Le poumon est un organe complexe, à la fois multi-physique et multi-échelle, dont les fonctions principales sont le transport de l’air de la bouche aux alvéoles pulmonaires et les échanges gazeux entre les gaz atmosphériques et les gaz du sang. Dans ce texte nous nous intéressons plus particulièrement à la ventilation et nous présentons une hiérarchie de modèles mathématiques permettant de la décrire et de la simuler. Les différents modèles peuvent être utilisés pour comprendre les phénomènes physiologiques en jeu, explorer différents scénarios ou encore développer des outils d’aide au diagnostique.
@incollection{XUPS_2022____1_0, author = {C\'eline Grandmont}, title = {Mod\'elisation math\'ematique et~num\'erique de la ventilation}, booktitle = {Math\'ematiques et biologie}, series = {Journ\'ees math\'ematiques X-UPS}, pages = {1--46}, publisher = {Les \'Editions de l{\textquoteright}\'Ecole polytechnique}, year = {2022}, doi = {10.5802/xups.2022-01}, language = {fr}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/xups.2022-01/} }
TY - JOUR AU - Céline Grandmont TI - Modélisation mathématique et numérique de la ventilation JO - Journées mathématiques X-UPS PY - 2022 SP - 1 EP - 46 PB - Les Éditions de l’École polytechnique UR - https://proceedings.centre-mersenne.org/articles/10.5802/xups.2022-01/ DO - 10.5802/xups.2022-01 LA - fr ID - XUPS_2022____1_0 ER -
%0 Journal Article %A Céline Grandmont %T Modélisation mathématique et numérique de la ventilation %J Journées mathématiques X-UPS %D 2022 %P 1-46 %I Les Éditions de l’École polytechnique %U https://proceedings.centre-mersenne.org/articles/10.5802/xups.2022-01/ %R 10.5802/xups.2022-01 %G fr %F XUPS_2022____1_0
Céline Grandmont. Modélisation mathématique et numérique de la ventilation. Journées mathématiques X-UPS (2022), pp. 1-46. doi : 10.5802/xups.2022-01. https://proceedings.centre-mersenne.org/articles/10.5802/xups.2022-01/
[All92] Grégoire Allaire Homogenization and two-scale convergence, SIAM J. Math. Anal., Volume 23 (1992) no. 6, pp. 1482-1518 | DOI | MR | Zbl
[BGM10] Leonardo Baffico; Céline Grandmont; Bertrand Maury Multiscale modeling of the respiratory tract, Math. Models Methods Appl. Sci., Volume 20 (2010) no. 1, pp. 59-93 | DOI | MR | Zbl
[Bre83] Haïm Brezis Analyse fonctionnelle : Théorie et applications, Collection Math. Appliquées pour la Maîtrise, Masson, Paris, 1983
[Bru21] M. Brunengo Étude d’un modèle visco-élastique du poumon humain et application à l’oscillation haute fréquence extra-thoracique, Thèse, Université Côte d’Azur (2021)
[CG15] Paul Cazeaux; Céline Grandmont Homogenization of a multiscale viscoelastic model with nonlocal damping, application to the human lungs, Math. Models Methods Appl. Sci., Volume 25 (2015) no. 6, pp. 1125-1177 | DOI | MR | Zbl
[Dev10] A. Devys Modélisation, analyse mathématique et simulation numérique de problèmes issus de la biologie, Thèse, Université Côte d’Azur (2010)
[dRMF + 06] L. de Rochefort; X. Maître; R. Fodil; L. Vial; B. Louis; D. Isabey; C. Croce; L. Darrasse; G. Sbirlea-Apiou; G. Caillibotte; J. Bittoun; E. Durand Phase contrast velocimetry with hyperpolarized helium-3 for in vitro and in vivo characterization of airflow, Magn Reson Med, Volume 55 (2006), pp. 1318-1325 | DOI
[dRVF + 07] L. de Rochefort; L. Vial; R. Fodil; X. Maître; B. Louis; G. Isabey; D. Caillibotte; M. Thiriet; J. Bittoun; E. Durand; G. Sbirlea-Apiou In vitro validation of computational fluid dynamic simulation in human proximal airways with hyperpolarized he magnetic resonance phase-contrast velocimetry, J. Appl. Physiol., Volume 102 (2007) no. 5, pp. 2012-2023 | DOI
[FGNQ01] L. Formaggia; J. F. Gerbeau; F. Nobile; A. Quarteroni On the coupling of 3D and 1D Navier-Stokes equations for flow problems in compliant vessels, Comput. Methods Appl. Mech. Engrg., Volume 191 (2001) no. 6-7, pp. 561-582 | DOI | MR | Zbl
[FMN07] Luca Formaggia; Alexandra Moura; Fabio Nobile On the stability of the coupling of 3D and 1D fluid-structure interaction models for blood flow simulations, ESAIM Math. Model. Numer. Anal., Volume 41 (2007) no. 4, pp. 743-769 | DOI | Numdam | MR | Zbl
[FMP + 05] C. Fetita; S. Mancini; D. Perchet; F. Prêteux; M. Thiriet; L. Vial An image-based computational model of oscillatory flow in the proximal part of tracheobronchial trees, Comput. Meth. Biomech. Biomed. Eng., Volume 8 (2005) no. 4, pp. 279-293 | DOI
[GM21] Céline Grandmont; Sébastien Martin Existence of solutions and continuous and semi-discrete stability estimates for 3D/0D coupled systems modelling airflows and blood flows, ESAIM Math. Model. Numer. Anal., Volume 55 (2021) no. 5, pp. 2365-2419 | DOI | MR | Zbl
[GMM06] Céline Grandmont; Bertrand Maury; Nicolas Meunier A viscoelastic model with non-local damping application to the human lungs, ESAIM Math. Model. Numer. Anal., Volume 40 (2006) no. 1, pp. 201-224 | DOI | Numdam | MR | Zbl
[GR86] Vivette Girault; Pierre-Arnaud Raviart Finite element methods for Navier-Stokes equations. Theory and algorithms, Springer Series in Computational Math., 5, Springer-Verlag, Berlin, 1986 | DOI
[KFH + 09] H.J. Kim; C.A. Figueroa; T.J.R. Hughes; K.E. Jansen; C.A. Taylor Augmented Lagrangian method for constraining the shape of velocity profiles at outlet boundaries for three-dimensional finite element simulations of blood flow, Comput. Methods Appl. Mech. Engrg., Volume 198 (2009) no. 45-46, pp. 3551-3566 | DOI | MR | Zbl
[Mau13] Bertrand Maury The respiratory system in equations, MS&A. Modeling, Simulation and Applications, 7, Springer-Verlag Italia, Milan, 2013 | DOI | MR
[MFWS04] B. Mauroy; M. Filoche; E. R. Weibel; B. Sapoval An optimal bronchial tree may be dangerous, Nature, Volume 427 (2004), pp. 633-636 | DOI
[MM14] S. Martin; B. Maury Notion de résistance de l’arbre pulmonaire bronchique dans la ventilation respiratoire humaine, Modéliser & simuler. Épistémologies et pratiques de la modélisation et de la simulation (Modélisations, simulations, systèmes complexes), Volume 2, Éditions Matériologiques, Paris, 2014, pp. 493-524 (hal-01362018)
[MR07] V. Maz’ya; J. Rossmann estimates of solutions to mixed boundary value problems for the Stokes system in polyhedral domains, Math. Nachr., Volume 280 (2007) no. 7, pp. 751-793 | DOI | MR | Zbl
[MSSM08] S. Martin; T. Similowski; C. Straus; B. Maury Impact of respiratory mechanics model parameters on gas exchange efficiency, Mathematical and numerical modelling of the human lung (ESAIM Proc.), Volume 23, EDP Sciences, Les Ulis, 2008, pp. 30-47 | MR | Zbl
[OS95] Matthias Orlt; Anna-Margarete Sändig Regularity of viscous Navier-Stokes flows in nonsmooth domains, Boundary value problems and integral equations in nonsmooth domains (Luminy, 1993) (Lecture Notes in Pure and Appl. Math.), Volume 167, Dekker, New York, 1995, pp. 185-201 | MR | Zbl
[Poz17] N. Pozin Multiscale lung ventilation modeling in health and disease, Thèse, Sorbonne Université (2017)
[PSS70] T. J. Pedley; R. C. Schroter; M. F. Sudlow Energy losses and pressure drop in models of human airways, Respiration Physiology, Volume 9 (1970) no. 3, pp. 371-386 | DOI
[QRV01] Alfio Quarteroni; Stefania Ragni; Alessandro Veneziani Coupling between lumped and distributed models for blood flow problems, Comput. Vis. Sci., Volume 4 (2001) no. 2, pp. 111-124 | DOI | MR | Zbl
[QV03] Alfio Quarteroni; Alessandro Veneziani Analysis of a geometrical multiscale model based on the coupling of ODEs and PDEs for blood flow simulations, Multiscale Model. Simul., Volume 1 (2003) no. 2, pp. 173-195 | DOI | MR | Zbl
[Van08] C. Vannier Modélisation mathématique du poumon humain, Thèse, Université de Paris XI–Orsay (2008)
[VC06] I. Vignon-Clementel A coupled multidomain method for computational modeling of blood flow, PhD thesis, Stanford University (2006)
[VCFJT06] I.E. Vignon-Clementel; C.A. Figueroa; K.E. Jansen; C.A. Taylor Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries, Comput. Methods Appl. Mech. Engrg., Volume 195 (2006) no. 29-32, pp. 3776-3796 | DOI | MR | Zbl
[VV05] A. Veneziani; C. Vergara Flow rate defective boundary conditions in haemodynamics simulations, Internat. J. Numer. Methods Fluids, Volume 47 (2005) no. 8-9, pp. 803-816 | DOI | MR | Zbl
[Wei63] Ewald R. Weibel Morphometry of the human lung, Springer, Berlin, Heidelberg, 1963 | DOI
Cité par Sources :