A level set projection model of lipid vesicles in general flows

A new numerical method to model the dynamic behavior of lipid vesicles under general flows is presented. A gradient-augmented level set method is used to model the membrane motion. To enforce the volume- and surface-incompressibility constraints a four-step projection method is developed to integrate the full Navier-Stokes equations. This scheme is implemented on an adaptive non-graded Cartesian grid. Convergence results are presented, along with sample two-dimensional results of vesicles under various flow conditions.

[1]  J. A. Sethian,et al.  Fast Marching Methods , 1999, SIAM Rev..

[2]  Peter Smereka,et al.  Semi-Implicit Level Set Methods for Curvature and Surface Diffusion Motion , 2003, J. Sci. Comput..

[3]  M. Minion,et al.  Accurate projection methods for the incompressible Navier—Stokes equations , 2001 .

[4]  Steven J. Ruuth,et al.  A simple embedding method for solving partial differential equations on surfaces , 2008, J. Comput. Phys..

[5]  Frédéric Gibou,et al.  A second order accurate projection method for the incompressible Navier-Stokes equations on non-graded adaptive grids , 2006, J. Comput. Phys..

[6]  Udo Seifert,et al.  Configurations of fluid membranes and vesicles , 1997 .

[7]  S. Osher,et al.  A Level Set Formulation of Eulerian Interface Capturing Methods for Incompressible Fluid Flows , 1996 .

[8]  Victor Steinberg,et al.  Dynamics of interacting vesicles and rheology of vesicle suspension in shear flow , 2008 .

[9]  Jian Zhang,et al.  Adaptive Finite Element Method for a Phase Field Bending Elasticity Model of Vesicle Membrane Deformations , 2008, SIAM J. Sci. Comput..

[10]  B MacdonaldColin,et al.  Level Set Equations on Surfaces via the Closest Point Method , 2008 .

[11]  C. Pozrikidis,et al.  Modeling and Simulation of Capsules and Biological Cells , 2003 .

[12]  J. Freund.  Leukocyte Margination in a Model Microvessel , 2006 .

[13]  Q. Du,et al.  Energetic variational approaches in modeling vesicle and fluid interactions , 2009 .

[14]  Colin B. Macdonald,et al.  The Implicit Closest Point Method for the Numerical Solution of Partial Differential Equations on Surfaces , 2009, SIAM J. Sci. Comput..

[15]  P. Olla,et al.  The behavior of closed inextensible membranes in linear and quadratic shear flows , 1999, chao-dyn/9906006.

[16]  George Biros,et al.  A numerical method for simulating the dynamics of 3D axisymmetric vesicles suspended in viscous flows , 2009, J. Comput. Phys..

[17]  V Steinberg,et al.  Dynamics of a vesicle in general flow , 2009, Proceedings of the National Academy of Sciences.

[18]  T. Biben,et al.  Tumbling of vesicles under shear flow within an advected-field approach. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  Victor Steinberg,et al.  Transition to tumbling and two regimes of tumbling motion of a vesicle in shear flow. , 2006, Physical review letters.

[20]  George Em Karniadakis,et al.  A semi-Lagrangian high-order method for Navier-Stokes equations , 2001 .

[21]  Thomas Podgorski,et al.  Lateral migration of vesicles in microchannels: effects of walls and shear gradient , 2009 .

[22]  J. Sethian,et al.  LEVEL SET METHODS FOR FLUID INTERFACES , 2003 .

[23]  Qiang Du,et al.  Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions , 2006, J. Comput. Phys..

[24]  Thierry Biben,et al.  Rheology of a dilute two-dimensional suspension of vesicles , 2010, Journal of Fluid Mechanics.

[25]  George Biros,et al.  A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2D , 2009, J. Comput. Phys..

[26]  J. Westerweel,et al.  In vivo whole-field blood velocity measurement techniques , 2007 .

[27]  Goldsmith Hl,et al.  Red cell motions and wall interactions in tube flow. , 1971 .

[28]  Gwennou Coupier,et al.  Noninertial lateral migration of vesicles in bounded Poiseuille flow , 2008, 0803.3153.

[29]  Colin B. Macdonald,et al.  Level Set Equations on Surfaces via the Closest Point Method , 2008, J. Sci. Comput..

[30]  Benjamin Seibold,et al.  A gradient-augmented level set method with an optimally local, coherent advection scheme , 2009, J. Comput. Phys..

[31]  V. V. Lebedev,et al.  Nearly spherical vesicles in an external flow , 2007, 0705.3543.

[32]  D. Marsh,et al.  Elastic curvature constants of lipid monolayers and bilayers. , 2006, Chemistry and physics of lipids.

[33]  C. Pozrikidis.  Axisymmetric motion of a file of red blood cells through capillaries , 2005 .

[34]  Petia M. Vlahovska,et al.  Monolayer slip effects on the dynamics of a lipid bilayer vesicle in a viscous flow , 2010, Journal of Fluid Mechanics.

[35]  Ann S. Almgren,et al.  An adaptive level set approach for incompressible two-phase flows , 1997 .

[36]  M Intaglietta,et al.  Capillary red blood cell velocity measurements in human nailfold by videodensitometric method. , 1975, Microvascular research.

[37]  S. Osher,et al.  Level set methods: an overview and some recent results , 2001 .

[38]  R. Skalak,et al.  Motion of a tank-treading ellipsoidal particle in a shear flow , 1982, Journal of Fluid Mechanics.

[39]  P. Vlahovska,et al.  Dynamics of a viscous vesicle in linear flows. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[40]  T. Oberholzer,et al.  Giant Vesicles as Microreactors for Enzymatic mRNA Synthesis , 2002, Chembiochem : a European journal of chemical biology.

[41]  David L. Chopp,et al.  Another Look at Velocity Extensions in the Level Set Method , 2009, SIAM J. Sci. Comput..

[42]  Manouk Abkarian,et al.  Vesicles and red blood cells in shear flow. , 2008, Soft matter.

[43]  Feng Feng,et al.  Finite element modeling of lipid bilayer membranes , 2006, J. Comput. Phys..

[44]  Prosenjit Bagchi,et al.  Mesoscale simulation of blood flow in small vessels. , 2007, Biophysical journal.

[45]  Vincent Noireaux,et al.  A vesicle bioreactor as a step toward an artificial cell assembly. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[46]  C. Misbah,et al.  Towards a thermodynamically consistent picture of the phase-field model of vesicles: local membrane incompressibility. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[47]  J A Sethian,et al.  A fast marching level set method for monotonically advancing fronts. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[48]  H Hinghofer-Szalkay,et al.  Continuous monitoring of blood volume changes in humans. , 1987, Journal of applied physiology.

[49]  P. Colella,et al.  An Adaptive Level Set Approach for Incompressible Two-Phase Flows , 1997 .

[50]  W. Helfrich.  Elastic Properties of Lipid Bilayers: Theory and Possible Experiments , 1973, Zeitschrift fur Naturforschung. Teil C: Biochemie, Biophysik, Biologie, Virologie.

[51]  M. Abkarian,et al.  Dynamics of vesicles in a wall-bounded shear flow. , 2005, Biophysical journal.

[52]  Udo Seifert.  Fluid membranes in hydrodynamic flow fields: Formalism and an application to fluctuating quasispherical vesicles in shear flow , 1999 .

[53]  Seifert,et al.  Fluid Vesicles in Shear Flow. , 1996, Physical review letters.

[54]  U. Seifert,et al.  Swinging and tumbling of elastic capsules in shear flow , 2007, Journal of Fluid Mechanics.

[55]  Ron Kimmel,et al.  Fast Marching Methods , 2004 .

[56]  P. Cullis,et al.  Drug Delivery Systems: Entering the Mainstream , 2004, Science.

[57]  David L. Chopp,et al.  Some Improvements of the Fast Marching Method , 2001, SIAM J. Sci. Comput..

[58]  Wei Lu,et al.  Stability and shape evolution of voids and channels due to surface misfit , 2008 .

[59]  F. Campelo,et al.  Dynamic model and stationary shapes of fluid vesicles , 2006, The European physical journal. E, Soft matter.