Section: New Results

Cancer

Participants : Luís Lopes Neves de Almeida, José Luis Avila Alonso [DISCO Inria team] , Catherine Bonnet [DISCO Inria team] , Rebecca Chisholm, Jean Clairambault, François Delhommeau [Hæmatology department, St Antoine Hospital, Paris] , Luna Dimitrio [former PhD student and Mamba member] , Ján Eliaš, Alexandre Escargueil [Cancer biology and therapeutics lab, St Antoine Hospital, Paris] , Pierre Hirsch [Hæmatology department, St Antoine Hospital, Paris] , Michal Kowalczyk [Univ. Santiago de Chile] , Annette Larsen [Cancer biology and therapeutics lab, St Antoine Hospital, Paris] , Tommaso Lorenzi, Alexander Lorz, Anna Marciniak-Czochra [Univ. Heidelberg] , Roberto Natalini [IAC-CNR, Univ. Tor Vergata, Rome] , Silviu Iulan Niculescu [DISCO Inria team] , Hitay Özbay [Bilkent Univ., Ankara] , Benoît Perthame, Andrada Maran, Fernando Quirós [Univ. Autónoma de Madrid] , Michèle Sabbah [Cancer biology and therapeutics lab, St Antoine Hospital, Paris] , Thomas Stiehl [Univ. Heidelberg] , Min Tang [Jiaotong University, Shanghai] , Emmanuel Trélat [LJLL, UPMC] , Nicolas Vauchelet, Romain Yvinec [INRA Tours] .

Drug resistance.

We have continued to develop our phenotypically based models of drug-induced drug resistance in cancer cell populations, representing their Darwinian evolution under drug pressure by integro-differential equations. In one of them [40] , a 1D space variable has been added to the phenotypic structure variable to account for drug diffusion in tumour spheroids. In another one [33] , where deterministic and agent-based modelling are processed in parallel, we have considered a physiologically based 2-dimensional phenotypic structure variable, in order to take account of previously published biological observations on (reversible) drug tolerance persistence in a population of non-small cell lung cancer (NSCLC) cells (Sharma et al., Cell, April 2010), reproducing the observations and assessing the model by testing biologically based hypotheses. Together with ongoing work with E. Trélat and A. Lorz on drug therapy optimisation, using such phenotype-based models to overcome drug resistance, this has represented a significant part of our work on the subject, which is conducted in close collaboration with the INSERM-UPMC team “Cancer biology and therapeutics” (A. Larsen, A. Escargueil, M. Sabbah) at St Antoine Hospital.

Reversible drug resistance and fractional killing in tumor cell population treatment.

We developed a model of drug resistance in TRAIL (TNF-Related Apoptosis Induced-Ligand) treatment in HeLa cell lines. The TRAIL signal transduction pathway is one of the best studied apoptosis pathways and hence permits detailed comparisons with data. Our model was able to explain experimental observations fractional killing and cell-to-cell variability, and predicted reversible resistance [3] . (Work in close collaboration with G. Batt and S. Stoma from the Inria team LIFEWARE.)

Radiotherapy.

Radiation is still a major treatment in cancer. We explored by extensive computer simulations using an agent-based model the consequences of spatially inhomogeneous irradiation. The model predicted that in the case of different competing sub-populations, namely cancer stem cells with unlimited division capacity, and cancer cells with limited division capacity, inhomogeneous radiation focusing higher doses at the tumour centre and lower doses at the tumour periphery should outperform homogeneous irradiation [12] . Cancer stem cells are believed to have a longer cell cycle duration than cancer cells, and are less radiosensitive than cancer cells, which is why they often survive radiation and lead to tumour relapse.

Intercellular interactions in epithelio-mesenchymal transition (EMT).

A PhD thesis on this subject, co-supervised by L. Almeida and M. Sabbah (INSERM team “Cancer biology and therapeutics”, St Antoine) has begun at Fall. It is also based on phenotype-structured modelling of Darwinian evolution in cancer cell populations.

Interactions between tumour cell populations and their cellular micro-environment.

A phenotype-structured model of the interactions between a brest cancer cell population (MCF7 cultured cells, collaboration with M. Sabbah, St Antoine Hospital) and its adipocyte stroma support cell population has been developed (T. Lorenzi, J. Clairambault), which, beyond submitted proposals (ANR, Emergence Paris-Sorbonne Universités call), will be studied and experimentally identified in a forthcoming internship (January-June 2015) and PhD thesis in applied mathematics.

Hele-Shaw model of tumour growth.

In the growing field of mathematical analysis of mechanical domain of tumor growth, we focus on the rigorous link between cells models, relying on mechanical properties of cells, and free boundary problem, where the tumor is described by the dynamics of its boundary. The latter model is referred to Hele-Shaw model [44] . Benoît Perthame, Min Tang and Nicolas Vauchelet have proved the rigorous derivation of a geometric model of the Hele-Shaw type for a model with viscoelastic forces, constructing analytically traveling wave solutions of the Hele-Shaw model of tumor growth with nutrient that explain theoretically the numerical results observed. The limiting model exhibits travelling waves, which have been investigated in [43] . Another interesting feature for this model is the transversal instability occurring when the spatial dimension is greater than 1. Together with Fernando Quirós (Univ. Autónoma de Madrid), the aforementioned have also formulated a Hele-Shaw type free-boundary problem for a tumor growing under the combined effects of pressure forces, cell multiplication and active motion, the latter being the novelty of this study [61] . In order to understand the emergence of instabilities in the Hele-Shaw model with nutrients, Michal Kowalczyk (Univ. Chile, Santiago), Benoît Perthame and Nicolas Vauchelet have studied a related model of thermo-reactive diffusion where they can study the spectrum of the linearized system around a traveling wave and in which they can compute the transition to instability in terms of a parameter related to the ratio between heat conduction and molecular diffusion. However, the rigorous study of such instabilities for the whole system of equations is not reachable for the moment; only a study for a simplified model has been performed in [39] .

Modelling and control of acute myeloblastic leukæmia (AML).

The collaboration with the Disco project-team has been continued, leading to one book chapter [25] , four conference proceedings [21] , [22] , [23] , [24] and JL Avila Alonso's PhD thesis defence.

In more detail:

Starting initially from a PDE model of hematopoiesis designed by Adimy et al. (Adimy, M., Crauste, F., El Abllaoui, A. Discrete maturity-structured model of cell differentiation with applications to acute myelogenous leukemia, J. Biol. Sys., 16(3):395-424, 2008), we have derived several models of healthy or cancer cell dynamics in hematopoiesis and performed several stability analyses.

We have proposed in [25] a new mathematical model of the cell dynamics in acute myeloid leukæmia (AML) which takes into account the four different phases of the proliferating compartment as well the fast self-renewal phenomenon frequently observed in AML. As was the case in [25] this model is transformed into a distributed delay system and was analyzed here with input-output techniques. Local stability conditions for an equilibrium point of interest are derived in terms of a set of inequalities involving the parameters of the mathematical model.

We have also studied a coupled delay model for healthy and cancer cell dynamics in AML consisting of two stages of maturation for cancer cells and three stages of maturation for healthy cells. For a particular healthy equilibrium point, locally stability conditions involving the parameters of the mathematical model have been obtained [22] , [23] .

We have performed in [21] a stability analysis of both the PDE model of healthy hæmatopoiesis and a coupled PDE model of healthy and cancer cell dynamics. The stability conditions obtained here in the time domain strengthen the idea that fast self-renewal plays an important role in AML.

A time-domain stability analysis by means of Lyapunov-Krasovskii functionals has been performed on the delay system modeling healthy hematopoiesis for a strictly positive equilibrium point of interest.

Furthermore, a working collaboration on AML modelling with Anna Marciniak-Czochra (Univ. Heidelberg) was also initiated by the end of 2014 by a visit of three of us (C. Bonnet, J. Clairambault, T. Lorenzi) to Heidelberg and a visit of T. Stiehl, A. Marciniak-Czochra PhD student, to Paris. The topics we plan to investigate are, beyond the role of fast self renewal in AML cell populations, the part played by clonal heterogeneity in leukæmic cell populations and the issues it raises in therapeutics, a well known clinical problem in clinical hæmatology.

Let us also mention that on the subject of early leukæmogenesis, Andrada Qillas Maran has undertaken a PhD thesis under the supervision of J. Clairambault and B. Perthame. Models relying on piecewise deterministic Markov processes (PDMPs), designed and studied by R. Yvinec (INRA Tours) for the single-cell part of the model under construction, will be used in collaboration with him. Our clinical referents in hæmatology for this PhD work are F. Delhommeau and P. Hirsch (St Antoine Hospital).

The p53 protein spatio-temporal dynamics.

The development of our molecular-based model of the spatio-temporal intracellular dynamics of the p53 protein (the so-called “guardian of the genome") has been continued  [55] , [9] , leading us also, more generally, to propose a modelling frame dedicated to the dynamics of intracellular proteins and their gene regulatory networks [8] .

Others.

In a collaboration with ANGE, B. Perthame has studied a data assimilation algorithm for multidimensional hyperbolic conservation laws using kinetic schemes and kinetic formulations.