ballesta, Cancéropole IdF-Inria postdoc has been detached (ending in March 2013) to identify parameters of a model of acute myeloblastic leukaemia (AML) in patient fresh cell cultures with and without anticancer drugs. This work has led to several presentations, and publications are in preparation.

3. Hybrid models Systems combining PDEs and discrete representations in hybrid models, with applications to cancer growth and therapy, in particular for AML, are the object of study of the ANR program Bimod, coordinated by V. Volpert (Lyon), associating CNRS (V. Volpert, Lyon), Bordeaux II University (P. Magal) and the Bang project-team.

4. Molecular model of the activity of the p53 protein. This work, the object of Luna Dimitrio's PhD thesis [1] , co-supervised by J. Clairambault and R. Natalini (Rome), has led to her PhD defence in September 2012 at UPMC, and to a first publication [18] , that should be followed by others. After L. Dimitrio's leave for the pharmaceutic industry, a new PhD student, Ján Eliš, has taken over this theme in September 2012 in a new PhD thesis at UPMC, under the supervision of J. Clairambault and B. Perthame

Physiological and pharmacological control of cell proliferation

Participants : Annabelle Ballesta, Frédérique Billy, Jean Clairambault, Sandrine Dulong [INSERM Villejuif (U 776)] , Olivier Fercoq [MaxPlus project-team] , Stéphane Gaubert [MaxPlus project-team] , Thomas Lepoutre [Dracula project-team] , Francis Lévi [INSERM Villejuif (U 776)] .

1. Periodic (circadian) control of cell proliferation in a theoretical model of the McKendrick type. This theme (cf. supra “transition kernels...”) has been continued [39] , [11] , [12] , [10] , [42] , [43] . Whereas transition kernels between cell cycle phases without control have been experimentally identified in cell cultures by FUCCI imaging [12] , their circadian control remains elusive and has been modelled on the basis of gating by plain cosines representing the influence exerted on these transition kernels by circadian clocks. To go further, it would be necessary to have access by cell imaging to the activity of the best physiological candidates to such gating, namely the cyclin-Cdk complexes, together with the activities of the clock-controlled proteins Wee1 and p21, which thus far have remained unavailable to us through biological experimentation with imaging.

2. Intracellular pharmacokinetic-pharmacodynamic (PK-PD) models for anticancer drugs. This theme has continued to be developed with new publications for the drugs irinotecan [40] , [44] , 5-fluorouracil and oxaliplatin [43] .

Optimisation of cancer chemotherapy

Participants : Annabelle Ballesta, Frédérique Billy, Frédéric Bonnans [Commands project-team] , Jean Clairambault, Sandrine Dulong [INSERM Villejuif (U 776)] , Xavier Dupuis [Commands project-team] , Olivier Fercoq [MaxPlus project-team] , Stéphane Gaubert [MaxPlus project-team] , Thomas Lepoutre [Dracula project-team] , Alexander Lorz, Francis Lévi [INSERM U 776, Villejuif] , Michael Hochberg [ISEM, CNRS, Montpellier] , Benoît Perthame.

Optimising cancer chemotherapy, in particular chronotherapy, is the final aim of the activities mentioned above. This theoretical activity has been continued, using the McKendrick paradigm in works involving the C5Sys network [12] , [42] , [43] , with numerical optimisation algorithms for the toxicity constraint, and also in more general settings taking into account another major issue of anticancer treatment, namely resistance to drugs in cancer cells. To this latter aim, we have developed another type of models based on integro-differential equations, which are inspired from those used in ecology for Darwinian evolution. These are aimed at studying another major issue in cancer therapy: appearance of resistances to treatment in tumour cell populations. Indeed, these cell populations, because of their heterogeneity and genomic instability, present an ability to adapt and evolve (in the Darwinian sense) that is much higher than in healthy cell populations [10] , [27] , [39] . The time scales under investigation, much shorter than in ecology, are still much longer than in microbiology, and are those of clinical treatments.

From a molecular point of view, studying drug resistance leads to the study of ABC transporters, which is one of the tracks followed by A. Ballesta, following her PhD thesis, in collaboration with F. Lévi's INSERM team in Villejuif [40] , [44] .

Underway is also the use of methods of optimal control developed by the Commands project-team (F. Bonnans, X. Dupuis) to optimise therapies in the treatment of Acute Myeloblastic Leukaemia (AML, cf. supra “Modelling haematopoiesis with applications to AML”).

Protein polymerisation and application to amyloid diseases (ANR grant TOPPAZ)

Participants : Annabelle Ballesta, Vincent Calvez [ENS Lyon] , Marie Doumic-Jauffret, Pierre Gabriel, Hadjer Wafaâ Haffaf, Benoît Perthame, Stéphanie Prigent [BPCP, INRA Jouy-en-Josas] , Human Rezaei [BPCP, INRA Jouy-en-Josas] , Léon Matar Tine [SIMPAF project-team, Inria Lille Nord-Europe] .

Published in PLoS One in collaboration with the biologists' team of H. Rezaei [29] , a new and very complete PDE model for protein polymerisation has been designed. Following F. Charles's work, A. Ballesta has applied this model to Huntington's disease (PolyQ expansion) and compared it with its ODE counterpart, leading to a better understanding of the leading mechanisms responsible for PolyQ fibrillisation. New applications of this framework model are in progress with H.W. Haffaf and S. Prigent.

The eigenvalue problem playing a major role in the representation of Prion proliferation dynamics and, in a more general way, of many fragmentation-coalescence phenomena, the article [15] published in J. de Math. Pur. Appl. investigated the dependency of the principal eigenvector and eigenvalue upon its parameters. We exhibited possible nonmonotonic dependency on the parameters, conversely to what would have been conjectured on the basis of some simple cases.

Inverse problem in growth-fragmentation equations

Participants : Marie Doumic-Jauffret, Marc Hoffmann [ENSAE] , Nathalie Krell [Univ. Rennes I] , Patricia Reynaud [CNRS, Nice Univ.] , Lydia Robert [UPMC] , Vincent Rivoirard [Paris IX Univ.] , Léon Matar Tine [SIMPAF project-team, Inria Lille Nord-Europe] .

In collaboration with statisticians (M. Hoffman, Professor at Université de Marne-la-Vallée, V. Rivoirard, MC at Université d'Orsay, and P. Reynaud, CR CNRS at Université de Nice), in the article [19] published in SIAM Num. Anal., we explored a statistical viewpoint on the cell division problem. In contrast to a deterministic inverse problem approach, we take the perspective of statistical inference. By estimating statistically each term of the eigenvalue problem and by suitably inverting a certain linear operator, we are able to construct an estimator of the division rate that achieves the same optimal error bound as in related deterministic inverse problems. Our procedure relies on kernel methods with automatic bandwidth selection. It is inspired by model selection and recent results of Goldenschluger and Lepski.

An extension of this work, which consists of the statistical estimation of a branching process modelling the same growth and fragmentation dynamics, has been submitted in [49] , in collaboration with N. Krell, M. Hoffmann and L. Robert.

In [20] , published in J. Math. Biol. with L. Matar Tine, we generalised the inverse techniques proposed previously in [53] , [57] , in order to adapt them to general fragmentation kernels and growth speeds. The potential applications of this problem are numerous, ranging from polymerisation processes to the cell division cycle. An extension of this work is in progress with M. Escobedo and T. Bourgeron.