A DLM/FD/IB Method for Simulating Compound Cell Interacting with Red Blood Cells in a Microchannel

Abstract

In this article, a computational model and related methodologies have been tested for simulating the motion of a malaria infected red blood cell (iRBC for short) in Poiseuille flow at low Reynolds numbers. Besides the deformability of the red blood cell membrane, the migration of a neutrally buoyant particle (used to model the malaria parasite inside the membrane) is another factor to determine the iRBC motion. Typically an iRBC oscillates in a Poiseuille flow due to the competition between these two factors. The interaction of an iRBC and several RBCs in a narrow channel shows that, at lower flow speed, the iRBC can be easily pushed toward the wall and stay there to block the channel. But, at higher flow speed, RBCs and iRBC stay in the central region of the channel since their migrations are dominated by the motion of the RBC membrane.

Keywords

Compound cell / Red blood cells / Elastic spring model / Fictitious domain method / Immersed boundary method / Microchannel

References

[1] Schmid-Schonbein G. W., Shih Y. Y., Chien S.. Morphometry of human leukocytes. Blood, 1980, 56: 866-875

[2] Diez-Silva M., Dao M., Han J. Shape and biomechanical characteristics of human red blood cells in health and disease. MRS Bull., 2010, 35: 382-388

[3] Glenister F. K., Coppel R. L., Cowman A. F. Contribution of parasite proteins to altered mechanical properties of malaria-infected red blood cells. Blood, 2002, 99: 1060-1063

[4] Veerapaneni S. K., Young Y.-N., Vlahovska P. M., Blawzdziewicz J.. Dynamics of a compound vesicle in shear flow. Phys. Rev. Lett., 2011, 106: 158103

[5] Kaoui B., Krüger T., Harting J.. Complex dynamics of a bilamellar vesicle as a simple model for leukocytes. Soft Matter, 2013, 9: 8057-8061

[6] Nash G. B., O’Brien E., Gordon-Smith E. C., Dormandy J. A.. Abnormalities in the mechanical properties of red blood cells caused by Plasmodium falciparum. Blood, 1989, 74: 855-861

[7] Imai Y., Kondo H., Ishikawa T. Modeling of hemodynamics arising from malaria infection. J. Biomech., 2010, 43: 1386-1393

[8] Wu T., Feng J. J.. Simulation of malaria-infected red blood cells in microfluidic channels: Passage and blockage. Biomicrofluidics, 2013, 7: 044115

[9] Shi L., Pan T.-W., Glowinski R.. Deformation of a single blood cell in bounded Poiseuille flows. Phys. Rev. E, 2012, 85: 016307

[10] Shi L., Pan T.-W., Glowinski R.. Lateral migration and equilibrium shape and position of a single red blood cell in bounded Poiseuille flows. Phys. Rev. E, 2012, 86: 056308

[11] Shi L., Pan T.-W., Glowinski R.. Numerical simulation of lateral migration of red blood cells in Poiseuille flows. Int. J. Numer. Methods Fluids, 2012, 68: 1393-1408

[12] Pan T.-W., Glowinski R.. Direct simulation of the motion of neutrally buoyant circular cylinders in plane Poiseuille flow. J. Comput. Phys., 2002, 181: 260-279

[13] Pan T.-W., Glowinski R.. Direct simulation of the motion of neutrally buoyant balls in a threedimensional Poiseuille flow. C. R. Mecanique, Acad. Sci. Paris, 2005, 333: 884-895

[14] Pan T.-W., Chang C.-C., Glowinski R.. On the motion of a neutrally buoyant ellipsoid in a threedimensional Poiseuille flow. Comput. Methods Appl. Mech. Engrg., 2008, 197: 2198-2209

[15] Pan T.-W., Huang S.-L., Chen S.-D. A numerical study of the motion of a neutrally buoyant cylinder in two dimensional shear flow. Computers & Fluids, 2013, 87: 57-66

[16] Pan T.-W., Shi L., Glowinski R.. A DLM/FD/IB method for simulating cell/cell and cell/particle interaction in microchannels. Chinese Annals of Mathematics, Series B, 2010, 31: 975-990

[17] Pan T.-W., Zhao S., Niu X., Glowinski R.. A DLM/FD/IB method for simulating compound vesicle motion under creeping flow condition. J. Comput. Phys., 2015, 300: 241-253

[18] Glowinski R., Pan T.W., Hesla T. A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: Application to particulate flow. J. Comput. Phys., 2001, 169: 363-427

[19] Desjardins B., Esteban M. J.. Existence of weak solutions for the motion of rigid bodies in a viscous fluid. Arch. Rational Mech. Anal., 1999, 146: 59-71

[20] Tsubota K., Wada S., Yamaguchi T.. Simulation study on effects of hematocrit on blood flow properties using particle method. J. Biomech. Sci. Eng., 2006, 1: 159-170

[21] Wang T., Pan T. W., Xing Z., Glowinski R.. Numerical simulation of rheology of red blood cell rouleaux in microchannels. Phys. Rev. E, 2009, 79: 041916

[22] Peskin C. S.. Numerical analysis of blood flow in the heart. J. Comput. Phys., 1977, 25: 220-252

[23] Peskin C. S., McQueen D. M.. Modeling prosthetic heart valves for numerical analysis of blood flow in the heart. J. Comput. Phys., 1980, 37: 11332

[24] Peskin C. S.. The immersed boundary method. Acta Numer., 2002, 11: 479-517

[25] Bristeau M. O., Glowinski R., Périaux J.. Numerical methods for the Navier-Stokes equations. applications to the simulation of compressible and incompressible viscous flow, Computer Physics Reports, 1987, 6: 73-187

[26] Glowinski R.. Ciarlet P. G., Lions J. L.. Finite element methods for incompressible viscous flow. Handbook of Numerical Analysis, 2003, Amsterdam: North-Holland 3-1176

[27] Girault V., Glowinski R.. Error analysis of a fictitious domain method applied to a Dirichlet problem. Japan J. Indust. Appl. Math., 1995, 12: 487-514

[28] Chorin A. J., Hughes T. J. R., McCracken M. F., Marsden J. E.. Product formulas and numerical algorithms. Comm. Pure Appl. Math., 1978, 31: 205-256

[29] Dean E. J., Glowinski R.. A wave equation approach to the numerical solution of the Navier-Stokes equations for incompressible viscous flow. C. R. Acad. Sc. Paris, Série 1, 1997, 325: 783-791

[30] Dean E. J., Glowinski R., Pan T. W.. De Santo J. A.. A wave equation approach to the numerical simulation of incompressible viscous fluid flow modeled by the Navier-Stokes equations. Mathematical and Numerical Aspects of Wave Propagation, 1998, Philadelphia: SIAM 65-74

[31] Alexeev A., Verberg R., Balazs A. C.. Modeling the interactions between deformable capsules rolling on a compliant surface. Soft Matter, 2006, 2: 499-509

[32] Fischer T. M., Stöhr-Liesen M., Schmid-Schönbein H.. The red cell as a fluid droplet: Tank tread-like motion of the human erythrocyte membrane in shear flow. Science, 1978, 202: 894-896

[33] Keller S. R., Skalak R.. Motion of a tank-treading ellipsoidal particle in a shear flow. J. Fluid Mech., 1982, 120: 27-47

[34] Beaucourt J., Rioual F., Séon T. Steady to unsteady dynamics of a vesicle in a flow. Phys. Rev. E, 2004, 9: 011906

[35] Li H. B., Yi H. H., Shan X. W., Fang H. P.. Shape changes and motion of a vesicle in a fluid using a lattice Boltzmann model. Europhysics Letters, 2008, 81: 54002

[36] Lai M.-C., Hu W. F., Lin W. W.. A fractional step immersed boundary method for stokes flow with an inextensible interface enclosing a solid particle. SIAM. J. Sci. Comput., 2012, 34: 692-710

[37] Segré G., Silberberg A.. Radial particle displacements in Poiseuille flow of suspensions. Nature, 1961, 189: 209-210