3.1 Introduction

Data and image analysis, statistical, ODEs, PDEs, and agent-based approaches are used either individually or in combination, with a strong focus on PDE analysis and agent-based approaches. Mamba was created in January 2014. It aims at developing models, simulations, numerical and control algorithms to solve questions from life sciences involving dynamics of phenomena encountered in biological systems such as protein intra-cellular spatio-temporal dynamics, cell motion, early embryonic development, multicellular growth, wound healing and liver regeneration, cancer evolution, healthy and tumor growth control by pharmaceuticals, protein polymerization occurring in neurodegenerative disorders, control of dengue epidemics, etc.

Another guideline of our project is to remain close to the most recent questions of experimental biology or medicine. In this context, we develop many close and fruitful collaborations with biologists and physicians.

We focus mainly on the creation, investigation and transfer of new mathematical models, methods of analysis and control, and numerical algorithms, but in selected cases software development as that of CellSys and TiQuant by D. Drasdo and S. Hoehme is performed. More frequently, the team develops “proof of concept” numerical codes in order to test the adequacy of our models to experimental biology.

We have organized the presentation of our research program in three methodological axes (Subsections 3.2, 3.3 and 3.4) and two application axes (Subsections 4.2 and 4.4). Evolving along their own logic in close interaction with the methodological axes, the application axes are considered as application-driven research axes in themselves. The methodological research axes are the following.

Axis 1 is devoted to work in physiologically-based design, analysis and control of population dynamics. It encompasses populations of bacteria, of yeasts, of cancer cells, of neurons, of aggregating proteins, etc. whose dynamics are represented by partial differential equations (PDEs), structured in evolving physiological traits, such as age, size, size-increment, time elapsed since last firing (neurons).

Axis 2 is devoted to reaction equations and motion equations of agents in living systems. It aims at describing biological phenomena such as tumor growth, chemotaxis and wound healing.

Axis 3 tackles questions of model and parameter identification, combining stochastic and deterministic approaches and inverse problem methods in nonlocal and multi-scale models.

3.2 Methodological axis 1: Analysis and control for population dynamics

Population dynamics is a field with varied and wide applications, many of them being in the core of MAMBA interests - cancer, bacterial growth, protein aggregation. Their theoretical study also brings a qualitative understanding on the interplay between individual growth, propagation and reproduction in such populations. In the past decades, many results where obtained in the BANG team on the asymptotic and qualitative behavior of such structured population equations, see e.g.  139, 70, 97, 83. Other Inria teams interested by this domain are Mycenae, Numed and Dracula, with which we are in close contacts. Among the leaders of the domain abroad, we can cite among others our colleagues Graeme Wake (New Zealand), Glenn Webb (USA), Jacek Banasiak (South Africa), Odo Diekmann (Netherlands), with whom we are also in regular contact. Most remarkably and recently, connections have also been made with probabilists working on Piecewise Deterministic Markov Processes (F. Malrieu at the university of Rennes, Jean Bertoin at the ETH in Zurich, Vincent Bansaye at Ecole Polytechnique, Julien Berestycki at Cambridge, Amaury Lambert at College de France, M. Hoffmann at Paris Dauphine, Alex Watson in UCL, London and J. Bertoin in Zurich), leading to a better understanding of the links between both types of results – see also the Methodological axis 3.

We divide this research axis, which relies on the study of structured population equations, according to four different applications, bringing their own mathematical questions, e.g., stability, control, or blow-up.

Time asymptotics for nucleation, growth and division equations

Following the many results obtained in the BANG team on the asymptotic and qualitative behavior of structured population equation, we put our effort on the investigation of limit cases, where the trend to a steady state or to a steady exponential growth described by the first eigenvector fails to happen. In  78, the case of equal mitosis (division into two equally-sized offspring) with linear growth rate was studied, and strangely enough, it appeared that the general relative entropy method could also be adapted to such a non-dissipative case. Many discussions and common workshops with probabilists, especially through the ANR project PIECE coordinated by F. Malrieu, have led both communities to work closer.

We also enriched the models by taking into account a nucleation term, modeling the spontaneous formation of large polymers out of monomers  145. We investigated the interplay between four processes: nucleation, polymerization, depolymerization and fragmentation.

New perspectives are now to consider not only one species but several interacting ones, which may exhibit complex interplays which may lead to damped oscillations or to infinite growth; these are in collaboration with C. Schmeiser and within the Vienna associated team MaMoCeMa (J. Delacour's Ph.D) and with K. Fellner from Graz (M. Mezache's Ph.D).

Cell population dynamics and its control

One of the important incentives for such model design, source of many theoretical works, is the challenging question of drug-induced drug resistance in cancer cell populations, described in more detail below in the Applicative axis 1, Cancer. The adaptive dynamics setting used consists of phenotype-structured integro-differential [or reaction-diffusion, when phenotype instability is added under the form of a Laplacian] equations describing the dynamic behavior of different cell populations interacting in a Lotka-Volterra-like manner that represents common growth limitation due to scarcity of expansion space and nutrients. The phenotype structure allows us to analyse the evolution in phenotypic traits of the populations under study and its asymptotics for two populations  127, 124, 123, 125. Space may be added as a complementary structure variable provided that something is known of the (Cartesian) geometry of the population  126, which is seldom the case.

Modelling Mendelian and non-Mendelian inheritances in density-dependent population dynamics

Classical strategies for controlling mosquitoes responsible of vector-borne disease are based on mechanical methods, such as elimination of oviposition sites; and chemical methods, such as insecticide spraying. Long term usage of the latter generates resistance 81, 110, transmitted to progeny according to Mendelian inheritance (in which each parent contributes randomly one of two possible alleles for a trait). New control strategies involve biological methods such as genetic control, which may either reduces mosquito population in a specific area or decreases the mosquito vector competence 61, 119, 157. Among the latter, infection of wild populations by the bacterium Wolbachia appears promising (see also Applicative axis 2 below). Being maternally-transmitted, the latter obeys non-Mendelian inheritance law. Motivated by the effects of the (possibly unwanted) interaction of these two types of treatment, we initiated the study of modelling of Mendelian and non-Mendelian inheritances in density-dependent population dynamics. First results are shown in 20.

Control and macroscopic limits of collective dynamics

The term self-organization is used to describe the emergence of complex organizational patterns from simple interaction rules in collective dynamics systems. Such systems are valuable tools to model various biological systems or opinion dynamics, whether it be the collective movement of animal groups, the organization of cells in an organism or the evolution of opinions in a large crowd. A special case of self-organization is given by consensus, i.e. the situation in which all agents' state variables converge. Another phenomenon is that of clustering, when the group is split into clusters that each converge to a different state. A natural question in this framework is that of control: can the system be guided to a desired predetermined configuration? In the case when self-organization is not achieved naturally by the system, can it be driven to it? On the contrary, in the case where consensus and clustering are situations to be avoided (for example in crowd dynamics), can we design control strategies to keep the system away from clustering?

Another natural question is that of the large population limit. When the number of agents tends to infinity, the previous system of equations becomes unmanageable, a problem well-known as the curse of dimensionality. A common answer to this issue consists of studying the macroscopic limit of the system. It is then crucial to understand whether the limit system retains the properties of the microscopic one.

Models of neural network

Mean field limits have been proposed by biophysicists in order to describe neural networks based on physiological models. The various resulting equations are called integrate-and-fire, time elapsed models, voltage-conductance models. Their specific nonlinearities and the blow-up phenomena make their originality which has led to develop specific mathematical analysis 136, followed by 133, 118, 137, 82. This field also yields a beautiful illustration for the capacity of the team to combine and compare stochastic and PDE modelling (see Methodological axis 3), in 87.

Models of interacting particle systems

The organisation of biological tissues during development is accompanied by the formation of sharp borders between distinct cell populations. The maintenance of this cell segregation is key in adult tissue homeostatis, and its disruption can lead tumor cells to spread and form metastasis. This segregation is challenged during tissue growth and morphogenesis due to the high mobility of many cells that can lead to intermingling. Therefore, understanding the mechanisms involved in the generation and maintain of cell segregation is of tremendous importance in tissue morphogenesis, homeostasis, and in the development of various invasive diseases such as tumors. In this research axis, we aim to provide a mathematical framework which enables to quantitatively link the segregation and border sharpening ability of the tissue to these cell-cell interaction phenomena of interest 4. As agent-based models do not enable precise mathematical analysis of their solutions due to the lack of theoretical results, we turn towards continuous -macroscopic- models and aim to provide a rigorous link between the different models 4.

Models of population dynamics structured in phenotype

The collaboration of Jean Clairambault with Emmanuel Trélat and Camille Pouchol (from September this year assistant professor at MAP5 Paris-Descartes, University of Paris), together now with Nastassia Pouradier Duteil, has been continued and presently leads us to a possible quantitative biological identification of the structuring phenotypes of the model developed in  144, through a beginning collaboration with an Indian systems biologist (Mohit Kumar Jolly, IIS Bangalore). Our motivation in this collaboration is to couple a physiologically based system of 6 ODEs developed by our Indian collaborator with our phenotype-structured cell population dynamics model  90, 91.

In the framework of the HTE project EcoAML 2016-2020, Thanh Nam Nguyen, Jean Clairambault, Delphine Salort and Benoît Perthame, in collaboration with Thierry Jaffredo at IBPS-SU, have designed a phenotype-structured integrodifferential model of interactions between haematopoietic stem cells (healthy or leukaemic) and their supporting stromal cells  130. In this model, without diffusion, to our relative astonishment, our postdoctoral fellow T.N. Nguyen predicts in particular that under special circumstances, a coexistence between healthy and leukaemic stem cell subpopulations is possible. The explanation of such possible theoretical coexistence still remains to be explained.

The idea of cooperation between cell subpopulations in a tumour is also studied using phenoytype-structured models of cell populations by Frank Ernesto Alvarez Borges, PhD student of Stéphane Mischler (Paris-Dauphine University), Mariano Rodríguez Ricard (University of Havana, Cuba) and Jean Clairambault, in collaboration with José Antonio Carrillo (Imperial College London). A feature of these models, in as much as conflicting continuous phenotypes (e.g., adhesivity vs. motility, or fecundity vs. viability, or fecundity vs. motility  1) are supposed to structure a unique cell population, is that they can also represent the emergence of multicellularity in such a cell population, when two subpopulations of the same population, i.e., endowed with the same genome and represented w.r.t. relevant heterogeneity in the cell population by such conflicting phenotypes, are determined by two different choices of the 2-d phenotype. In a simplified representation when the two phenotypes are just extreme values of a 1-d continuous phenotype (e.g., 0 for total adhesivity and no motility, 1 for no adhesivity and complete motility) this situation may be related to the previously described case, developed in  130, in which two extreme values of a convex function linked to proliferation are occupied by the two extreme phenotype values (0 and 1), leading to the coexistence of two cell subpopulations.

Collaborations

• Nucleation, growth and fragmentation equations: Klemens Fellner, university of Graz, Austria; Piotr Gwiązda, Polish Academy of Sciences, Poland; Christian Schmeiser, university of Vienna, through the associated team MaMoCeMa.
• Cell population dynamics and its control: Tommaso Lorenzi, former Mamba postdoc, now at the University of St. Andrews, Scotland, maintains a vivid collaboration with the Mamba team. He is in particular an external member of the HTE program MoGlImaging (see also Applicative axis 1). Emmanuel Trélat, Sorbonne Université professor, member of LJLL and of the CAGE Inria team, is the closest Mamba collaborator for optimal control. Benedetto Piccoli, Professor at Rutgers University (Camden, New Jersey), is collaborating on the analysis and control of collective dynamics. Nathalie Ayi, Sorbonne University, is participating in the development of graph-limit methods.
• Mendelian inheritance and resistance in density-dependent population dynamics: Pastor Pérez-Estigarribia, Christian Schaerer, Universidad Nacional de Asunción, Paraguay.
• Neural networks: Delphine Salort, Professor Sorbonne Université, Laboratory for computations and quantification in biology, and Patricia Reynaud, University of Nice, Maria Cáceres, University of Granada.
• Models of interacting particle systems: Pierre Degond, Imperial College London; Julien Barré, APMO, Orléans; Ewelina Zatorska, University College London; Sara Merino from the university of Vienna (through the associated team MaMoCeMa).

3.3 Methodological axis 2: Reaction and motion equations for living systems

The Mamba team had initiated and is a leader on the works developed in this research axis. It is a part of a consortium of several mathematicians in France through the ANR Blanc project Kibord, which involves in particular members from others INRIA team (DRACULA, COMMEDIA). Finally, we mention that from Sept. 2017 on, Mamba benefited from the ERC Advanced Grant ADORA (Asymptotic approach to spatial and dynamical organizations) of Benoît Perthame.

We divide this research axis, which relies on the study of partial differential equations for space and time organisation of biological populations, according to various applications using the same type of mathematical formalisms and methodologies: asymptotic analysis, weak solutions, numerical algorithms.

Aggregation equation

In the mathematical study of collective behavior, an important class of models is given by the aggregation equation. In the presence of a non-smooth interaction potential, solutions of such systems may blow up in finite time. To overcome this difficulty, we have defined weak measure-valued solutions in the sense of duality and its equivalence with gradient flows and entropy solutions in one dimension  116. The extension to higher dimensions has been studied in  85. An interesting consequence of this approach is the possibility to use the traditional finite volume approach to design numerical schemes able to capture the good behavior of such weak measure-valued solutions  109, 117.

Identification of the mechanisms of single cell motion

In this research axis, we aim to study the mechanisms of single cell adhesion-based and adhesion free motion. This work is done in the frame of the recently created associated team MaMoCeMa (see Section 9) with the WPI, Vienna. In a first direction 148 with N. Sfakianakis (Heidelberg University), we extended the live-cell motility Filament Based Lamellipodium Model to incorporate the forces exerted on the lamellipodium of the cells due to cell-cell collision and cadherin induced cell-cell adhesion. We took into account the nature of these forces via physical and biological constraints and modelling assumptions. We investigated the effect these new components had in the migration and morphology of the cells through particular experiments. We exhibit moreover the similarities between our simulated cells and HeLa cancer cells.

In a second work done in collaboration with the group of biologist at IST (led by Michael Sixt Austria), we developed and analyzed a two-dimensional mathematical model for cells migrating without adhesion capabilities 16. Cells are represented by their cortex, which is modelled as an elastic curve, subject to an internal pressure force. Net polymerization or depolymerization in the cortex is modelled via local addition or removal of material, driving a cortical flow. The model takes the form of a fully nonlinear degenerate parabolic system. An existence analysis is carried out by adapting ideas from the theory of gradient flows. Numerical simulations show that these simple rules can account for the behavior observed in experiments, suggesting a possible mechanical mechanism for adhesion-independent motility.

Free boundary problems for tumor growth

Fluid dynamic equations are now commonly used to describe tumor growth with two main classes of models: those which describe tumor growth through the dynamics of the density of tumoral cells subjected to a mechanical stress; those describing the tumor through the dynamics of its geometrical domain thanks to a Hele-Shaw-type free boundary model. The first link between these two classes of models has been rigorously obtained thanks to an incompressible limit in  135 for a simple model. This result has motivated the use of another strategy based on viscosity solutions, leading to similar results, in  120.

Since more realistic systems are used in the analysis of medical images, we have extended these studies to include active motion of cells in  134, viscosity in  140 and proved regularity results in  128. The limiting Hele-Shaw free boundary model has been used to describe mathematically the invasion capacity of a tumour by looking for travelling wave solutions, in  138, see also Methodological axis 3. It is a fundamental but difficult issue to explain rigorously the emergence of instabilities in the direction transversal to the wave propagation. For a simplified model, a complete explanation is obtained in  121.

Coupling of diffusion and growth

The growth of an organism is triggered by signaling molecules called morphogens that diffuse in the organism during its development. Meanwhile, the diffusion of the morphogens is itself affected by the changes in shape and size of the organism. In other words, there is a complete coupling between the diffusion of the morphogens and the evolution of the shapes. We are working on the elaboration of a mathematical framework for diffusion equations on time-evolving manifolds, both theoretically and in collaboration with developmental biologists, for the special case of the diffusion of Gurken during the oogenesis of Drosophila.

Migration of cells in extracellular matrix

A single cell based model has been developed that reproduces a large set of experimental observations of cells migrating in extracellular matrix based on physical mechanisms with mimimal internal cell dynamics. This includes individually migrating cells in micro-channels of different size, and their collective dynamics in case of many cells, as well as the impact of cell division and growth. The model explicitly mimics the extracellular matrix as the cells as deformable objects with explicit filopodia.

Collaborations

• Shanghai Jiao Tong University, joint publications with Min Tang on bacterial models for chemotaxis and free boundary problems for tumor growth.
• Imperial College London, joint works with José Antonio Carrillo on aggregation equation.
• University of Maryland at College Park, UCLA, Univ. of Chicago, Univ. Autónoma de Madrid, Univ. of St. Andrews (Scotland), joint works on mathematics of tumor growth models.
• Joint work with Francesco Rossi (Università di Padova, Italy) and Benedetto Piccoli (Rutgers University, Camden, New Jersey, USA) on Developmental PDEs.
• Cooperation with Shugo Yasuda (University of Hyogo, Kobe, Japan) and Vincent Calvez (EPI Dracula) on the subject of bacterial motion.
• Cooperation with Nathalie Ferrand (INSERM), Michèle Sabbah (INSERM) and Guillaume Vidal (Centre de Recherche Paul Pascal, Bordeaux) on cell aggregation by chemotaxis.
• Nicolas Vauchelet, Université Paris 13

3.4 Methodological axis 3: Model and parameter identification combining stochastic and deterministic approaches in nonlocal and multi-scale models

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. 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 101, TRAIL treatment of HeLa cells 72, growth of multicellular spheroids 115, blood detoxification after drug-induced liver damage 147, 106.

This naturally leads to increasingly combine methods from various fields: image analysis, statistics, probability, numerical analysis, PDEs, ODEs, agent-based modeling 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 variability 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 Moireau (Inria M3DISIM) in inverse problems and data assimilation, and with numerous experimentalists.

Estimation methods for growing and dividing populations

In this domain, all originated in two papers in collaboration with J.P. Zubelli in 2007  141, 99, 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  78. 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)  94. 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  98, a work which inspired also very recently other groups in statistics and probability  73, 112 and was the basis for Adélaïde Olivier's Ph.D thesis  132, 114 and of more recent work  13114 (see also axis 5).

Data assimilation and stochastic modeling 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  62, 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  103, was devoted to the stochastic modeling and analysis of protein aggregation, compared both with the deterministic approach traditionally developed in Mamba  145 and with experiments.

Parameter identification in multi-level and multi-scale models of liver

Several projects are pursued on multiscale, multilevel modeling of liver regeneration and its consequences with integration of an increasingly amount of data. So far the most promising strategy working was for every additional data set, first testing whether the model would be able to simulate it without any modifications, and to modify the model if necessary by inclusion of further biological mechanisms or information. A key unsolved problem is that biological data seem often not perfectly reproducable, and measurements at different times may differ from each other. This can result from slightly different experimental settings or conditions, or different measurement methods. While for testing of qualitative mechanisms this is usually sufficient, the quantitative difference is sometimes of the order of the effect which makes a quantitative modeling very challenging. For ammonia detoxification during fibrosis, extensive simulations have been performed varying multiple clinically relevant parameters. The basis model needed to integrate multiple data sets and could only be modelled if modifications in tissue microarchitecture, adaptations of intracellular enzyme activities, and possible aging effects were taken into account (ongoing project close to finalization).

Collaborations

• Marc Hoffmann, Université Paris-Dauphine, for the statistical approach to growth and division processes, Miguel Escobedo, Bilbao and Magali Tournus, Marseille, for the deterministic approach.
• Philippe Moireau, Inria M3DISIM, for the inverse problem and data assimilation aspects  68, 63

4.1 Introduction

The team has three main application-driven research axes. Applicative axis 1 focuses on cancer, an application on which almost all team members work, with various approaches. A main focus of the team is to study cancer as a Darwinian evolutionary phenomenon in phenotype-structured cell populations. Optimal control methods take into account the two main pitfalls of clinical cancer therapeutics, namely unwanted toxic side effects in healthy cell populations and drug resistance in cancer cell populations. Other studies concern telomere shortening, and multi-scale models. Applicative axis 2 is devoted to growth, evolution and regeneration in populations and tissues. It involves protein aggregation and fragmentation models for neurodegenerative diseases (prion, Alzheimer), organ modeling, mainly of the liver, its damages induced by toxic molecules, and its regeneration after toxic insult. Applicative axis 3 is new and encompasses works related to epidemiology, both for infectious and vector-borne diseases.

4.2 Applicative axis 1: Focus on cancer

The MAMBA team designs and analyses mathematical models of tumor growth and therapy, at the cell population level, using agent-based or partial differential equations, with special interest in methodologies for therapeutic optimization using combined anticancer drug treatments. Rather than, or not only, modeling the effect of drugs on molecular targets, we represent these effects by their functional consequences on the fate of healthy and cancer cell populations: proliferation (velocity of the cell division cycle, decreasing it, e.g., by antagonizing growth factor receptors), apoptosis, cell death or senescence. Our goal in doing this is to circumvent the two main issues of anticancer therapy in the clinic, namely unwanted toxic side effects in populations of healthy cells and emergence of drug-induced drug resistance in cancer cell populations. This point of view leads us to take into account phenomena of transient and reversible resistance, observed in many cancer cell populations, by designing and analyzing models of cell populations structured in continuous phenotypes, relevant for the description of the behavior of cell populations exposed to drugs: either degree of resistance to a given drug, or potential of resistance to drug-induced stress, proliferation potential, and plasticity. Such modeling options naturally lead us to to take into account in a continuous way (i.e., by continuous-valued phenotype or relevant gene expression) the wide phenotypic heterogeneity of cancer cell populations. They also lead us to adopt the point of view of adaptive dynamics according to which characteristic traits of cell populations evolve with tumor environmental pressure (drugs, cytokines or metabolic conditions, mechanical stress and spatial conditions), in particular from drug sensitivity to resistance. This position is original on the international scene of teams dealing with drug resistance in cancer.

Modeling Acute Myeloid Leukemia (AML) and its control by anticancer drugs by PDEs and Delay Differential equations

In collaboration with Catherine Bonnet (Inria DISCO, Saclay) and François Delhommeau (St Antoine hospital in Paris), together with DISCO PhD students José Luis Avila Alonso and Walid Djema, this theme has led to common published proceedings of conferences: IFAC, ACC, CDC, MTNS  65, 66, 67, 77, 93, 64. These works study the stability of the haematopoietic system and its possible restabilization by combinations of anticancer drugs with functional targets on cell populations: proliferation, apoptosis, differentiation.

Adaptive dynamics setting to model and circumvent evolution towards drug resistance in cancer by optimal control

We tackle the problem to represent and inhibit - using optimal control algorithms, in collaboration with Emmanuel Trélat, proposed Inria team CAGE - drug-induced drug resistance in cancer cell populations. This theme, presently at the core of our works on cancer modeling with a evolutionary perspective on tumor heterogeneity, is documented in a series of articles  88, 89, 123, 124, 126. Taking into account the two main pitfalls of cancer therapy, unwanted side effects on healthy cells and evolution towards resistance in cancer cells, it has attracted to our team the interest of several teams of biologists, with whom we have undertaken common collaborative works, funded by laureate answers to national calls (see ITMO Cancer HTE call).

This theme is also at the origin of methodological developments (see Research axis 1). In collaboration with Shensi Shen from Institut Gustave Roussy and Francois Vallette from Université de Nantes, we aim to develop simple non-spatial models to understand the mechanisms of drug resistance acquisition -and loss- in melanoma and glioblastoma. The models are systematically compared with in vitro and in vivo data generated by our collaborators and treated via image processing techniques developed in the team.

Senescence modeling by telomere shortening

In many animals, aging tissues accumulate senescent cells, a process which is beneficial to protect from cancer in the young organism. In collaboration with Teresa Teixeira and Zhou Xu from IBCP, we proposed a mathematical model based on the molecular mechanisms of telomere replication and shortening and fitted it on individual lineages of senescent Saccharomyces cerevisiae cells, in order to decipher the causes of heterogeneity in replicative senescence  79.

Biomechanically mediated growth control of cancer cells other cell types

Model simulations indicate that the response of growing cell populations on mechanical stress follows a simple universal functional relationship and is predictable over different cell lines and growth conditions despite the response curves look largely different. We developed a hybrid model strategy in which cells were represented by coarse-grained individual units calibrated in a high resolution cell model and parameterized each model cell by measurable biophysical and cell-biological parameters. Cell cycle progression in our model is controlled by volumetric strain, the latter being derived from a bio-mechanical relation between applied pressure and cell compressibility. After parameter calibration from experiments with mouse colon carcinoma cells growing against the resistance of an elastic alginate capsule, the model adequately predicts the growth curve in i) soft and rigid capsules, ii) in different experimental conditions where the mechanical stress is generated by osmosis via a high molecular weight dextran solution, and iii) for other cell types with different growth kinetics. Our model simulation results suggest that the growth response of cell population upon externally applied mechanical stress is the same, as it can be quantitatively predicted using the same growth progression function 122. This model has now been extended to compare the efficiency of different culturing methods, monolayer growth, multicellular spheroids growth and growth within elastic capsules. The methodology of culturing is relevant in terms of cell yield and cell homogeneity.

Bio-mechanical models of tissue growth

The degenerate Cahn-Hilliard equation is a standard model to describe living tissues. It takes into account cell populations undergoing short-range attraction and long-range repulsion effects. In this framework, we consider the usual Cahn-Hilliard equation with a singular single-well potential and degenerate mobility. These degeneracy and singularity induce numerous difficulties, in particular for its numerical simulation. To overcome these issues, we propose in [hal-02274417] a relaxation system formed of two second order equations which can be solved with standard packages. This system is endowed with an energy and an entropy structure compatible with the limiting equation. Here, we study the theoretical properties of this system; global existence and convergence of the relaxed system to the degenerate Cahn-Hilliard equation. We also study the long-time asymptotics which interest relies on the numerous possible steady states with given mass.

Free boundary multiphase models of tumor growth

Multiphase mechanical models are now commonly used to describe living tissues including tumour growth. The specific model we study here consists of two equations of mixed parabolic and hyperbolic type which extend the standard compressible porous media equation, including cross-reaction terms. We study the incompressible limit, when the pressure becomes stiff, which generates a free boundary problem. We establish the complementarity relation and also a segregation result. Several major mathematical difficulties arise in the two species case which are addressed in 9. Firstly, the system structure makes comparison principles fail. Secondly, segregation and internal layers limit the regularity available on some quantities to BV. Thirdly, the Aronson-Bénilan estimates cannot be established in our context. We are lead, as it is classical, to add correction terms. This procedure requires technical manipulations based on BV estimates only valid in one space dimension. Another novelty is to establish an $L^1$ version in place of the standard upper bound.

Philosophy of cancer

The quite natural idea that cancer is a disease of the control of coherent multicellularity, expressed when cohesion of tissues and coherence of (unknown, except maybe for the case of a centralised circadian clock) synchronising signals fail to ensure it, by a regression towards unicellularity, stopping in this “reverse evolution path” at a coarse, incoherent multicellularity state  2 continues to be developed and popularised by Jean Clairambault in seminars and workshops, and published in review articles  90, 91. This view, and the investigation of the immune system in the design of such coherence of all multicellular organisms  3 is naturally inscribed in a philosophy of cancer perspective, and from a mathematical viewpoint, to multicellularity genes - and links between them and unicellularity genes - seen as a hyperstructure  4 above structures consisting of the genes of unicellularity, i.e., those that make a single cell a coherent living system, such hyperstructure being failed in cancer; this view is presently under development with colleagues from universities of the Paris region, together with Nils Baas at NTNU, Trondheim, Norway). This perspective, that makes use of category theory as a structuring point of view to apprehend multicellularity and cancer, is also meant to endow us with an innovative methodology to apply topological data analysis (TDA) to investigate cancer genome data.

Modelling of TMZ induced drug resistance

Temozolomide (TMZ) is a standard chemotherapy treatment in patients with glioblastoma. Resistance to this drug is correlated to the presence of a specific enzyme, which activity in cancer cells creates a drug-induced cell death resistant phenotype. Understanding the transition of cancer cells to a resistant phenotype is still a topic of research where multiple hypothesis have been studied: From an adaptive process to an inherent resistance to treatment. It has been recently shown that if TMZ treatment does not significantly induce cell death in glioblastoma, it still generates a response in terms of the spatial arrangement of cell aggregates. Moreover, the coupling of TMZ with irradiation has been shown to generate a better response in patients compared with using irradiation alone. Therefore, understanding the mechanisms of glioblastoma reaction to TMZ treatment could open new therapeutic avenues. In the frame of the post-doctorate of Gissell Estrada Rodriguez, we developed a 2D mathematical model in 33, suggesting a new possible mechanism for TMZ induced rearrangement of cancer cells (see section new results).

Modelling of the Epithelial-Mesenchymal Transition (EMT)

Understanding cell-fate decisions remains a major research challenge in developmental biology. In particular, the forward and backward epithelial-mesenchymal cellular transitions (EMT-MET) play a crucial role in embryonal development, tissue repair and cancer metastasis. The epithelial cell phenotype (E) is characterized by strong cell-to-cell adhesion, while the mesenchymal phenotype (M) is characterized by a strong cellular motility. Recent research has shown that there even exists a third hybrid phenotype (E/M) with mixed characteristics, that enables collective cell migration. EMT and MET play a crucial role in cancer metastasis, for instance when cancer cells from a primary tumor gain the ability to migrate through the bloodstream or lymph system to distant organs and then recover their adhesion to form secondary tumors. Thus, understanding the dynamics of MET and EMT is crucial for decoding metastasis and for designing effective therapeutics.

Collaborations

• AML modelling: Catherine Bonnet, DISCO Inria team, Saclay, and François Delhommeau, INSERM St Antoine (also collaborator in the INSERM HTE laureate project EcoAML, see below).
• INSERM HTE laureate project MoGlImaging, headed by E. Moyal (Toulouse): François Vallette, CRCNA and INSERM Nantes
• INSERM HTE laureate project EcoAML, headed by François Delhommeau, INSERM St Antoine: François Delhommeau, Thierry Jaffredo (IBPS), Delphine Salort (LCQB-IBPS)

• Adaptive dynamics to model drug resistance and optimal control to circumvent it:

  Alexandre Escargueil, Michèle Sabbah (1 PhD thesis in common), St Antoine Hospital, Paris

  Emmanuel Trélat (1 PhD thesis in common) at Inria team CAGE and Laboratoire Jacques-Louis Lions at Sorbonne Université.

  Frédéric Thomas at CREEC, Montpellier.

  Tommaso Lorenzi (Univ. of St Andrews).

• Telomere shortening: Teresa Teixeira and Zhou Xu (IBCP, Paris).
• Biomechanical control of cancer cells: Pierre Nassoy, Bioimaging and Optofluidics Group, LP2N – UMR 5298. IOGS, CNRS & University of Bordeaux; TreeFrog Pharmaceutics, 30 Avenue Gustave Eiffel Bâtiment A, 33600 Pessac
• EMT: Camille Pouchol (Université de Paris), Mohit Kumar Jolly (Indian Institute of Science, Bengalore)

4.3 Applicative axis 2: Growth, evolution and regeneration in populations and tissues

The applications in this category span very different subjects from amyloid diseases, wound healing, liver regeneration and toxicity, up to bacterial growth and development of organisms. As the applications, the methods span a wide range. Those concerning identification of models and parameters with regard to data have partially been outlined in axis 3. Focus in this axis is on the model contribution to the biologically and/or medically relevant insights and aspects.

Liver-related modelling is partially performed within the INRIA team MIMESIS (Strasbourg) with the focus on real-time, patient-specific biomechanical liver models to guide surgery and surgeons. Internationally, spatial temporal liver related models are developed in Fraunhofer MEVIS (Bremen), by T. Ricken (TU Dortmund), and P. Segers group (Leuven).

Different from these, Mamba has a strong focus on spatial-temporal modeling on the histological scale, integration of molecular processes in each individual cell, and single-cell (agent) based models  100. Works by Schliess  147, 106 have been highlighted in editorials.

Mathematical modeling of protein aggregation is a relatively recent domain, only a few other groups have emerged yet; among them we can cite the Inria team Dracula, with whom we are in close contact, and e.g., the work by Jean-Michel Coron (Sorbonne Université) and Monique Chyba (Hawaii, USA) in control, and Suzanne Sindi (USA) for the modeling of the yeast prion. We have interactions with all these groups and organized a workshop in June 2017, gathering both the biophysics and applied mathematics communities.

Amyloid disease

Application to protein aggregation in amyloid diseases is a long-standing interest of Mamba, dating back to 2010  84, and developed through the collaboration with n rHuman Rezaei's team at Inra. More recently, with Wei-Feng Xue in Canterbury, we investigated the intrinsic variability among identical experiments of nucleation  95, 104, Sarah Eugène's Ph.D subject (co-supervised by Philippe Robert)  103.

In collaboration with Tom Banks first  69, 68 and then Philippe Moireau, we developed quantitative comparisons between model and data. Through data assimilation and statistical methods  62, we proposed new models and mechanisms.

Wound healing: adipose tissues

After injury, if regeneration can be observed in hydra, planaria and some vertebrates, regeneration is rare in mammals and particularly in humans. In this research axis, we investigated the mechanisms by which biological tissues recover after injury. We explored this question on adipose tissue, using the mathematical framework recently developed in 143. Our assumption is that simple mechanical cues between the Extra-Cellular Matrix (ECM) and differentiated cells can explain adipose tissue morphogenesis and that regeneration requires after injury the same mechanisms. We validated this hypothesis by means of a two-dimensional Individual Based Model (IBM) of interacting adipocytes and ECM fiber elements 142. The model successfully generated regeneration or scar formation as functions of few key parameters, and seemed to indicate that the fate of injury outcome could be mainly due to ECM rigidity.

Following these encouraging results, the team is currently taking a step further in the model validation and confrontation to experimental data. The first direction concerns the development of a 3D framework to validate the mechanisms observed in 2D, in the frame of the PhD of P. Chassonnery, co-directed by D. Peurichard and L. Casteilla (RESTORE, Toulouse).

Influence of cell mechanics in embryonic bile duct lument formation: insight from quantitative modeling

In vitro construction of hepatic tissue for regenerative therapy consists in recapitulating mechanisms of embryonic development. However, implementing those mechanisms in a spatially and temporally coordinated way remains difficult. Specifically, the construction of bile ducts and in particular the controlled formation of luminal structures formed by cholangiocytes is a challenge. The team works on a high resolution individual-based computational model which can help in unravelling the mechanisms of initial bile duct lumen formation. Guided by the quantification of morphological features and expression of genes in developing bile ducts from embryonic mouse liver, hypotheses for the mechanisms of biliary lumen formation were generated and tested with the model. Our simulations with a hybrid simulation technology as developed in ref.   122 suggest that successful bile duct lumen formation primarily requires the simultaneous contribution of several mechanisms discussed in the literature.

Mathematical modelling of axolotl regeneration

Tissue response after injury/amputation induces one or two alternatives: scar formation versus regeneration (complete recovery of tissue shape and functions). In most mammals, regeneration is considered largely impaired for the benefit of a fibrotic scar after injury automatically associated with dysfunctions, but complete regeneration has been largely described and investigated in animal models such as zebra fish, salamander, or axolotl. Despite several processes regulating regeneration have been identified at different scales -from diffusing molecules and cellular gene expression patterns up to tissue mechanics-, how these mechanisms individually or collectively play a role in the regulation of regenerative processes remains poorly understood. In order to give insights into the mechanisms of tissue regeneration, Valeria Caliaro started an Inria PhD project in october 2019, in collaboration with Osvaldo Chara, internationally recognized group leader of SysBio in Argentina. This project focuses on the role of cell proliferation in space and time along the two first phases of regeneration after injury: (i) initiation of a regeneration response, (ii) tissue patterning during regenerate growth. The first part of the project aims at building an agent-based model featuring few key mechanisms regulating cell proliferation after injury. By introducing heuristic rules which rely on Prof O. Chara expertise, we propose a 2D-ABM using methodologies borrowed from socio-dynamics and collective behavior studies (based on many interacting agent systems). While the focus is made on proliferation-based mechanisms, other mechanisms responsible for collective behavior such as volume exclusion, diffusion or aggregation are taken into account. The resulting model will provide a synthetic tissue model which will serve to investigate regeneration in cellular systems, focusing on cell proliferation properties. The second part of the PhD will be devoted to the derivation of continuous models from the agent-based formalism. This will provide a large scale ‘synthetic tissue’ model to explore the role of large scale effects in general tissue models.

Quantitative cell-based model predicts mechanical stress response of growing tumor spheroids

Model simulations indicate that the response of growing cell populations on mechanical stress follows the same functional relationship and is predictable over different cell lines and growth conditions despite experimental response curves look largely different. We developed a hybrid model strategy in which cells are represented by coarse-grained individual units calibrated with a high resolution cell model and parameterized by measurable biophysical and cell-biological parameters. Cell cycle progression in our model is controlled by volumetric strain, the latter being derived from a bio-mechanical relation between applied pressure and cell compressibility. After parameter calibration from experiments with mouse colon carcinoma cells growing against the resistance of an elastic alginate capsule, the model adequately predicts the growth curve in i) soft and rigid capsules, ii) in different experimental conditions where the mechanical stress is generated by osmosis via a high molecular weight dextran solution, and iii) for other cell types with different growth kinetics from the growth kinetics in absence of external stress. Our model simulation results suggest a generic, even quantitatively same, growth response of cell populations upon externally applied mechanical stress, as it can be quantitatively predicted using the same growth progression function (5.

Bacterial population growth

We exploited all the methods developed to estimate the division rate of a population (see axis 3) to address a seminal question of biology: is it a size-sensing or a timing mechanism which triggers bacterial growth? In  146, we showed that a sizer model is robust and fits the data well. Several studies from other groups came at the same time, showing a renewed interest on a question dated back to Jacques Monod's PhD thesis (1941). Of special interest is the “adder” model, for which we are currently developing new estimation methods 14.

A new model for the emergence of blood capillary networks

In 58, we propose a new model for the emergence of blood capillary networks. We assimilate the tissue and extra cellular matrix as a porous medium, using Darcy's law for describing both blood and intersticial fluid flows. Oxygen obeys a convection-diffusion-reaction equation describing advection by the blood, diffusion and consumption by the tissue. Discrete agents named capillary elements and modelling groups of endothelial cells are created or deleted according to different rules involving the oxygen concentration gradient, the blood velocity, the sheer stress or the capillary element density. Once created, a capillary element locally enhances the hydraulic conductivity matrix, contributing to a local increase of the blood velocity and oxygen flow. No connectivity between the capillary elements is imposed. The coupling between blood, oxygen flow and capillary elements provides a positive feedback mechanism which triggers the emergence of a network of channels of high hydraulic conductivity which we identify as new blood capillaries. We provide two different, biologically relevant geometrical settings and numerically analyze the influence of each of the capillary creation mechanism in detail. All mechanisms seem to concur towards a harmonious network but the most important ones are those involving oxygen gradient and sheer stress. This work offers a new paradigm for capillary network creation by placing the flow of blood at the central place in the process. The model proposed in 3 provides a proof of concept of this approach and elaborates a road map by which the model can be gradually improved towards a fully fledged simulator of blood capillary network formation. Such simulator would have huge potential for biological or clinical applications in cancer, wound healing, tissue engineering and regeneration.

A quantitative high resolution computational mechanics cell model for growing and regenerating tissues

Mathematical models are increasingly designed to guide experiments in biology, biotechnology, as well as to assist in medical decision making. They are in particular important to understand emergent collective cell behavior. For this purpose, the models, despite still abstractions of reality, need to be quantitative in all aspects relevant for the question of interest. Considered was as showcase example the regeneration of liver after drug-induced depletion of hepatocytes, in which the surviving and dividing hepatocytes must squeeze in between the blood vessels of a network to refill the emerged lesions. Here, the cells’ response to mechanical stress might significantly impact the regeneration process. We present a 3D high-resolution cell-based model integrating information from measurements in order to obtain a refined and quantitative understanding of the impact of cell-biomechanical effects on the closure of drug-induced lesions in liver. Our model represents each cell individually and is constructed by a discrete, physically scalable network of viscoelastic elements, capable of mimicking realistic cell deformation and supplying information at subcellular scales. The cells have the capability to migrate, grow, and divide, and the nature and parameters of their mechanical elements can be inferred from comparisons with optical stretcher experiments. Due to triangulation of the cell surface, interactions of cells with arbitrarily shaped (triangulated) structures such as blood vessels can be captured naturally. Comparing our simulations with those of so-called center-based models, in which cells have a largely rigid shape and forces are exerted between cell centers, we find that the migration forces a cell needs to exert on its environment to close a tissue lesion, is much smaller than predicted by center-based models. To stress generality of the approach, the liver simulations were complemented by monolayer and multicellular spheroid growth simulations. In summary, our model can give quantitative insight in many tissue organization processes, permits hypothesis testing in silico, and guide experiments in situations in which cell mechanics is considered important 122.

Liver regeneration and disease: towards a full virtual liver model at histological scale

In our work towards a full virtual liver model at histological level, a number of steps were performed. The models under points (1)-(4) focus on either a single or a few liver lobules. A liver lobule is the smallest repetitive functional and anatomical building block of liver, while (5) addresses a much larger organisational building block of the liver, a liver lobe that consists of thousands to hundreds of thousands of lobules depending on the species. A second strand (6), (7) addresses image analysis, which in most cases forms the entrance to modeling as it provides the data necessary to generate model hypotheses and to parameterize a model.

(1) Cell types: In a former work by Hoehme et. al. ( 113) a model of liver regeneration after drug-induced damage was established considering hepatocytes and blood vessels. This model has now been expanded to include all relevant cell types, including hepatocytes, blood vessels, hepatic stellate cells, Kupffer cells, invading macrophages and other immune cells. Thereby it is now possible to study perturbations in the temporal scenario of damage and regeneration after signaling events or cells types are knocked down individually or collectively. This model is currently compared to respective perturbation experiments. In addition, alternative mechanisms at the level of molecularly intermediated cell-cell communication discussed in the vast medical and biological literature have been implemented and are systematically assessed for their biological consequence at the tissue level. This permits an in-silico testing of alternative hypotheses contributing to a more efficient identification of informative future experiments.

(2) Liver disease: Degenerative liver diseases such as liver fibrosis and cirrhosis develop out of a disturbed balance of degenerative and regenerative processes. The model under (1) has thereby been extended by the formation of extracellular matrix, mimicked as fiber networks, to capture the disease process leading to liver fibrosis. In that process characteristic streets form that modify the mechanics, perfusion behavior and detoxification capacity of the liver. The model is now used to simulate disease pathways emerging from different administration schemes of drugs that are knowning to long-term lead to hepatocellular cancer.

(3) Consequence of liver fibrosis: Whole-slide scans from fibrotic liver in a mouse model has been analysed at different time points after emergence of the disease with regard to the degree of excess matrix to mimic the possible consequences of fibrotic inclusions on perfusion and function of liver within a multiscale model that considers ammonia detoxification in each individual hepatocyte as well as blood flow and transport processes in the liver lobule. This model has now be confronted on multimodal data in healthy liver, liver after a toxic dose of a drug, and fibrosis. The requirement to explain simultaneously all data sets in the same model imposes significant challenges for which solutions are currently explored.

(4) Bile flux: Bile flux has been for decades believed to be controlled by convection at the level of liver lobules as well as at the level of the entire organ. By a methodology based on correlative imaging for quantitative intravital flux analysis no directed advection was detectable in bile canaliculi at the resolution limit. Instead, after active transport across hepatocyte membranes bile salts within the liver lobules are transported in the canaliculi by a diffusion-dominated process. Only in the interlobular ducts i.e., at super-lobular level, diffusion is augmented by advection. In silico simulations of bile transport in real 3D bile network microarchitectures can quantitatively explain the data assuming diffusive transport as sole mechanism.

(5) Liver regeneration after partial hepatectomy (partial organ removal): Partial hepatectomy is an adequate therapy in case of diseases or events that destructed only part of the liver. A typical case is a primary tumor or a metastasis affecting only a single liver lobe. Within an biophysical agent-based model capturing many aspects of the cell mechanics we studied regrowth of liver after partial organ removal in mouse calibrated with multivariate experimental data. Our model predicts characteristic proliferation pattern that change from small animals (as mouse) to large animals (as pig).

(6) Bile duct ligation: Bile duct ligation (BDL) is an experimental procedure that mimics obstructive cholestatic disease. One of the early consequences of BDL in rodents is the appearance of so-called bile infarcts that correspond to Charcot-Gombault necrosis in human cholestasis. The mechanisms causing bile infarcts and their pathophysiological relevance are unclear. Therefore, intravital two photon–based imaging of BDL mice was performed with fluorescent bile salts (BS) and non-BS organic anion analogues. Key findings were followed up by matrix-assisted laser desorption ionization imaging, clinical chemistry, immunostaining, and gene expression analyses. Our group performed analysis of intravital imaging. The key finding is that bile microinfarcts occur in the acute phase after BDL in a limited number of dispersed hepatocytes followed by larger infarcts involving neighboring hepatocytes, and they allow leakage of bile from the BS-overloaded biliary tract into blood, thereby protecting the liver from BS toxicity; in the chronic phase after BDL, reduced sinusoidal BS uptake is a dominant protective factor, and the kidney contributes to the elimination of BS until cholemic nephropathy sets in 107.

(7) Periportalisation during liver fibrosis formation: Within a liver lobule, the function of hepatocytes is zonated i.e., certain functions are only executed by either hepatocytes close to the center (pericentral region) or hepatocytes in the periphery of the lobule (periportal region). Little is known about how liver fibrosis influences lobular zonation. To address this question, three mouse models of liver fibrosis were used, CCl4 administration repeated for 2, 6 and 12 months to induce pericentral damage, as well as bile duct ligation (21 days) and a particular mdr2-mouse model to study periportal fibrosis. Analyses were performed by RNA-sequencing, immunostaining of zonated proteins and image analysis. Image analysis was performed by our group. The key result was that liver fibrosis leads to strong alterations of lobular zonation, where the pericentral region adopts periportal features. Beside adverse consequences, periportalization supports adaptation to repeated doses of hepatotoxic compounds 108.

Toxicity extrapolation from in vitro to in vivo

In vivo toxicity prediction from in vitro data is a major objective in toxicology as it permits bypassing animal experiments, and as the predictive power of animal experiments for human is limited. Objective was the prediction of paracetamol (acetaminophen)-induced hepatotoxicity from in vitro experiments. For this purpose, numerous iterations between in vitro experiments, in vivo experiments and simulations were performed for mouse. Using a recent thesis (Géraldine Cellière'ns PhD thesis  86) as a start point, two candidate mechanisms could be identified both explaining the in vivo data after calibration of the in silico model with in vitro toxicity data.

Relating imaging on microscopic scales with imaging on macroscopic scales: From Diffusion-Weighted MRI Calibrated With Histological Data: an Example From Lung Cancer

Diffusion-weighted magnetic resonance imaging (DWI) is a key non-invasive imaging technique for cancer diagnosis and tumor treatment assessment, reflecting Brownian movement of water molecules in tissues. Since densely packed cells restrict molecule mobility, tumor tissues produce usually higher signal (less attenuated signal) on isotropic maps compared with normal tissues. However, no general quantitative relation between DWI data and the cell density has been established. In order to link low-resolution clinical cross-sectional data with high resolution histological information, we developed an image processing and analysis chain, which was used to study the correlation between the diffusion coefficient (D value) estimated from DWI and tumor cellularity from serial histological slides of a resected non-small cell lung cancer tumor. Color deconvolution followed by cell nuclei segmentation was performed on digitized histological images to determine local and cell-type specific 2d (two-dimensional) densities. From these, the 3d cell density was inferred by a model-based sampling technique, which is necessary for the calculation of local and global 3d tumor cell count. Next, DWI sequence information was overlaid with high resolution CT data and the resected histology using prominent anatomical hallmarks for co-registration of histology tissue blocks and non-invasive imaging modalities’ data. The integration of cell numbers information and DWI data derived from different tumor areas revealed a clear negative correlation between cell density and D value. Importantly, spatial tumor cell density can be calculated based on DWI data. In summary, our results demonstrate that tumor cell count and heterogeneity can be predicted from DWI data, which may open new opportunities for personalized diagnosis and therapy optimization  158. The work of that paper has been further advanced to adapt the procedures for clinical use (in preparation).

Collaborations

• Protein aggregation in amyloid diseases: Human Rezae's team at Inra Jouy-en-Josas (France) and W-F Xue's team in at university of Kent (Great Britain); Tom Banks at the North Carolina State University (USA) and Philippe Moireau (M3DISIM)
• Bacterial growth and division: Lydia Robert, Sorbonne Université (France)
• Liver research & toxicology: JG. Hengstler group (IfADo, Dortmund, Germany); R. Gebhardt (Univ. Leipzig); U. Klingmueller (DKFZ, Heidelberg); Irène Vignon-Clementel (INRIA, COMMEDIA)
• Growth in capsules and biomechanics: Pierre Nassoy, Institut dOptique Graduate School, Talence, France; Josef Kaes, Peter Debye Institute for Soft Matter Physics, Physics, Univ. Leipzig, Germany.
• Wound healing: (Adipose tissue regeneration) team of L. Casteilla (StromaLab, Toulouse). (Axolotl regeneration) team of O. Chara, SysBio group, Argentina.
• Diffusion of morphogen: Center for Computational and Integrative Biology, Rutgers University (Camden, New Jersey), joint work with Professor Nir Yakoby's Drosophila Laboratory
• Linking micro and macro-image information: Oliver Sedlaczek, Univ. and DKFZ Heidelberg, Kai Breuhahn, Univ. Heidelberg.

4.4 Applicative axis 3: Modelling and control in mathematical epidemiology

This axis is new and encompasses different works related to epidemiology, both for infectious and vector-borne diseases. The team was working since several years on the modeling, analysis and control of the propagation of vector-borne diseases such as dengue fever. Ordinary or partial differential equations of reaction-diffusion are used, and various (optimal or not) control strategies. In parallel and with the acknowledged opportunity of the onset and spreading of the Covid-19 pandemic, we expanded our interest to issues related to infectious diseases, using similar evolution systems.

Biological control of arboviroses

Sterile Insect Technique (SIT) 102 is a biological control method relying on massive releases of sterile male insects into the wild. The latter compete with wild males to mate with the females, and induce no offspring to the latter, thus reducing the next generation's population. This can result in a progressive reduction, or even disparition, of the target population.

A related technique is based on the infection by Wolbachia111. This symbiotic bacterium is maternally transmitted from infected females to their offspring, but induces cytoplasmic incompatibility149, 80: mating between infected males and uninfected females gives no offspring. Releases of Wolbachia infected males alone is thus comparable to classical SIT.

On the other hand, releasing both infected males and females in sufficient quantity may result in infection of the wild population. This gives rise to an interesting new control principle, as Wolbachia has been shown to severely reduce the insect vectorial ability to transmit dengue, zika or chikungunya, indirectly by lifespan and fertility reduction, and directly by reducing the ability of the viruses to proliferate within the organism 129.

We proposed new insights on the practical and theoretical issues raised by the implementation of the previous methods. Concerning the SIT, we obtained control synthesis results through impulsive periodic release of controlled amplitude 75, and through optimal control approach 76. Concerning Wolbachia technique, we investigated general control principles 6 capable of spreading the infection.

We also considered the effects of hindrances to these strategies 20, 48.

Mathematical epidemiology of infectious diseases

The current outbreak of Covid-19 resulted in the appearance of many novel experiences at individual and collective, biological and social, national and international levels, making this pandemic a full epistemological experience as well. Motivated by the great number of questions raised by this global event, some members of the team devoted part of their time to exploring more or less closely related scientific issues. One should notice however that this evolution constitutes indeed the continuation of a movement already initiated previously, and only accelerated by the current events.

The issues raised by the effective implementation of the social distancing measures largely implemented on the Earth’s surface during the whole year 2020, have been the focus of intense reflection. We contributed to this debate by studying optimal control policies aiming at reducing the total number of infected people during the whole epidemic outbreak, the so-called epidemic final size. In another research line, we established the equation fulfilled by the epidemic final size for a fully general SEIR model in a heterogenous population characterized by some trait in a discrete or continuous subset, and studied the uniqueness of its solution. This allowed to extend the use and meaningfulness of the classical concept of next-generation operator introduced by O. Diekmann et al.  in 1990 92. Last, in cooperation with the Inria team NeCS (Inria Grenoble-Rhône-Alpes), we studied in a control theory perspective the effects of the testing policies in the dynamics and in the control of the epidemic.

Collaborations

• Biological control of arboviroses: Nicolas Vauchelet (Université Paris 13); Yannick Privat (Université de Strasbourg); D. Villela, C. Struchiner (Fiocruz, Brazil); Jorge Zubelli (IMPA, Brazil); Alain Rapaport (INRA-Montpellier), Y. Dumont (CIRAD-Montpellier); Ch. Schaerer, P. Pérez-Estigarribia (UNA, Paraguay), O. Vasilieva (Universidad del Valle, Cali, Colombia), D. Cardona-Salgado (Universidad Autónoma de Occidente, Cali, Colombia); Hervé Bossin (ILM, Papeete); René Gato and Misladys Rodriguez (Inst. Pedro Kouri, La Havane)
• Mathematical epidemiology of infectious diseases: Nicolas Vauchelet (Université Paris 13); Michel Duprez (Inria Nancy - Grand Est); Yannick Privat (Université de Strasbourg); Carlos Canudas de Wit (Inria Grenoble - Rhône-Alpes and CNRS); Alain Kibangou (Université Grenoble-Alpes).

6.1 New software

6.2 New platforms

7.1 Direct and inverse Problems in Structured-population equations

The many results obtained during the last years have oriented us towards new research directions in the wide field of structured population equations: the study of the direct and inverse problem in the newly-proposed "adder model"  154; oscillatory behaviours of such equations; the study of models for heterogeneous aggregation, i.e. where the aggregates are formed out of several monomeric species.

7.2 Stochastic Models of Biological Systems

7.3 Analysis and control of populations of mosquitoes

7.4 Modelling and control in epidemiology

7.5 Analysis and numerics for mechanical models of tumor growth

Several class of models have been devised to describe the macroscopic dynamics of growing tumors, depending on the mechanical behaviour of the tissue. The team has progressed on several aspects: analysis of models and asymptotic analysis towards free boundary problem, numerical methods compatible with energy properties.

New directions of research have been initiated for the analysis of PDE models arising in biology. Firstly, in contact with CNR in Roma, we have studied the use of the Kedem-Katchalsky conditions to represent the effect of a membrane, the analysis is presented in 10 based on Pierre's method for parabolic systems as improved in 18. Secondly, because living tissues can be seen as multiphasic mixtures (different cells have different mechanical properties), the degenerate and singular Cahn-Hilliard equation is widely used in the domain to represent the proportion of cancer cells. In 21, we proposed a relaxation approach, compatible for the energy decay, which allows to reduce this fourth order equation to a system of two second order equations and thus to use standard finite elements softwares. It is also possible to keep two cell densities rather than a proportion and one arrives to systems of porous media equations (cells move in anextra-cellular matrix with properties well describes by a porous media model). In this context, the pressure law is stiif and it is relevant to study the limiting free boundary problem. The first result in this direction is derived in 9. When the nutrients are included in the model, specific theoretical difficulties arise and this is treated in 38 thanks to a new remarkable estimate on the porous media equation.

From the numerical side, to preserve energy properties, the Scalar Auxiliary Variable method is very efficient and has been proposed by J. Chen and his collaborators a few years ago. We have used this method in the context of chemotaxis 49, and extended to the nonlinear Schroedinger equation in 50.

7.6 Focus on cancer

7.7 Single Cell-based Modeling, biomechanics, Liver regeneration, and liver function

7.8 The role of actin protrusion dynamics in cell migration through a degradable viscoelastic extracellular matrix: a computational model

Actin protrusion dynamics plays an important role in the regulation of three-dimensional (3D) cell migration 15. We present a computational model of cell migration through a degradable viscoelastic ECM. The cell is modeled as an active deformable object that captures the viscoelastic behavior of the actin cortex and the subcellular processes underlying 3D cell migration. The ECM is regarded as a viscoelastic material, with or without anisotropy due to fibrillar strain stiffening, and modeled by means of the meshless Lagrangian smoothed particle hydrodynamics (SPH) method. ECM degradation is captured by local fluidization of the material and permits cell migration through the ECM. By simulations, we demonstrate that changes in ECM stiffness and cell strength affect cell migration and are accompanied by changes in number, lifetime and length of protrusions.

7.9 Macroscopic limit and control of collective dynamics

8.1 Bilateral grants with industry

9.1 International initiatives

9.2 National initiatives

Mamba (Marie Doumic and Philippe Robert) participates to the GDR "MeDyna" (mechanisms and dynamics of assemblies of peptides and proteins), coordinated by Stéphane Bressanelli from IBPC.

10.1 Promoting scientific activities

10.2 Teaching - Supervision - Juries

10.3 Popularization

• Emma Leschiera and her advisor Chloé Audebert participated in a speed meeting with students from a middle school in Nanterre.
• In a book chapter 57 (in French) two subfields of in-silico biology is described: QSAR modelling and the modelling of virtual organs down to micro-architectural level as a complement to animal experimentation, because they enable experiments to be interpreted, guide the optimisation of their design, and facilitate the extrapolation from in vitro to in vivo toxicity. The chapter positions the tools from these two disciplines in the context of the Adverse Outcome Pathways (AOP) which is a recently established toxicological construct that connects, in a formalized, transparent and quality-controlled way, mechanistic information to apical endpoints for regulatory purposes.

11.1 Major publications

• 1 unpublishedL. Almeida, P.-A. Bliman, G. Nadin, B. Perthame and N. Vauchelet. Final size and convergence rate for an epidemic in heterogeneous populationOctober 2020, working paper or preprint
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• 2 articleM. Doumic, S. Hecht and D. Peurichard. A purely mechanical model with asymmetric features for early morphogenesis of rod-shaped bacteria micro-colonyMathematical Biosciences and Engineering176October 2020, http://aimspress.com/article/doi/10.3934/mbe.2020356
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11.2 Publications of the year

11.3 Other

11.4 Cited publications

• 58 articleP. Aceves-Sanchez, B. Aymard, D. Peurichard, P. Kennel, A. Lorsignol, F. Plouraboue, L. Casteilla and P. Degond. A new model for the emergence of blood capillary networksNetworks and Heterogeneous Media162021, 91-138
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• 59 incollectionL. Almeida, R. Chisholm, J. Clairambault, T. Lorenzi, A. Lorz, C. Pouchol and E. Trélat. Why Is Evolution Important in Cancer and What Mathematics Should Be Used to Treat Cancer? Focus on Drug ResistanceTrends in Biomathematics: Modeling, Optimization and Computational Problems: Selected works from the BIOMAT Consortium Lectures, Moscow 2017Springer International PublishingAugust 2018, 107-120
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• 60 articleL. Almeida, Y. Privat, M. Strugarek and N. Vauchelet. Optimal releases for population replacement strategies: application to wolbachiaSIAM Journal on Mathematical Analysis5142019, 3170--3194
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• 61 article L. Alphey. Genetic control of mosquitoes Annual review of entomology 59 2014
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• 62 article A. Armiento, M. Doumic, P. Moireau and H. Rezaei. Estimation from Moments Measurements for Amyloid Depolymerisation Journal of Theoretical Biology March 2016
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• 63 phdthesis A. Armiento. Inverse problems and data assimilation methods applied to protein polymerisation Université Paris 7 - Diderot January 2017
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• 64 incollectionJ. Avila Alonso, C. Bonnet, J. Clairambault, H. Ozbay, S.-I. Niculescu, F. Merhi, A. Ballesta, R. Tang and J.-P. Marie. Analysis of a New Model of Cell Population Dynamics in Acute Myeloid LeukemiaDelay Systems : From Theory to Numerics and Applications1Advances in Delays and DynamicsSpringerJanuary 2014, 315-328
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• 65 inproceedings J. Avila Alonso, C. Bonnet, E. Fridman, F. Mazenc and J. Clairambault. Stability analysis of PDE's modelling cell dynamics in Acute Myeloid Leukemia 53rd IEEE Conference on Decision and Control Los Angeles, United States December 2014
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• 66 inproceedings J. Avila Alonso, C. Bonnet, H. Ozbay, J. Clairambault and S.-I. Niculescu. A coupled model for healthy and cancer cells dynamics in Acute Myeloid Leukemia The 19th World Congress of the International Federation of Automatic Control Cape Town, Souh Africa August 2014
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• 67 inproceedings J. Avila Alonso, C. Bonnet, H. Ozbay, J. Clairambault and S.-I. Niculescu. A discrete-maturity Interconnected Model of Healthy and Cancer Cell Dynamics in Acute Myeloid Leukemia Mathematical Theory of Networks and Systems Groningen, Netherlands July 2014
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• 68 articleH. Banks, M. Doumic, C. Kruse, S. Prigent and H. Rezaei. Information Content in Data Sets for a Nucleated-Polymerization ModelJournal of Biological Dynamics91June 2015, 26
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• 69 articleH. Banks, M. Doumic-Jauffret and C. Kruse. A numerical scheme for the early steps of nucleation-aggregation ModelsJournal of Mathematical Biology741-2January 2017, 259-287
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• 70 articleF. Bekkal Brikci, J. Clairambault and B. Perthame. Analysis of a molecular structured population model with possible polynomial growth for the cell division cycleMath. Comput. Modelling477-82008, 699--713
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• 71 articleE. Bernard, M. Doumic and P. Gabriel. Cyclic asymptotic behaviour of a population reproducing by fission into two equal partsKinetic and Related Models 123June 2019, 551-571
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• 72 articleF. Bertaux, S. Stoma, D. Drasdo and G. Batt. Modeling Dynamics of Cell-to-Cell Variability in TRAIL-Induced Apoptosis Explains Fractional Killing and Predicts Reversible ResistancePLoS Computational Biology10102014, 14
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• 73 article J. Bertoin and A. Watson. Probabilistic aspects of critical growth-fragmentation equations Advances in Applied Probability 9 2015
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• 74 article P.-A. Bliman, M. Aronna, F. Coelho and M. da Silva. Ensuring successful introduction of Wolbachia in natural populations of Aedes aegypti by means of feedback control Journal of Mathematical Biology August 2017
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• 75 articleP.-A. Bliman, D. Cardona-Salgado, Y. Dumont and O. Vasilieva. Implementation of control strategies for sterile insect techniquesMathematical biosciences3142019, 43--60
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• 76 article P.-A. Bliman, D. Cardona-Salgado, Y. Dumont and O. Vasilieva. Optimal control approach for implementation of sterile insect techniques arXiv preprint arXiv:1911.00034 2019
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• 77 inproceedings C. Bonnet, J. Avila Alonso, H. Ozbay, J. Clairambault, S.-I. Niculescu and P. Hirsch. A Discrete-Maturity Interconnected Model of Healthy and Cancer Cell Dynamics in Acute Myeloid Leukemia The 10th AIMS Conference on Dynamical Systems,Differential Equations and Applications Madrid, Spain July 2014
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• 78 articleT. Bourgeron, M. Doumic and M. Escobedo. Estimating the division rate of the growth-fragmentation equation with a self-similar kernelInverse Problems302Jan 2014, 025007URL: http://dx.doi.org/10.1088/0266-5611/30/2/025007
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• 79 articleT. Bourgeron, Z. Xu, M. Doumic and M. Teixeira. The asymmetry of telomere replication contributes to replicative senescence heterogeneityScientific Reports5October 2015, 15326
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• 80 incollection K. Bourtzis. -Based Technologies for Insect Pest Population Control' Advances in Experimental Medicine and Biology 627 Springer, New York, NY 02 2008
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• 81 articleA. Brown. Insecticide resistance in mosquitoes: a pragmatic review.Journal of the American Mosquito Control Association221986, 123--140
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• 82 articleM. Caceres and B. Perthame. Beyond blow-up in excitatory integrate and fire neuronal networks: refractory period and spontaneous activityJournal of Theoretical Biology3502014, 81-89
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• 83 articleV. Calvez, M. Doumic and P. Gabriel. Self-similarity in a general aggregation--fragmentation problem. Application to fitness analysisJournal de Mathématiques Pures et Appliquées9812012, 1 - 27URL: http://www.sciencedirect.com/science/article/pii/S002178241200013X
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• 84 articleV. Calvez, N. Lenuzza, M. Doumic, J.-P. Deslys, F. Mouthon and B. Perthame. Prion dynamic with size dependency - strain phenomenaJ. of Biol. Dyn.412010, 28--42
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• 85 articleJ. Carrillo, F. James, F. Lagoutière and N. Vauchelet. The Filippov characteristic flow for the aggregation equation with mildly singular potentialsJournal of Differential Equations260133 pages2016, 304-338
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• 86 phdthesis G. Cellière. Multi-scale modeling of hepatic drug toxicity and its consequences on ammonia detoxification Université Paris 6 - Pierre et Marie Curie July 2017
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• 87 articleJ. Chevallier, M. Caceres, M. Doumic and P. Reynaud-Bouret. Microscopic approach of a time elapsed neural modelMathematical Models and Methods in Applied SciencesDecember 2015, 2669
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• 88 articleR. Chisholm, T. Lorenzi and J. Clairambault. Cell population heterogeneity and evolution towards drug resistance in cancer: Biological and mathematical assessment, theoretical treatment optimisationBBA - General Subjects1860June 2016, 2627 - 2645
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• 89 articleR. Chisholm, T. Lorenzi, A. Lorz, A. Larsen, L. de Almeida, A. Escargueil and J. Clairambault. Emergence of Drug Tolerance in Cancer Cell Populations: An Evolutionary Outcome of Selection, Nongenetic Instability, and Stress-Induced AdaptationCancer Research756March 2015, 930-939
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• 90 articleJ. Clairambault. An evolutionary perspective on cancer, with applications to anticancer drug resistance modelling and perspectives in therapeutic controlJournal of Mathematical Study2019, 1 - 21
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• 91 articleJ. Clairambault and C. Pouchol. A survey of adaptive cell population dynamics models of emergence of drug resistance in cancer, and open questions about evolution and cancerBIOMATH81Copyright: 2019 Clairambault et al. This article is distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.May 2019, 23
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• 92 articleO. Diekmann, J. Heesterbeek and J. Metz. On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populationsJournal of mathematical biology2841990, 365--382
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• 93 inproceedings W. Djema, F. Mazenc, C. Bonnet, J. Clairambault, P. Hirsch and F. Delhommeau. Stability of a Delay System Coupled to a Differential-Difference System Describing the Coexistence of Ordinary and Mutated Hematopoietic Stem Cells Conference on Decision and Control Las Vegas, United States December 2016
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• 94 article M. Doumic, M. Escobedo and M. Tournus. Estimating the division rate and kernel in the fragmentation equation Annales de l'Institut Henri Poincaré (C) Non Linear Analysis https://arxiv.org/abs/1804.08945 2018
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• 95 articleM. Doumic, S. Eugene and P. Robert. Asymptotics of Stochastic Protein Assembly ModelsSIAM Journal on Applied Mathematics766November 2016, 20
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• 96 articleM. Doumic, K. Fellner, M. Mezache and H. Rezaei. A bi-monomeric, nonlinear Becker--Döring-type system to capture oscillatory aggregation kinetics in prion dynamicsJournal of theoretical biology4802019, 241--261
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• 97 articleM. Doumic and P. Gabriel. Eigenelements of a General Aggregation-Fragmentation ModelMathematical Models and Methods in Applied Sciences205May 2010, 757-783URL: https://hal.archives-ouvertes.fr/hal-00408088
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• 98 unpublishedM. Doumic, M. Hoffmann, N. Krell and L. Robert. Statistical estimation of a growth-fragmentation model observed on a genealogical treeOctober 2012, 46 pages, 4 figures
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• 99 articleM. Doumic, B. Perthame and J. Zubelli. Numerical Solution of an Inverse Problem in Size-Structured Population DynamicsInverse Problems2542009, 045008
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• 100 incollectionD. Drasdo, A. Buttenschön and P. Van Liedekerke. Agent-Based Lattice Models of Multicellular SystemsNumerical Methods and Advanced Simulation in Biomechanics and Biological ProcessesElsevier2018, 223-238
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• 101 articleD. Drasdo, S. Hoehme and J. Hengstler. How predictive quantitative modeling of tissue organization can inform liver disease pathogenesisJournal of Hepatology614October 2014, 951--956
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• 102 bookV. Dyck, J. Hendrichs and A. Robinson. The Sterile Insect Technique, Principles and Practice in Area-Wide Integrated Pest ManagementSpringer, Dordrecht2006, 787
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• 103 phdthesis S. Eugene. Stochastic modelling in molecular biology: a probabilistic analysis of protein polymerisation and telomere shortening UPMC LJLL September 2016
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• 104 articleS. Eugene, W.-F. Xue, P. Robert and M. Doumic-Jauffret. Insights into the variability of nucleated amyloid polymerization by a minimalistic model of stochastic protein assemblyJournal of Chemical Physics14417May 2016, 12
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• 105 article P. Gabriel and H. Martin. Steady distribution of the incremental model for bacteria proliferation arXiv preprint arXiv:1803.04950 2018
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• 106 articleA. Ghallab, G. Cellière, S. Henkel, D. Driesch, S. Hoehme, U. Hofmann, S. Zellmer, P. Godoy, A. Sachinidis, M. Blaszkewicz, R. Reif, R. Marchan, L. Kuepfer, D. Häussinger, D. Drasdo, G. Gebhardt and J. Hengstler. Model-guided identification of a therapeutic strategy to reduce hyperammonemia in liver diseasesJournal of Hepatology644November 2015, 860--871
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• 107 article A. Ghallab, U. Hofmann, S. Sezgin, N. Vartak, R. Hassan, A. Zaza, P. Godoy, K. Schneider, G. Guenther, Y. Ahmed, A. Abbas, V. Keitel, L. Kuepfer, S. Dooley, F. Lammert, C. Trautwein, M. Spiteller, D. Drasdo, A. Hofmann, P. Jansen, J. Hengstler and R. Reif. Bile Microinfarcts in Cholestasis Are Initiated by Rupture of the Apical Hepatocyte Membrane and Cause Shunting of Bile to Sinusoidal Blood Hepatology August 2018
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• 108 articleA. Ghallab, M. Myllys, C. H Holland, A. Zaza, W. Murad, R. Hassan, Y. A Ahmed, T. Abbas, E. A Abdelrahim, K. Schneider and others. Influence of liver fibrosis on lobular zonationCells8122019, 1556
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• 109 article L. Gosse and N. Vauchelet. Hydrodynamic singular regimes in 1+1 kinetic models and spectral numerical methods Journal of Mathematical Analysis and Applications 2016
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• 110 articleJ. Hemingway and H. Ranson. Insecticide resistance in insect vectors of human diseaseAnnual review of entomology4512000, 371--391
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• 111 articleM. Hertig and S. Wolbach. Studies on rickettsia-like micro-organisms in insectsThe Journal of medical research4431924, 329
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• 112 phdthesis V. Hoang. Adaptive estimation for inverse problems with applications to cell divisions Université de Lille 1 -- Sciences et Technologies November 2016
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• 113 articleS. Hoehme, M. Brulport, A. Bauer, E. Bedawy, W. Schormann, M. Hermes, V. Puppe, R. Gebhardt, S. Zellmer, M. Schwarz, E. Bockamp, T. Timmel, J. Hengstler and D. Drasdo. Prediction and validation of cell alignment along microvessels as order principle to restore tissue architecture in liver regenerationProceedings of the National Academy of Sciences107232010, 10371--10376
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• 114 article M. Hoffmann and A. Olivier. Nonparametric estimation of the division rate of an age dependent branching process Stochastic Processes and their Applications December 2015
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• 115 articleN. Jagiella, B. Müller, M. Müller, I. Vignon-Clementel and D. Drasdo. Inferring Growth Control Mechanisms in Growing Multi-cellular Spheroids of NSCLC Cells from Spatial-Temporal Image DataPLoS Computational Biology1222016, e1004412
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• 116 articleF. James and N. Vauchelet. Equivalence between duality and gradient flow solutions for one-dimensional aggregation equationsDiscrete and Continuous Dynamical Systems - Series A3632016, 1355-1382
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• 117 articleF. James and N. Vauchelet. Numerical methods for one-dimensional aggregation equationsSIAM Journal on Numerical Analysis5322015, 895-916
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• 118 articleM.-J. Kang, B. Perthame and D. Salort. Dynamics of time elapsed inhomogeneous neuron network modelComptes Rendus Mathématique353September 2015, 1111-1115
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• 119 articleJ. Kean, S. Rainey, M. McFarlane, C. Donald, E. Schnettler, A. Kohl and E. Pondeville. Fighting arbovirus transmission: natural and engineered control of vector competence in Aedes mosquitoesInsects612015, 236--278
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• 120 articleI. Kim, B. Perthame and P. Souganidis. Free boundary problems for tumor growth: a viscosity solutions approachNonlinear Analysis: Theory, Methods and Applications1382016, 207-228
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• 121 articleM. Kolwalczyk, B. Perthame and N. Vauchelet. Transversal instability for the thermodiffusive reaction-diffusion systemChinese Annals of Mathematics - Series B36513 pages2015, 871-882
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• 122 unpublishedP. Liedekerke, J. Neitsch, T. Johann, E. Warmt, I. Gonzàlez-Valverde, S. Hoehme, S. Grosser, J. Kaes and D. Drasdo. A quantitative high resolution computational mechanics cell model for growing and regenerating tissuesDecember 2018, working paper or preprint
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• 123 articleT. Lorenzi, R. Chisholm and J. Clairambault. Tracking the evolution of cancer cell populations through the mathematical lens of phenotype-structured equationsBiology Direct111December 2016, 43
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• 124 articleT. Lorenzi, R. Chisholm, L. Desvillettes and B. Hughes. Dissecting the dynamics of epigenetic changes in phenotype-structured populations exposed to fluctuating environmentsJournal of Theoretical Biology386September 2015, 166-176
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• 125 misc T. Lorenzi, R. Chisholm and A. Lorz. Effects of an advection term in nonlocal Lotka-Volterra equations December 2015
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• 126 articleA. Lorz, T. Lorenzi, J. Clairambault, A. Escargueil and B. Perthame. Modeling the effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumorsBulletin of Mathematical Biology771January 2015, 1-22
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• 127 articleA. Lorz, T. Lorenzi, M. Hochberg, J. Clairambault and B. Perthame. Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapiesESAIM: Mathematical Modelling and Numerical AnalysisMarch 2013, 23
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• 128 unpublishedA. Mellet, B. Perthame and F. Quiros. A Hele-Shaw Problem for Tumor GrowthDecember 2015, working paper or preprint
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• 129 articleL. Moreira, I. Iturbe-Ormaetxe, J. Jeffery, G. Lu, A. Pyke, L. Hedges, B. Rocha, S. Hall-Mendelin, A. Day, M. Riegler, L. Hugo, K. Johnson, B. Kay, E. McGraw, A. van den Hurk, P. Ryan and S. O'Neill. A Symbiont in aegypti Limits Infection with Dengue, Chikungunya, and Plasmodium'Cell13972009, 1268 - 1278
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• 130 articleT. Nguyen, J. Clairambault, T. Jaffredo, B. Perthame and D. Salort. Adaptive dynamics of hematopoietic stem cells and their supporting stroma: A model and mathematical analysisMathematical Biosciences and Engineering1605May 2019, 4818--4845
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• 131 articleA. Olivier. How does variability in cells aging and growth rates influence the malthus parameter?Kinetic and Related Models 102June 2017, 481-512
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• 132 phdthesis A. Olivier. Statistical analysis of growth-fragmentation models Université Paris Dauphine - Paris IX November 2015
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• 133 articleK. Pakdaman, B. Perthame and D. Salort. Adaptation and Fatigue Model for Neuron Networks and Large Time Asymptotics in a Nonlinear Fragmentation EquationJournal of Mathematical Neuroscience412014, 14
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• 134 unpublishedB. Perthame, F. Quirós, M. Tang and N. Vauchelet. Derivation of a Hele-Shaw type system from a cell model with active motionJuly 2013,
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• 135 articleB. Perthame, F. Quirós and J.-L. Vázquez. The Hele-Shaw asymptotics for mechanical models of tumor growthArchive for Rational Mechanics and Analysis2122014, 93-127
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• 136 articleB. Perthame and D. Salort. On a voltage-conductance kinetic system for integrate and fire neural networksKinetic and Related Models 64December 2013, 841-864
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• 137 articleB. Perthame, D. Salort and G. Wainrib. Distributed synaptic weights in a LIF neural network and learning rulesPhysica D: Nonlinear Phenomena353-3542017, 20-30
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• 138 articleB. Perthame, M. Tang and N. Vauchelet. Traveling wave solution of the Hele-Shaw model of tumor growth with nutrientMathematical Models and Methods in Applied Sciences241325 pages2014, 2601-2626
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• 139 bookB. Perthame. Transport equations in biologyFrontiers in MathematicsBaselBirkhäuser Verlag2007, x+198
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• 140 articleB. Perthame and N. Vauchelet. Incompressible limit of mechanical model of tumor growth with viscosityPhilosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences (1934--1990)37316 pages2015, 20140283
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• 141 articleB. Perthame and J. Zubelli. On the inverse problem for a size-structured population modelInverse Problems2332007, 1037--1052
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• 142 articleD. Peurichard. Extra-cellular matrix rigidity may dictate the fate of injury outcomeJ. Theor Biol4692019, 127-136
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• 143 articleD. Peurichard. Simple mechanical cues could explain adipose tissue morphologyJ. Theor Biol2017, URL: https://doi.org/10.1016/j.jtbi.2017.06.030
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• 144 unpublishedC. Pouchol, J. Clairambault, A. Lorz and E. Trélat. Asymptotic analysis and optimal control of an integro-differential system modelling healthy and cancer cells exposed to chemotherapyDecember 2016, accepted in the J. Math. Pures et App.
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• 145 articleS. Prigent, A. Ballesta, F. Charles, N. Lenuzza, P. Gabriel, L. Tine, H. Rezaei and M. Doumic. An efficient kinetic model for assemblies of amyloid fibrils and its application to polyglutamine aggregation.PLoS ONE7112012, e43273
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• 146 articleL. Robert, M. Hoffmann, N. Krell, S. Aymerich, J. Robert and M. Doumic. Division in Escherichia coli is triggered by a size-sensing rather than a timing mechanismBMC Biology1212014, 17
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• 147 articleF. Schliess, S. Hoehme, S. Henkel, A. Ghallab, D. Driesch, J. Böttger, R. Guthke, M. Pfaff, J. Hengstler, R. Gebhardt, D. Häussinger, D. Drasdo and S. Zellmer. Integrated metabolic spatial-temporal model for the prediction of ammonia detoxification during liver damage and regenerationHepatology606December 2014, 2040--2051
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• 148 articleN. Sfakianakis, D. Peurichard, A. Brunk and C. Schmeiser. Modelling cell-cell collision and adhesion with the Filament Based Lamellipodium ModelBIOMATH2019, URL: http://dx.doi.org/10.11145/j.biomath.2018.11.097
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• 149 articleS. Sinkins. and cytoplasmic incompatibility in mosquitoes'Insect Biochemistry and Molecular Biology347Molecular and population biology of mosquitoes2004, 723 - 729
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• 150 unpublishedM. Strugarek, H. Bossin and Y. Dumont. On the use of the sterile insect technique or the incompatible insect technique to reduce or eliminate mosquito populationsMay 2018, https://arxiv.org/abs/1805.10150 - working paper or preprint
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• 151 unpublishedM. Strugarek, L. Dufour, N. Vauchelet, L. Almeida, B. Perthame and D. Villela. Oscillatory regimes in a mosquito population model with larval feedback on egg hatchingJanuary 2018, https://arxiv.org/abs/1801.03701 - working paper or preprint
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• 152 unpublishedM. Strugarek and H. Ji. Sharp seasonal threshold property for cooperative population dynamics with concave nonlinearitiesApril 2018, https://arxiv.org/abs/1804.07641 - working paper or preprint
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• 153 phdthesis M. Strugarek. Mathematical modeling of population dynamics, applications to vector control of Aedes spp. (Diptera:Culicidae) Sorbonne Université , UPMC September 2018
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• 154 article S. Taheri-Araghi, S. Bradde, J. Sauls, N. Hill, P. Levin, J. Paulsson, M. Vergassola and S. Jun. Cell-Size Control and Homeostasis in Bacteria Current Biology 11679 1-7 2015
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• 155 articleP. Van Liedekerke, J. Neitsch, T. Johann, K. Alessandri, P. Nassoy and D. Drasdo. Quantitative agent-based modeling reveals mechanical stress response of growing tumor spheroids is predictable over various growth conditions and cell linesPLoS computational biology1532019, e1006273
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• 156 unpublishedP. Van Liedekerke, J. Neitsch, T. Johann, K. Alessandri, P. Nassoy and D. Drasdo. Quantitative modeling identifies robust predictable stress response of growing CT26 tumor spheroids under variable conditionsDecember 2016, working paper or preprint
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• 157 articleT. Walker and S. Sinkins. Biological control of arbovirus vectorsArboviruses: Molecular Biology, Evolution and Control. Caister Academic Press, Norfolk, UK2016, 291--302
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• 158 article Y. Yin, O. Sedlaczek, B. Müller, A. Warth, M. González-Vallinas, B. Lahrmann, N. Grabe, H.-U. Kauczor, K. Breuhahn, I. Vignon-Clementel and D. Drasdo. Tumor cell load and heterogeneity estimation from diffusion-weighted MRI calibrated with histological data: an example from lung cancer IEEE Transactions on Medical Imaging 2017
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• 159 articleM. de Langlard, F. Lamadie, S. Charton and J. Debayle. Bayesian Inference of a Parametric Random Spheroid from its Orthogonal ProjectionsMethodology and Computing in Applied Probability2020, 1--19
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