Section: New Results

Proliferation dynamics and its control

The part of this activity that is related to cell proliferation in health and cancer is organised according to 3 axes: (i) proliferation dynamics in physiologically structured cell populations, (ii) its physiological and pharmacological control at the level of cell populations and in a whole organism, and (iii) pharmacokinetic-phamacodynamic representation of drug control with application to optimisation of cancer chemotherapy.

A new activity has recently emerged in this field, related to neurodegenerative disorders: modelling amyloid protein (Alzheimer) and prion proliferation dynamics.

Cell division dynamics in structured cell populations

Participants : Mostafa Adimy [ Anubis project-team ] , Annabelle Ballesta, Houda Benjelloun [ INSA Rouen ] , Catherine Bonnet [ DISCO project-team INRIA Saclay IdF ] , Jean Clairambault, Fabien Crauste [ CNRS Lyon, UMR5208 Institut Camille Jourdan ] , Marie Doumic-Jauffret, Vladimir Flores [ CONICET (OCSID) Institut Beppo Levi, Rosario -Argentina- ] , Stéphane Gaubert [ MaxPlus project-team ] , Germain Gillet [ IBCP, Université Cl. Bernard Lyon 1 ] , Erwan Hingant [ Université Rennes 1 ] , Peter Kim [ University of Utah, Salt Lake City ] , Thomas Lepoutre, Jean-Pierre Marie [ INSERM Paris (Eq.18 de l'UMR 872) Hôtel-Dieu, Paris ] , Hitay Özbay [ Bilkent University, Ankara, Turkey ] , Benoît Perthame, Melina Rapacioli [ CONICET ] , Edmundo Rofman [ CONICET ] , Rafael Verdes [ CONICET, Université Favaloro, Buenos Aires, Argentine ] , Vitaly Volpert [ CNRS Lyon, UMR5208 Institut Camille Jordan ] .

1. Integrated model of the cell division cycle. Starting from models designed in the Bang project-team from 2003 on, based on physiologically structured PDEs, a convergence model has been produced, coupling a linear cell division cycle model of the McKendrick type with a nonlinear proliferation/quiescence model. Exchanges between phases G0 and G1 are represented by nonlinear functions, allowing a rich behaviour (equilibria, exponential decay, exponential or polynomial increase) for the cell populations that can be either healthy or tumoral. This has been the object of E. Hingant's M2 internship in Spring 2009, and will be continued. This activity aims at representing simultaneously the proliferation dynamics of healthy and tumoral tissues that will subsequently be physiologically or pharmacologically controlled.

2. Modelling haematopoiesis with applications to CML and AML. A PDE model of haematopoiesis, physiologically structured in age and maturity, has been developed (M. Doumic-Jauffret, P. Kim) with applications to Chronic Myelogenous Leukaemia (CML) [19] . The stability of another model, designed by M. Adimy and F. Crauste, structured by a discrete differentiation variable and multiple delays, with applications to Acute Myeloblastic Leukaemia (AML, clinical aviser: J.-P. Marie) has been studied with possible therapeutic implications (C. Bonnet, J. Clairambault, H. Özbay) in a conference paper that has been presented at the CDC conference in December 2008 in Cancun [53] , and is presently the object of H. Benjelloun's internship and H. Özbay's visiting professor work. New developments about this stability analysis will be presented in 2010 in an IFAC conference (accepted paper, invited session in “Time Delay Systems 2010” in Prague).

3. Molecular model of apoptosis. With G. Gillet (prof. at IBCP/Lyon), we are currently designing a mathematical ODE model for the mitochondrial pathway of apoptosis, focused on the early phase of apoptosis (before the cytochrome c release). We aim to justify our modeling choices, analyse (theoretically and numerically) the behaviour of our system and compare numerically its results with experimental results obtained by G. Gillet, to answer biological issues such that: on which protein is it more efficient to act in order to reduce/induce apoptosis ? Which therapeutic strategy can result from this ?

4. Developmental model of the Optic Tectum in the chick embryo. This work aims at validating a transport and diffusion system of PDEs as a model to describe the spatially organized operation of the proliferative neuroepithelial cells activity during the optic tectum corticogenesis. It is led in collaboration with 2 Argentinian teams, one at the Mathematics Institute Beppo Levi (Rosario), and another at the Favoloro University (Buenos Aires) and gathers theoreticians and experimentalists to produce a physiologically based model of neuroembryogenesis based on a transport equation with spatial diffusion. It has been presented by V. Flores and E. Rofman at the 24th IFIP TC7 conference held in July 2009 in Buenos Aires, and a common article (Verdes-Flores-Rapacioli-Rofman-Perthame-Clairambault) on this subject is in preparation.

Physiological and pharmacological control of cell proliferation

Participants : Annabelle Ballesta, Jean Clairambault, Sandrine Dulong [ INSERM Villejuif (U 776) ] , Stéphane Gaubert [ MaxPlus project-team ] , Erwan Hingant, Thomas Lepoutre, Francis Lévi [ INSERM Villejuif (U 776) ] , Sylvain Soliman [ Contraintes project-team ] .

This activity develops along 2 parallel axes: a theoretical (mathematical) one (J. Clairambault, S. Gaubert, T. Lepoutre) and a more experimental, or experimentally based, one. In the former are studied in a theoretical way the structural properties of periodic (circadian or pharmacological) controls on proliferation dynamics, measured by a Malthus-like exponent (first eigenvalue of the underlying differential operator). In the latter are examined, with experimental identification of parameters, the influence (pharmacodynamics) exerted by anticancer drugs on the cell division cycle; new developments include the representation of biological mechanisms of drug resistance in cancer cells.

1. Periodic (circadian) control of cell proliferation in a theoretical model of the McKendrick type. The influence exerted by a periodic function alternatively blocking and enhancing transition rates between phases of the cell division cycle, and similarly acting on apoptosis rates, that had been initiated by studies published in 2006, 2007 and 2008, has been continued, resulting in an article published in Mathematical Modelling of Natural Phenomena [11] , to which must be added another one submitted [12] , and T. Lepoutre's PHD thesis [2] .

2. Intracellular pharmacokinetic-pharmacodynamic (PK-PD) models for anticancer drugs This activity takes place within the framework of the European projects BioSim and Tempo (both ended at the end of 2009). New developments include a PK-PD model for 5FU+Leucovorin delivery (described in [26] ), and an intracellular PK-PD model for Irinotecan, with identification of parameters on Caco2 cell cultures (A. Ballesta's PhD thesis work under J. Clairambault's supervision, working on experimental aspects with S. Dulong in F. Lévi's laboratory), both models having been presented in different conferences or workshops, including for A. Ballesta's a presentation at the SFBT annual meeting in Saint-Flour, for which she was granted the Delattre prize of the best PhD student presentation.

3. Whole body physiologically based model of anticancer drug pharmacokinetics. The application of molecular PK-PD principles to whole body modelling is necessary to make possible future optimisation of drug delivery schedules with respect to unwanted toxicity side effects in different physiological compartments, simultaneously with therapeutic effects on the tumour compartment. In an INRIA internship internal report, H. Gayrard had studied in 2008 such a whole-body model, structured in compartmental ODEs, for Irinotecan, from infusion in the general circulation until its delivery in the intracellular medium. A. Ballesta has taken over this subject as part of her PhD thesis.

Optimisation of cancer chemotherapy

Participants : Annabelle Ballesta, Jean Clairambault, Thomas Lepoutre, Francis Lévi [ INSERM Villejuif (U 776) ] .

Optimising cancer chemotherapy, especially chronotherapy, is the final aim of the activities mentioned above. This has been lately discussed in T. Lepoutre's PhD thesis [2] and in [12] . Until now had been taken into account as constraints in optimisation strategies only the unwanted toxic side effects of anticancer drugs on healthy cells. More recently, another issue of anticancer treatment has been considered, namely the different mechanisms of resistance to drugs in cancer cells. This has led to include the effect of ABC transporters (active efflux pumps, as is the P-glycoprotein) in the intracellular PK-PD models mentioned above [26] in A. Ballesta's PhD joint work with F. Lévi, to a review article on cell proliferation modelling for therapeutic optimisation in cancer [10] and a common European research position paper [33] .

Prion proliferation dynamics and protein polymerization

Participants : Vincent Calvez [ ENS Lyon ] , Marie Doumic-Jauffret, Pierre Gabriel, Thierry Goudon [ SIMPAF project-team, INRIA Lille Nord-Europe ] , Thomas Lepoutre, Benoît Perthame.

In collaboration with biologists from INRA/BCBP, Jouy (H. Rezaei) and CEA/DSV (N. Lenuzza and F. Mouthon)

Since spring 2007, a collaboration with CEA/DSV has been initiated by B. Perthame and V. Calvez. It has first lead to two articles in 2008 [9] , [8] and several ones in 2009 (two already accepted [17] , [18] , one submitted [21] and two in progress). Prion pathology (Bovine Spongiform Encephalopathy, commonly known as Mad-Cow Disease, or Creutzfeldt-Jakob Disease for instance) and Alzheimer Disease are both characterized by accumulation of large protein polymers, so-called fibrils , in the brain. The objective of our work is a mathematical modelling, numerical analysis, and comparison between simulations and experiments for prion and Alzheimer amyloid aggregation phenomena. It is a very promising field and can provide a deeper understanding of biological phenomena. It also adresses new and profound mathematical issues in the field of fragmentation equations (which are also found to describe the cell division cycle, see [17] , [18] ) and its inverse problem (see [20] ).

Inverse problem in structured populations and fragmentation equations

Participants : Marie Doumic-Jauffret, Pedro Maia [ IMPA, Brazil ] , Benoît Perthame, Jorge P. Zubelli [ IMPA, Brazil ] , Frédérique Charles.

We have continued to investigate the identification of coefficients in the models used in structured populations modeling. With J. Zubelli (IMPA, Rio de Janeiro), we have shown that this is theoretically possible by regularization/denoising methods and have applied them to experimental data with Pedro Maia (IMPA, Rio de Janeiro). The comparison of various algorithms and their convergence analysis has been investigated, and has lead to a published article [20] and a submitted one [34] .

We intend to apply and extend these methods for the study of prion proliferation equations, in order to recover parameter functions of the equations from aggregates size distribution. It will be studied by Frédérique Charles during her 18 months post-doctoral position, beginning in November 2009.

Moreover, 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 of Nice), we explore a statistical viewpoint on the cell-division problem.