3D Simulation of Active thin Structures in a Viscous Fluid and Application to Mucociliary Transport

Astrid Decoene¹ ; Sébastien Martin² ; Chabane Méziane³

¹ Université de Bordeaux, Institut de Mathématiques de Bordeaux (CNRS-UMR 5251), 351 cours de la Libération, 33405 Talence, France
² Université Paris Cité, Laboratoire MAP5 (CNRS-UMR 8145), 45 rue des Saints-Pères, 75270 Paris cedex 06, France
³ Université Paris-Saclay, Laboratoire de Mathématiques d’Orsay (CNRS-UMR 8628), Bâtiment 307, rue Michel Magat, Faculté des Sciences d’Orsay, 91405 Orsay cedex, France

Mathematical modelling of natural phenomena, Tome 19 (2024), article no. 12.

Résumé

We propose a hierarchy of mathematical models for the numerical simulation of active thin structures in a viscous fluid and its application to mucociliary clearance. Our aim is to simulate large forests of cilia and analyze the collective dynamics arising in the flow, as well as their impact on the efficiency of the mucus transport. In a 3D model we describe the cilia individually and study their joint actions on the fluid. The model is built upon a 3D Stokes problem with singular source terms that represent the action of the 1D cilia on the fluid, including the influence of the background flow (making the problem nonlocal). Surface tension between the periciliary layer and the mucus is taken into account. From the 3D model we also derive a 1D space averaged model, describing the dynamics of the mean velocity of the mucus that is propelled by the cilia, hence allowing lower computational costs and still providing useful characterization of the efficiency of the transport. Mathematical properties of the models (existence and uniqueness of solutions in suitable functional spaces) are analyzed. Numerical simulations highlight the influence of critical parameters on the efficiency of the mucociliary transport in the case of dense forests of cilia.

Détail

DOI : 10.1051/mmnp/2024010

Bibliographie

[1] S.K. Lai, Y.Y. Wang, D. Wirtz, J. Hanes Micro- and macrorheology of mucus Adv. Drug Deliv. Rev. 2009 86 100

[2] M.R. Knowles, R.C. Boucher Mucus clearance as a primary innate defense mechanism for mammalian airways J. Clin. Invest. 2002 571 577

[3] P.J. Basser, T.A. Mcmahon, P. Griffith The mechanism of mucus clearance in cough Trans. ASME, J. Biomech. Eng. 1989 288 297

[4] A.M. Lucas, L.C. Douglas Principles underlying ciliary activity in the respiratory tract: II. A comparison of nasal clearance in man, monkey and other mammals Arch. Otolaryngol. 1934 518 541

[5] I.R. Gibbons Cilia and flagella of eukaryotes J. Cell. Biol. 1981 107 124

[6] D.R. Mitchell, The Evolution of Eukaryotic Cilia and Flagella as Motile and Sensory Organelles. Eukaryotic Membranes and Cytoskeleton: Origins and Evolution. Advances in Experimental Medicine and Biology, Vol. 607. Springer New York (2007) 130–140, chapter 11.

[7] N. Mizuno, M. Taschner, B.D. Engel, E. Lorentzen Structural studies of ciliary components J. Mol. Biol. 2012 163 180

[8] B. Prevon, J.M. Scholey, E.J.G. Peterman Intraflagellar transport: mechanisms of motor action, cooperation, and cargo delivery FEBS J. 2017 2905 2931

[9] E.M. Purcell Life at low Reynolds number Am. J. Phys. 1977 3 11

[10] M.J. Sanderson, E.R. Dirksen A versatile and quantitative computer-assisted photoelectronic technique used for the analysis of ciliary beat cycles Cell Motil. 1985 267 292

[11] D.R. Brumley, K.Y. Wan, M. Polin, R.E. Goldstein Flagellar synchronization through direct hydrodynamic interactions eLife 2014 e02750

[12] S. Enault, D. Lombardi, P. Poncet, M. Thiriet Mucus dynamics subject to air and wall motion ESAIM: Proc. 2010 125 141

[13] S. Mitran Continuum-kinetic-microscopic model of lung clearance due to core-annular fluid entrainment J. Comput. Phys. 2013 193 211

[14] B. Mauroy, C. Fausser, D. Pelca, J. Merckx, P. Flaud Toward the modeling of mucus draining from the human lung: role of the geometry of the airway tree Phys. Biol. 2011 056006

[15] D.J. Smith, E.A. Gaffney, J.R. Blake A viscoelastic traction layer model of muco-ciliary transport Bull. Math. Biol. 2007 289 327

[16] P. Kurbatova, N. Bessonov, V. Volpert, H.A.W.M. Tiddens, C. Cornu, P. Nony, D. Caudri Model of mucociliary clearance in cystic fibrosis lungs J. Theor. Biol. 2015 81 88

[17] A. Choudhury, M. Filoche, N.M. Ribe, N. Grenier, G.F. Dietze On the role of viscoelasticity in mucociliary clearance: a hydrodynamic continuum approach J. Fluid Mech. 2023 A33

[18] M. Bottier, M. Peña Fernández, G. Pelle, D. Isabey, B. Louis, J.B. Grotberg, M. Filoche A new index for characterizing micro-bead motion in a flow induced by ciliary beating: Part II, modeling PLOS Comput. Biol. 2017 1 21

[19] A. Decoene, S. Martin and F. Vergnet, A continuum active structure model for the interaction of cilia with a viscous fluid. Z. Angew. Math. Mech. (2023) e202100534

[20] R.H. Dillon, L.J. Fauci, C. Omoto, X. Yang Fluid dynamic models of flagellar and ciliary beating Ann. N. Y. Acad. Sci. 2007 494 505

[21] S. Gueron, K. Levit-Gurevich A three-dimensional model for ciliary motion based on the internal 9 + 2 structure Proc. Biol. Sci. 2001 599 607

[22] S. Gueron, N. Liron Ciliary motion modeling, and dynamic multicilia interactions Biophys. J. 1992 1045 1058

[23] S. Gueron, N. Liron Simulations of three-dimensional ciliary beats and cilia interactions Biophys. J. 1993 499 507

[24] M.H. Sedaghat, M.M. Shahmardan, M. Norouzi, M. Heydari Effect of cilia beat frequency on muco-ciliary clearance J. Biomed. Phys. Eng. 2016 265 278

[25] M.H. Sedaghat, S. Sadrizadeh, O. Abouali Three-dimensional simulation of mucociliary clearance under the ciliary abnormalities J. Non-Newton. Fluid Mech. 2023 105029

[26] S. Mitran Metachronal wave formation in a model of pulmonary cilia Comput. Struct. 2007 763 774

[27] D. Oriola, H. Gadêlha, J. Casademunt Nonlinear amplitude dynamics in flagellar beating R. Soc. Open Sci. 2017 160698

[28] Y. Man, F. Ling, E. Kanso Cilia oscillations Philos. Trans. R. Soc. B 2019 20190157

[29] B. Chakrabarti, D. Saintillan Hydrodynamic synchronization of spontaneously beating filaments Phys. Rev. Lett. 2019 208101

[30] B. Chakrabarti, S. Fürthauer, M.J. Shelley A multiscale biophysical model gives quantized metachronal waves in a lattice of beating cilia Proc. Natl. Acad. Sci. U.S.A. 2022 e2113539119

[31] R.G. Cox The motion of long slender bodies in a viscous fluid. Part 1. General theory J. Fluid Mech. 1970 791 810

[32] Y. Mori, L. Ohm, D. Spirn Theoretical justification and error analysis for slender body theory Commun. Pure Appl. Math. 2020 1245 1314

[33] N. Liron, S. Mochon The discrete-cilia approach to propulsion of ciliated micro-organisms J. Fluid Mech. 1976 593 607

[34] G.R. Fulford, J.R. Blake Force distribution along a slender body straddling an interface J. Austral. Math. Soc. Ser. B 1986 295 315

[35] D.J. Smith, E.A. Gaffney, J.R. Blake Discrete cilia modelling with singularity distributions: application to the embryonic node and the airway surface liquid Bull. Math. Biol. 2007 1477 1510

[36] W.L. Lee, P.G. Jayathilake, Z. Tan, D.V. Le, H.P. Lee, B.C. Khoo Muco-ciliary transport: effect of mucus viscosity, cilia beat frequency and cilia density Comput. Fluids 2011 214 221

[37] S. Chateau, J. Favier, U. D’Ortona, S. Poncet Transport efficiency of metachronal waves in 3D cilium arrays immersed in a two-phase flow J. Fluid Mech. 2017 931 961

[38]

[39] O.K. Matar, P.D.M. Spelt Dynamics of thin free films with reaction-driven density and viscosity variations Phys. Fluids 2005 122102

[40] S. Bertoluzza, A. Decoene, L. Lacouture, S. Martin Local error analysis for the Stokes equations with a punctual source term Numer. Math. 2018 677 701

[41] L. Lacouture, Modélisation et simulation du mouvement de structures fines dans un fluide visqueux : application au transport mucociliaire. Thèse de doctorat de l’université Paris-Sud (2016).

[42] G.R. Fulford, J.R. Blake Muco-ciliary transport in the lung J. Theor. Biol. 1986 381 402

[43] M.J. Sanderson, M.A. Sleigh Ciliary activity of cultured rabbit tracheal epithelium: beat pattern and metachrony J. Cell Sci. 1981 331 347

[44] L. Gheber, A. Korngreen, Z. Priel Effect of viscosity on metachrony in mucus propelling cilia Cell Motil. Cytoskel. 1998 9 20

[45] T.L. Hayden Representation theorems in reflexive Banach spaces Math. Z. 1968 405 406

[46] C.G. Simader, On Dirichlet’s Boundary value Problem. An Lp-theory based on a generalization of Garding’s inequality. Vol. 268 of Lecture Notes in Mathematics. Springer-Verlag, Berlin-Heidelberg-New York (1972).

[47] F. Hecht J. Numer. Math. 2012 251 265

[48] W.M. Foster, E. Langenback, E.H. Bergofsky Measurement of tracheal and bronchial mucus velocities in man: relation to lung clearance J. Appl. Physiol. 1980 965 971

[49] E. Kilgour, N. Rankin, S. Ryan, P. Rodger Mucociliary function deteriorates in the clinical range of inspired air temperature and humidity Intensive Care Med. 2004 1491 1494

[50] D.B. Yeates, N. Aspin, H. Levison, M.T. Jones, A.C. Bryan Mucociliary tracheal transport rates in man J. Appl. Physiol. 1975 487 495

[51] R. Trawoger, T. Kolobow, M. Cereda, M.E. Sparacino Tracheal mucus velocity remains normal in healthy sheep intubated with a new endotracheal tube with a novel laryngeal seal Anesthesiology 1997 1140 1144

[52] L. Morgan, M. Pearson, R. De Iongh, D. Mackey, H. Van Der Wall, M. Peters, J. Rutland Scintigraphic measurement of tracheal mucus velocity in vivo Eur. Respir. J. 2004 518 522

[53] A. Decoene, S. Martin, B. Maury Direct simulation of rigid particles in a viscoelastic fluid J. Non-Newton. Fluid Mech. 2018 1 25