A few advances in biological modeling and asymptotic analysis

Clair Poignard 1
1 MC2 - Modélisation, contrôle et calcul
Inria Bordeaux - Sud-Ouest, UB - Université de Bordeaux, CNRS - Centre National de la Recherche Scientifique : UMR5251
Abstract : This thesis consists of a synthetized presentation of my research in order to get the French diploma “Habilitation à diriger des recherches”. It is organized into four chapters that constitute the four main topics I focused on since I got my permanent position at Inria, in September 2008. These research axes have been developed within the framework of the Inria team MC2, leaded by T. Colin. This thesis is the result of collaborations with colleagues of Bordeaux and elsewhere (Karlsruhe, Lyon, Rennes, Villejuif) as well as with Phd students and postdoctoral fellows I co-supervised. Therefore I choose to use we to present the results. Chapter 1 is devoted to cell electropermeabilization modeling. Electropermeabilization (also called electroporation) is a significant increase in the electrical conductivity and permeability of cell membrane that occurs when pulses of large amplitude (a few hundred volts per centimeter) are applied to the cells: due to the electric field, the cell membrane is permeabilized, and then nonpermeant molecules can easily enter the cell cytoplasm by transport (active and passive) through the electropermeabilized membranes. If the pulses are too long, too numerous or if their amplitude is too high, the cell membrane is irreversibly destroyed and the cells are killed. However, if the pulse duration is sufficiently short (a few milliseconds or a few microseconds, depending on the pulse amplitude), the cell membrane reseals within several tens of minutes: such a reversible electroporation preserves the cell viability and is used in electrochemotherapy to vectorize the drugs into cancer cells. In Chapter 1, I present the modeling we derived in tight collaboration with biologists, namely the L.M. Mir’s group at the IGR, which is one of the world’s leader in this field, as well as the numerical schemes and the comparisons of the numerical simulations with the experimental data. Interestingly, our modeling that uncouples electric and permeable behaviours of the cell membrane makes it possible to explain the strange observation of cell desensitization, that has been reported very recently by A. Silve et al. [100]. This desensitization consists of a less degree of cell permeabilization after a few successive electric pulses than for the same number of pulses but with a delay between each electric pulse delivery. This phenomenon is counter-intuitive and was not predictable by the previous models of the literature. Chapter II is devoted to cell migration modeling and more precisely to the endothelial cell migration on micropatterned polymers. The goal is to provide models based on the experimental data in order to describe the cell migration on micropatterned polymers. The long–term goal is to provide tools for the optimization of such a migration, which is crucial in tissue engineering. We develop a continuous model of Patlak-KellerSegel type, which makes it possible to provide qualitative results in accordance with the experiments, and we analyse the mathematical properties of this model. Then, we provide an agent-based model, based on a classical mechanics approach. Strikingly, this very simple model has been quantitatively fitted with the experimental data provided by our colleagues of the biological institute IECB, in terms of cell orientation and cell migration. I conclude the chapter by on-going works on the invadopodia modeling, in collaboration with T. Suzuki from Osaka University and M. Ohta from Tokyo University of Sciences. Chapter III is devoted to a very recent activity I started in 2013 on tumor growth models, therefore this chapter is based on only one submitted preprint. I present the results on ductal carcinoma growth modeling. Originally confined to the milk duct, these breast cancers may become invasive and agressive after the degradation of the duct membrane, and the main features of our model is to describe the membrane degradation thanks to a non-linear Kedem–Katchalsky condition that describes the jump of pressure across the duct membrane. More precisely, the membrane permeability is given as a non-linear function of specific enzymes (MMPs) that degrade the membrane. We also provide some possible explanation of heterogeneity of tumor growth by modeling the influence of the micro-environment and the emergence of specific cell types. I eventually conclude by Chapter IV, which consists of a few advances in asymptotics analysis for domains that are singular or asymptotically singular, in the following of my PhD thesis. The results can be split into two parts: first I present approximate transmission conditions through a periodically rough thin layer, and how we characterize the influence of such a layer on the polarization tensor in the sense of Capdeboscq and Voeglius [19]. Then I focus on the numerical treatment of the eddy current problem in domains with corner singularity. Each chapter is organized into a description of the results, a few perspectives for forthcoming research and a list of the published papers related to the topic of the chapter. Before presenting the results we obtained, I give in the next part a brief summary in French
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Analysis of PDEs [math.AP]. Universite de Bordeaux, 2014
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Clair Poignard. A few advances in biological modeling and asymptotic analysis. Analysis of PDEs [math.AP]. Universite de Bordeaux, 2014. 〈tel-01113037〉

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