# Toward a mathematical modelling of white blood cells filtration

1 BANG - Nonlinear Analysis for Biology and Geophysical flows
LJLL - Laboratoire Jacques-Louis Lions, Inria Paris-Rocquencourt
Abstract : The aim of this work is to set up mathematical tools (mathematical models and numerical methods) to investigate the white blood cells filtration.

In the first part, we set up specific mathematical models to represent the physical phenomena that are involved in the filtration process.

The second part is devoted to the mathematical analysis of partial differential equations modelling the filtration. First, we consider a semilinear parabolic-hyperbolic system with a degenerate and anisotropic diffusion. We study the well-posedness of the system using a $L^{1}$ theory; we consider in particular the existence and the uniqueness of a solution and we investigate other mathematical properties such as maximum principle. Then, we establish the relaxation limit as the reaction constant becomes large. We prove that the system converges to a nonlinear parabolic-hyperbolic equation that generalizes the Stefan problem. We also study the flow through fibrous media using homogenization techniques. The fiber network under study is the one already used by M. Briane in the context of heat conduction of biological tissues. We derive and justify the effective Darcy equation and the permeability tensor for such fibrous media. The theoretical results on the permeability are illustrated by some numerical simulations. Finally, the low solid fraction limit is considered. Applying results by G. Allaire to our setting, we justify rigorously the leading order term in the empirical formulas for the effective permeability used in engineering. The results are also confirmed by a direct numerical calculation of the permeability, in which the small diameter of the fibers requires high accuracy approximations.

In part 3 we present the construction of suitable numerical methods to compute solutions of the considered models. Precisely, we discuss the mixed hybrid finite element formulation for the space discretization of the Darcy problem. For the discretization of the transport equation, we use a splitting technique for the space discretization and the Euler method for the time discretization.
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https://tel.archives-ouvertes.fr/tel-00011977
Contributor : Mohamed Belhadj <>
Submitted on : Saturday, March 18, 2006 - 5:45:49 PM
Last modification on : Wednesday, December 9, 2020 - 3:09:34 PM
Long-term archiving on: : Saturday, April 3, 2010 - 9:16:08 PM

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• HAL Id : tel-00011977, version 1

### Citation

Mohamed Belhadj. Toward a mathematical modelling of white blood cells filtration. Mathematics [math]. Université Pierre et Marie Curie - Paris VI, 2005. English. ⟨tel-00011977⟩

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