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Margination of white blood cells: a computational approach by a hydrodynamic phase field model

Published online by Cambridge University Press:  03 February 2016

Wieland Marth ,
Sebastian Aland  and
Axel Voigt
Wieland Marth
Affiliation:
Institute of Scientific Computing, TU Dresden, 01062 Dresden, Germany
Sebastian Aland
Affiliation:
Institute of Scientific Computing, TU Dresden, 01062 Dresden, Germany
Axel Voigt*
Affiliation:
Institute of Scientific Computing, TU Dresden, 01062 Dresden, Germany Center for Systems Biology Dresden, Pfotenhauerstr. 108, 01307 Dresden, Germany
*
†Email address for correspondence: [email protected]
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Abstract

We numerically investigate margination of white blood cells and demonstrate the dependency on a number of conditions including haematocrit, the deformability of the cells and the Reynolds number. The approach, which is based on a mesoscopic hydrodynamic Helfrich-type model, reproduces previous results, e.g. a decreasing tendency for margination with increasing deformability and a non-monotonic dependency on haematocrit. The consideration of inertia effects, which may be of relevance in various parts of the cardiovascular system, indicates a decreasing tendency for margination with increasing Reynolds number. The effect is discussed by analysing inertial and non-inertial lift forces for single cells under different flow conditions and large-scale two-dimensional simulations of interacting red blood cells and white blood cells in an idealized blood vessel.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Biological Fluid Dynamics: Blood flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 790 , 10 March 2016 , pp. 389 - 406
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Copyright
© 2016 Cambridge University Press 

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References

Abbitt, K. B. & Nash, G. B. 2003 Rheological properties of the blood influencing selectin-mediated adhesion of flowing leukocytes. Am. J. Physiol. Heart Circ. Physiol. 285, H229–H240.CrossRefGoogle ScholarPubMed
Aland, S., Egerer, S., Lowengrub, J. & Voigt, A. 2014 Diffuse interface models of locally inextensible vesicles in a viscous fluid. J. Comput. Phys. 277, 32–47.CrossRefGoogle Scholar
Asmolov, E. S. 1999 The inertial lift on a spherical particle in plane Poiseuille flow at large Reynolds number. J. Fluid Mech. 381, 63–87.CrossRefGoogle Scholar
Bark, D. L. & Ku, D. N. 2010 Wall shear over high degree stenoses pertinent to atherothrombosis. J. Biomech. 43, 2970–2977.CrossRefGoogle ScholarPubMed
Beaucourt, J., Rioual, F., Seon, T., Biben, T. & Misbah, C. 2004 Steady to unsteady dynamics of a vesicle in a flow. Phys. Rev. E 69, 011906.CrossRefGoogle ScholarPubMed
Biben, T., Kassner, Kl. & Misbah, C. 2005 Phase-field approach to three-dimensional vesicle dynamics. Phys. Rev. E 72, 041921.CrossRefGoogle ScholarPubMed
Biben, T. & Misbah, C. 2003 Tumbling of vesicles under shear flow within an advected-field approach. Phys. Rev. E 67, 031908.CrossRefGoogle ScholarPubMed
Bonito, A., Nochetto, R. H. & Pauletti, M. S. 2011 Dynamics of biomembranes: effect of the bulk fluid. Math. Model. Nat. Phenom. 6, 25–43.CrossRefGoogle Scholar
Bretherton, F. P. 1962 The motion of rigid particles in a shear flow at low Reynolds number. J. Fluid Mech. 14, 284–304.CrossRefGoogle Scholar
Campelo, F. & Hernández-Machado, A. 2007 Shape instabilities in vesicles: a phase-field model. Eur. Phys. J. Special Topics 143, 101–108.CrossRefGoogle Scholar
Carlo, D. Di, Humphry, E. J., Stone, H. & Toner, M. 2009 Particle segregation and dynamics in confined flows. Phys. Rev. Lett. 102, 094503.CrossRefGoogle ScholarPubMed
Du, Q., Li, M. & Liu, C. 2007 Analysis of a phase field Navier–Stokes vesicle–fluid interaction model. J. Discrete Continuous Dyn. Syst. B 8, 539–556.CrossRefGoogle Scholar
Du, Q., Liu, C., Ryham, R. & Wang, X. 2005 A phase field formulation of the Willmore problem. Nonlinearity 18, 1249–1267.CrossRefGoogle Scholar
Du, Q., Liu, C. & Wang, X. 2006 Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions. J. Comput. Phys. 212, 757–777.CrossRefGoogle Scholar
Fedosov, D. A., Fornleitner, J. & Gompper, G. 2012 Margination of white blood cells in microcapillary flow. Phys. Rev. Lett. 108, 028104.CrossRefGoogle ScholarPubMed
Fedosov, D. A. & Gompper, G. 2014 White blood cell margination in microcirculation. Soft Matt. 10, 2961–2970.CrossRefGoogle ScholarPubMed
Fischer, T. M., Stöhr-Liesen, M. & Schmid-Schönbein, H. 1978 The red-cell as a fluid droplet: tank tread-like motion of the human erythocyte-membrane in shear flow. Science 202, 894–896.CrossRefGoogle ScholarPubMed
Formaggiam, L., Quarteroni, A. & Veneziani, A. 2000 Cardiovascular Mathematics: Modeling and Simulation of the Circulatory System. Springer.Google Scholar
Freund, J. B. 2007 Leukocyte margination in a model microvessel. Phys. Fluids 19 (2), 023301.CrossRefGoogle Scholar
Geislinger, T. M. & Franke, T. 2014 Hydrodynamic lift of vesicles and red blood cells in flow – from Fåhraeus & Lindqvist to microfluidic cell sorting. Adv. Colloid Interface Sci. 208, 161–176.CrossRefGoogle ScholarPubMed
Ghigliotti, G., Biben, T. & Misbah, C. 2010 Rheology of a dilute two-dimensional suspension of vesicles. J. Fluid Mech. 653, 489–518.CrossRefGoogle Scholar
Goldsmith, H. L. & Mason, S. G. 1961 Axial migration of particles in Poiseuille flow. Nature 190, 1095–1096.CrossRefGoogle Scholar
Gu, R., Wang, X. & Gunzburger, M. 2014 Simulating vesicle–substrate adhesion using two phase field functions. J. Comput. Phys. 275, 626–641.CrossRefGoogle Scholar
Haußer, F., Li, S., Lowengrub, J., Marth, W., Rätz, A. & Voigt, A. 2013 Thermodynamically consistent models for two-component vesicles. Intl J. Biomath. Biostat. 2 (1), 19–48.Google Scholar
Helfrich, W. 1973 Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturforsch. c 28, 693–703.CrossRefGoogle ScholarPubMed
Ho, B. P. & Leal, L. G. 1974 Inertial migration of rigid spheres in two-dimensional unidirectional flows. J. Fluid Mech. 65, 365–400.CrossRefGoogle Scholar
Jain, A. & Munn, L. L. 2009 Determinants of leukocyte margination in rectangular microchannels. PLoS ONE 4 (9), e7104.CrossRefGoogle ScholarPubMed
Kim, Y. & Lai, M.-C. 2010 Simulating the dynamics of inextensible vesicles by the penalty immersed boundary method. J. Comput. Phys. 229, 4840–4853.CrossRefGoogle Scholar
Kraus, M., Wintz, W., Seifert, U. & Lipowsky, R. 1996 Fluid vesicle in shear flow. Phys. Rev. Lett. 77, 3685–3688.CrossRefGoogle ScholarPubMed
Krüger, T., Kaoui, B. & Harting, J. 2014 Interplay of inertia and deformability on rheological properties of a suspension of capsules. J. Fluid Mech. 751, 725–745.CrossRefGoogle Scholar
Ku, D. 1997 Blood flow in arteries. Annu. Rev. Fluid Mech. 29, 399–434.CrossRefGoogle Scholar
Kumar, A. & Graham, M. D. 2012 Margination and segregation in confined flows of blood and other multicomponent suspensions. Soft Matt. 8, 10536–10548.CrossRefGoogle Scholar
Laadhari, A., Saramito, P. & Misbah, C. 2012 Vesicle tumbling inhibited by inertia. Phys. Fluids 24, 031901.CrossRefGoogle Scholar
Ling, S., Marth, W., Praetorius, S. & Voigt, A.2015 An adaptive finite element multi-mesh approach for interacting deformable objects in flow. Comput. Meth. Appl. Math. doi:10.1515/cmam-2016-0003.CrossRefGoogle Scholar
Marth, W. & Voigt, A. 2014 Signaling networks and cell motility: a computational approach using a phase field description. J. Math. Biol. 69, 91–112.CrossRefGoogle ScholarPubMed
Matas, J.-P., Morris, J. F. & Guazzelli, E. 2004 Inertial migration of rigid spherical particles in Poiseuille flow. J. Fluid Mech. 515, 171–195.CrossRefGoogle Scholar
Meßlinger, S., Schmidt, B., Noguchi, H. & Gompper, G. 2009 Dynamical regimes and hydrodynamic lift of viscous vesicles under shear. Phys. Rev. E 80, 011901.CrossRefGoogle ScholarPubMed
N’Dri, N. A., Shyy, W. & Tran-Son-Tay, R. 2003 Computational modeling of cell adhesion and movement using a continuum-kinetics approach. Biophys. J. 85, 2273–2286.CrossRefGoogle ScholarPubMed
Pearson, M. J. & Lipowsky, H. H. 2000 Influence of erythrocyte aggregation on leukocyte margination in postcapillary venules of rat mesentery. Am. J. Physiol. Heart Circ. Physiol. 279, H1460–H1471.CrossRefGoogle ScholarPubMed
Prothero, J. W. & Burton, A. C. 1962 The physica of blood flow in capillaires. Biophys. J. 2, 199–212.CrossRefGoogle Scholar
Salac, D. & Miksis, M. J. 2011 A level set projection model of lipid vesicles in general flows. J. Comput. Phys. 230, 8192–8215.CrossRefGoogle Scholar
Salac, D. & Miksis, M. J. 2012 Reynolds number effects on lipid vesicles. J. Fluid Mech. 711, 122–146.CrossRefGoogle Scholar
Schonberg, J. & Hinch, E. 1989 Inertial migration of a sphere in Poiseuille flow. J. Fluid Mech. 203, 517–524.CrossRefGoogle Scholar
Segre, G. & Silberberg, A. 1961 Radial particle displacements in Poiseuille flows of suspensions. Nature 189, 209–210.CrossRefGoogle Scholar
Sohn, J. S., Tseng, Y.-H., Li, S., Voigt, A. & Lowengrub, J. 2010 Dynamics of multicomponent vesicles in a viscous fluid. J. Comput. Phys. 229, 119–144.CrossRefGoogle Scholar
Takeishi, N., Imai, Y., Nakaaki, K., Yamaguchi, T. & Ishikawa, T. 2014 Leukocyte margination at arteriole shear rate. Physiol. Rep. 2, e12037.CrossRefGoogle ScholarPubMed
Tanaka, H. & Araki, T. 2000 Simulation method of colloidal suspensions with hydrodynamic interactions: fluid particle dynamics. Phys. Rev. Lett. 85, 1338–1341.CrossRefGoogle ScholarPubMed
Veerapaneni, S. K., Gueyffier, D., Zorin, D. & Biros, G. 2009 A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2D. J. Comput. Phys. 228, 2334–2353.CrossRefGoogle Scholar
Vennemann, P., Lindken, R. & Westerweel, J. 2007 In vivo whole-field blood velocity measurement techniques. Exp. Fluids 42, 495–511.CrossRefGoogle Scholar
Vey, S. & Voigt, A. 2007 AMDiS: adaptive multidimensional simulations. Comput. Visual. Sci. 10, 57–67.CrossRefGoogle Scholar
Witkowski, T., Ling, S., Praetorius, S. & Voigt, A. 2015 Software concepts and numerical algorithms for a scalable adaptive parallel finite element method. Adv. Comput. Math 41, 1145–1177.CrossRefGoogle Scholar
Zhang, J., Das, S. & Du, Q. 2009 A phase field model for vesicle–substrate adhesion. J. Comput. Phys. 228, 7837–7849.CrossRefGoogle Scholar
Zhao, H. & Shaqfeh, E. S. G. 2011 The dynamics of a vesicle in simple shear flow. J. Fluid Mech. 674, 578–604.CrossRefGoogle Scholar

Marth Supplementary Material

The video file "movie1.avi" shows WBC margination in a vessel for $H_t=0.293$ considering a rigid WBC and adopting the simulation parameters Re$=5\cdot10^{-2}$, Be$_{RBC}=5.3$, In$=0.1$ and $Q=15$.

Download Marth Supplementary Material(Video)
Video 11.3 MB