Section: New Results

Proliferation dynamics and its control

Cell division dynamics in structured cell populations

Participants : José Luis Avila Alonso [DISCO project-team, INRIA Saclay IdF] , Annabelle Ballesta, Houda Benjelloun [INSA Rouen] , 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] , Faten Merhi [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. A focus has been set on transitions between phases of the cell division cycle. The underlying biological question is: “Is desynchronisation between cells in proliferating cell populations a hallmark of cancer?”. It has been considered by relating in a natural way transition kernels with the probability density functions of transition times in the cell population. It has been shown -which was expected, but never proved to our knowledge so far- that the more desynchronised cells are with respect to cell cycle phase transitions, the higher is the growth exponent of the cell population [48] , otherwise said: desynchronised cell populations grow faster. This has been proven when transition kernels are time-independent, i.e., when no external controlled is exerted on transitions. The same question is currently experimentally investigated by our biologist partners in the European network ERASysBio+ C5Sys, coordinated by F. Lévi (Villejuif) and D. Rand (Warwick). Simulations using experimentally identified transition kernels in proliferating cell cultures controlled by theoretical time-dependent (circadian) control functions have verified the relevance of this mathematical result for theoretical cancer treatment optimisation (cf. infra “Periodic (circadian) control of cell proliferation in a theoretical model of the McKendrick type”).

2. Modelling haematopoiesis with applications to AML. The stability of a delay system based on a PDE 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 advisers: J.-P. Marie and P. Hirsch; technical adviser: RP Tang) is studied with possible therapeutic implications [36] . This model is currently experimentally investigated, with the aim to identify its parameters in leukaemic cells, in the DIGITEO project ALMA (cf. infra “DIGITEO and Cancéropôle IdF” in “Regional initiatives”), coordinated by C. Bonnet (DISCO team, INRIA Saclay IdF) and in the recently launched DIGITEO project ALMA2 (coordinated by J. Clairambault), that takes over the combined experimental-modelling activity in ALMA. Two INRIA postdocs, F. Merhi (in ALMA, 2010-2011) and A. Ballesta (in ALMA2, 2011-2013) have been devoted to this task. From a theoretical point of view, the Adimy-Crauste model has been modified so as a) to include quick self-renewal of cells in each stage of maturation and b) to represent each phase of the proliferating compartment (i.e., $G_1$, $S$, $G_2$ and $M$) separately. For the time being, only the $M$ phase is supposed to have a fixed time duration as it is generally admitted that the short time (typically half an hour if the total proliferating phase duration is normalised to 24 hours) necessary to perform mitosis is hardly submitted to any variation.

   In a complementary manner, a new model for cell differentiation was introduced and analysed in [17] , in collaboration with A. Marciniak and J.P. Zubelli. It assumed that differentiation of progenitor cells is a continuous process. From the mathematical point of view, it is based on partial differential equations of transport type. Specifically, it consists of a structured population equation with a nonlinear feedback loop. This models the signaling process due to cytokines, which regulate the differentiation and proliferation process. We compared the continuous model to its discrete counterpart, a multicompartmental model of a discrete collection of cell subpopulations recently proposed by Marciniak-Czochra et al. [Stem Cells Dev., 18 (2009), pp. 377–386] to investigate the dynamics of the hematopoietic system. We obtained uniform bounds for the solutions, characterized steady state solutions, and analyzed their linearized stability. We showed how persistence or extinction might occur according to values of parameters that characterize the stem cells' self-renewal. We also performed numerical simulations and discuss the qualitative behavior of the continuous model vis-à-vis the discrete one.

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 apoptosis.

   With G. Gilllet (prof. at IBCP/Lyon), we have designed a mathematical ODE model for the mitochondrial pathway of apoptosis, focused on the early phase of apoptosis (before the cytochrome C release). We have validated it with experimental data carried out in G. Gillet's lab and applied it to propose new therapeutic strategies against cancer. This work has led to a nearly submitted article [47] .

5. Molecular model of the activity of the p53 protein. Following her first year of PhD in Rome with R. Natalini, working on cytoplasmic transport along microtubules presented in [38] , L. Dimitrio has begun her third PhD year, going on studying at INRIA nucleocytoplasmic transport with applications to p53 activity. Her PhD thesis work is supervised in co-tutela between Sapienza University in Rome (R. Natalini) and INRIA (J. Clairambault). The protein p53 plays a capital part as “guardian of the genome”, arresting the cell cycle and launching cell apoptosis or DNA repair in case of DNA damage. Results expected from this newly developed theme will provide a rational link between molecular pharmacokinetics-pharmacodynamics (cf. infra) of anticancer drugs and modelling of the cell division cycle in proliferating cell populations. L. Dimitrio has presented her ongoing work in different meetings in France and in Italy, and a paper in preparation will be submitted in 2012.

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. The impact of a periodic control exerted on cell cycle phase transitions has continued to be studied [16] with the collaboration of S. Gaubert (MaxPlus INRIA project-team, Saclay IdF) and T. Lepoutre (Dracula INRIA project-team, Lyon) and is currently investigated experimentally in the new C5Sys European network (cf. supra “Transition kernels in a McKendrick model of the cell division cycle” and “). Thanks to the work of Frédérique Billy (Postdoc in Bang) and Olivier Fercoq (PhD student in MaxPlus), together with permanent members of Bang, Dracula and MaxPlus teams, it has led to three publications [37] , [39] , [48] .

2. Intracellular pharmacokinetic-pharmacodynamic (PK-PD) models for anticancer drugs. This theme is actively worked on in collaboration, mainly with the teams of F. Lévi and J.-P. Marie (cf. supra “ Transition kernels in a McKendrick model of the cell division cycle” and “Modelling haematopoiesis with applications to AML”). After a PK-PD model for 5-FU with folinic acid [86] , it has led for the anticancer drug Irinotecan, the main object of A. Ballesta's PhD thesis [1] , to an article published in PLoS Computational Biology [8] , reporting a combined modelling and experimental approach to the effects of a combination of mathematical modelling and experimentation in cell cultures, and to another one [7] , focusing on drug delivery optimisation.

3. Whole body physiologically based model of anticancer drug pharmacokinetics. This theme has also been studied in A. Ballesta's PhD thesis. The use of identification, in genetically different laboratory mouse strains, of parameters characterising an ODE model of the action of Irinotecan (cf. supra “Intracellular pharmacokinetic-pharmacodynamic (PK-PD) models for anticancer drugs”) in cell cultures, transposed at the whole-body level, has been designed as a proof of concept for individual adaptation of drug delivery in the context of (future) personalised medicine, a perspective sketched in [15] and among other collaborative contexts linking mathematics and medicine in [14] , [3] .

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, especially chronotherapy, is the final aim of the activities mentioned above. This has been lately discussed in [16] and also in works involving the C5Sys network [37] , [48] , and in the more general review [39] . 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 [86] , in A. Ballesta's PhD joint work with F. Lévi's team [1] , [8] , [7] , and to a perspective paper [15] .

In project 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”).

Another way to represent and overcome drug resistance in cancer from a cell Darwinian point of view using concepts of adaptive dynamics in proliferating cell populations is also currently being investigated along the line of other recent works [28] and is currently developed within the multidisciplinary GDR DarEvCan coordinated by M. Hochberg, Montpellier (cf. infra “GDR DarEvCan” in “National initiatives”) and in a proposed ANR project also coordinated by M. Hochberg.

An open question that should have therapeutic implications consists of the interrogation: Is the emergence of drug resistance in cell populations a genetic (resulting from mutations at mitosis) or an epigenetic phenomenon (resulting from amplification of physiological mechanisms, such as ABC transport, which has nothing to do with genetic mutations)? And is it a reversible or irreversible phenomenon? These questions will be studied both theoretically and experimentally within the DarEvCan consortium and could result in new developments in the so-called Darwinian medicine.

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

Participants : Annabelle Ballesta, Vincent Calvez [ENS Lyon] , Frédérique Charles, 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] .

With H. Rezaei, a new and very complete PDE model for protein polymerisation has been designed. Following F. Charles' 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 fibrillization. This part is nearly submitted.

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 [11] investigated the dependency of the principal eigenvector and eigenvalue upon its parameters. We exhibited possible non-monotonic 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] , Patricia Reynaud [CNRS, Nice Univ.] , 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 [18] we have 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. This work is accepted in SIAM J. Num. Anal..

With L. Matar Tine, in [53] we have generalized the inverse techniques proposed in [88] , [67] and [66] , 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. This work is submitted.