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Section: New Results
Non conservative transport equations for cell population dynamics
Dimensional reduction of a multiscale model based on long time asymptotics
Participants : Frédérique Clément, Frédéric Coquel [CMAP] , Marie Postel, Kim Long Tran.
We have finalized the study on the dimensional reduction of our multiscale model of terminal follicle development, which has now been published [17]. We have considered a class of kinetic models for which a moment equation has a natural interpretation. We have shown that, depending on their velocity field, some models lead to moment equations that enable one to compute monokinetic solutions economically. We have detailed the example of a multiscale structured cell population model, consisting of a system of 2D transport equations. The reduced model, a system of 1D transport equations, is obtained from computing the moments of the 2D model with respect to one variable. The 1D solution is defined from the solution of the 2D model starting from an initial condition that is a Dirac mass in the direction removed by reduction. For arbitrary initial conditions, we have compared 1D and 2D model solutions in asymptotically large time. Finite volume numerical approximations of the 1D reduced model can be used to compute the moments of the 2D solution with proper accuracy, both in the conservative and non conservative framework. The numerical robustness is studied in the scalar case, and a full scale vector case is presented.
Analysis and calibration of a linear model for structured cell populations with unidirectional motion : application to the morphogenesis of ovarian follicles
Participants : Frédérique Clément, Frédérique Robin, Romain Yvinec [INRA] .
We have analyzed a multi-type age dependent model for cell populations subject to unidirectional motion, in both a stochastic and deterministic framework [23]. Cells are distributed into successive layers; they may divide and move irreversibly from one layer to the next. We have adapted results on the large-time convergence of PDE systems and branching processes to our context, where the Perron-Frobenius or Krein-Rutman theorem can not be applied. We have derived explicit analytical formulas for the asymptotic cell number moments, and the stable age distribution. We have illustrated these results numerically and we have applied them to the study of the morphodynamics of ovarian follicles. We have proven the structural parameter identifiability of our model in the case of age independent division rates. Using a set of experimental biological data, we have estimated the model parameters to fit the changes in the cell numbers in each layer during the early stages of follicle development.
This work has been undergone in the framework of the PhD of Frédérique Robin. It has been the matter of a poster at ReprosSciences2017 [24] (April 10-12) and of an oral presentation (Dynamiques de populations cellulaires structurées) at the annual meeting (September 27-29) of GDR MaMovi (Mathématiques Appliquées à la MOdélisation du VIvant).
Mathematical modeling of progenitor cell populations in the mouse cerebral cortex
Participants : Frédérique Clément, Alice Karam [IBPS] , Matthieu Perez, Marie Postel, Sylvie Schneider-Maunoury [IBPS] .
We have finalized the study of our PDE-based model of structured cell populations during the development of cerebral cortex. The model accounts for three main cell types: apical progenitors (APs), intermediate progenitors (IPs), and neurons. Each cell population is structured according to the cell age distribution. Since the model describes the different phases of the cell division cycle, we could derive the numeric equivalents of many of the experimental indexes measured in experimental setups, including classical mitotic or labeling indexes targeting the cells in phase S or mitosis, and more elaborated protocols based on double labeling with fluorescent dyes. We have formulated a multi-criterion objective function which enables us to combine experimental observations of different nature and to fit the data acquired in the framework of the NeuroMathMod project (Sorbonne-Universités Émergence call with IBPS, Institut de Biologie Paris Seine). Great efforts have been put on the experimental side to provide the model with the quantitative values of cell numbers for both progenitors and neurons. With the retrieved parameters, the model can provide useful information not supplied by the data, such as the cell origin of neurons (direct neurogenesis from AP or IPgenic neurogenesis) and the proportion of IPs cells undergoing several rounds of cell cycles. In addition, we have compared the cell dynamics patterns observed in wild-type mice with respect to mutant mice used as an animal model of human ciliopathies.
In the framework of the internship of Matthieu Perez (INSA Rouen, co-supervised by Frédérique Clément and Marie Postel), we have investigated numerically the link between our deterministic, PDE-based model of progenitor and neuron cell dynamics, and possible stochastic counterparts inspired from previous work in the team [31]. The deterministic approach is averaged with respect to the deterministic one, since it does not account for the trajectories of individual cells, yet it describes in more details the progression of cells within the cell cycle since it explicitly embeds the structuring of the cell cycle into different phases. The work has consisted in comparing the main model outputs (numbers of progenitors and neurons as a function of time) obtained by numerical simulations based on characteristics, on the deterministic side, or Gillespie algorithms, on the stochastic side. A proper strategy had to be settled to deal with the main difficulties raised by this comparison, namely the time-varying rates involved in the stochastic transition rates from one cell type to another, and the matching between the average stochastic rates and the deterministic rates ruling cell kinetics, especially the cell cycle duration.