Research Papers

An Enhanced Spring-Particle Model for Red Blood Cell Structural Mechanics: Application to the Stomatocyte–Discocyte–Echinocyte Transformation

[+] Author and Article Information
Mingzhu Chen

School of Mechanical & Design Engineering,
Dublin Institute of Technology,
Bolton Street, Dublin 1,
Dublin D01K822, Ireland
e-mail: 452413@dit.ie

Fergal J. Boyle

School of Mechanical & Design Engineering,
Dublin Institute of Technology,
Bolton Street, Dublin 1,
Dublin D01K822, Ireland
e-mail: fergal.boyle@dit.ie

1Corresponding author.

Manuscript received May 30, 2017; final manuscript received July 13, 2017; published online September 28, 2017. Assoc. Editor: Guy M. Genin.

J Biomech Eng 139(12), 121009 (Sep 28, 2017) (11 pages) Paper No: BIO-17-1229; doi: 10.1115/1.4037590 History: Received May 30, 2017; Revised July 13, 2017

Red blood cells (RBCs) are the most abundant cellular element suspended in blood. Together with the usual biconcave-shaped RBCs, i.e., discocytes, unusual-shaped RBCs are also observed under physiological and experimental conditions, e.g., stomatocytes and echinocytes. Stomatocytes and echinocytes are formed from discocytes and in addition can revert back to being discocytes; this shape change is known as the stomatocyte–discocyte–echinocyte (SDE) transformation. To-date, limited research has been conducted on the numerical prediction of the full SDE transformation. Spring-particle RBC (SP-RBC) models are commonly used to numerically predict RBC mechanics and rheology. However, these models are incapable of predicting the full SDE transformation because the typically employed bending model always leads to numerical instability with severely deformed shapes. In this work, an enhanced SP-RBC model is proposed in order to extend the capability of this model type and so that the full SDE transformation can be reproduced. This is achieved through the leveraging of an advanced bending model. Transformed vesicle and RBC shapes are predicted for a range of reduced volume and reduced membrane area difference (MAD), and very good agreement is obtained in the comparison of predicted shapes with experimental observations. Through these predictions, vesicle and SDE transformation phase diagrams are developed and, importantly, in the SDE case, shape boundaries are proposed for the first time relating RBC shape categories to RBC reduced volume and reduced MAD.

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Sherwood, L. , 2010, Human Physiology: From Cells to Systems, Brooks/Cole, Cengage Learning, Belmont, Australia.
Skalak, R. , Ozkaya, N. , and Skalak, T. C. , 1989, “ Biofluid Mechanics,” Annu. Rev. Fluid Mech., 21(1), pp. 167–204. [CrossRef]
Ruef, P. , and Linderkamp, O. , 1999, “ Deformability and Geometry of Neonatal Erythrocytes With Irregular Shapes,” Pediatr. Res., 45(1), pp. 114–119. [CrossRef] [PubMed]
Wolanskyj, A. P. , 2013, “ Benign Hematology,” Mayo Clinic Internal Medicine Board Review, R. D. Ficalora , ed., Oxford University Press, Oxford, UK, pp. 497–511. [CrossRef]
Chen, Y. , Cai, J. , and Zhao, J. , 2002, “ Diseased Red Blood Cells Studied by Atomic Force Microscopy,” Int. J. Nanosci., 1(5&6), pp. 683–688. [CrossRef]
Zehnder, L. , Schulzki, T. , Goede, J. S. , Hayes, J. , and Reinhart, W. H. , 2008, “ Erythrocyte Storage in Hypertonic (SAGM) or Isotonic (PAGGSM) Conservation Medium: Influence on Cell Properties,” Vox Sang., 95(4), pp. 280–287. [CrossRef] [PubMed]
Chabanel, A. , Reinhart, W. , and Chien, S. , 1987, “ Increased Resistance to Membrane Deformation of Shape-Transformed Human Red Blood Cells,” Blood, 69(3), pp. 739–743. [PubMed]
Waugh, R. E. , 1996, “ Elastic Energy of Curvature-Driven Bump Formation on Red Blood Cell Membrane,” Biophys. J., 70(2), pp. 1027–1035. [CrossRef] [PubMed]
Gedde, M. M. , Davis, D. K. , and Huestis, W. H. , 1997, “ Cytoplasmic pH and Human Erythrocyte Shape,” Biophys. J., 72(3), pp. 1234–1246. [CrossRef] [PubMed]
Jan, K. M. , and Chien, S. , 1973, “ Role of Surface Electric Charge in Red Blood Cell Interactions,” J. Gen. Physiol., 61(5), pp. 638–654. [CrossRef] [PubMed]
Hsu, R. , Kanofsky, J. R. , and Yachnin, S. , 1980, “ The Formation of Echinocytes by the Insertion of Oxygenated Sterol Compounds Into Red Cell Membranes,” Blood, 56(1), pp. 109–117. [PubMed]
Lim, H. W. G. , Wortis, M. , and Mukhopadhyay, R. , 2002, “ Stomatocyte-Discocyte-Echinocyte Sequence of the Human Red Blood Cell: Evidence for the Bilayer-Couple Hypothesis From Membrane Mechanics,” Proc. Natl. Acad. Sci. U. S. A., 99(26), pp. 16766–16769. [CrossRef] [PubMed]
Skalak, R. , Tozeren, A. , Zarda, R. P. , and Chien, S. , 1973, “ Strain Energy Function of Red Blood Cell Membranes,” Biophys. J., 13(3), pp. 245–264. [CrossRef] [PubMed]
Khairy, K. , and Howard, J. , 2011, “ Minimum-Energy Vesicle and Cell Shapes Calculated Using Spherical Harmonics Parameterization,” Soft Matter, 7(5), pp. 2138–2143. [CrossRef]
Fedosov, D. A. , Caswell, B. , and Karniadakis, G. E. , 2010, “ A Multiscale Red Blood Cell Model With Accurate Mechanics, Rheology, and Dynamics,” Biophys. J., 98(10), pp. 2215–2225. [CrossRef] [PubMed]
Dupin, M. M. , Halliday, I. , Care, C. M. , and Munn, L. L. , 2008, “ Lattice Boltzmann Modelling of Blood Cell Dynamics,” Int. J. Comput. Fluid Dyn., 22(7), pp. 481–492. [CrossRef]
Imai, Y. , Nakaaki, K. , Kondo, H. , Ishikawa, T. , Teck Lim, C. , and Yamaguchi, T. , 2011, “ Margination of Red Blood Cells Infected by Plasmodium Falciparum in a Microvessel,” J. Biomech., 44(8), pp. 1553–1558. [CrossRef] [PubMed]
Dao, M. , Li, J. , and Suresh, S. , 2006, “ Molecularly Based Analysis of Deformation of Spectrin Network and Human Erythrocyte,” Mater. Sci. Eng. C, 26(8), pp. 1232–1244. [CrossRef]
Fedosov, D. A. , Caswell, B. , Suresh, S. , and Karniadakis, G. E. , 2011, “ Quantifying the Biophysical Characteristics of Plasmodium-Falciparum-Parasitized Red Blood Cells in Microcirculation,” Proc. Natl. Acad. Sci. U. S. A., 108(1), pp. 35–39. [CrossRef] [PubMed]
Bow, H. , Pivkin, I. V. , Diez-Silva, M. , Goldfless, S. J. , Dao, M. , Niles, J. C. , Suresh, S. , and Han, J. , 2011, “ A Microfabricated Deformability-Based Flow Cytometer With Application to Malaria,” Lab Chip, 11(6), pp. 1065–1073. [CrossRef] [PubMed]
Fedosov, D. A. , Caswell, B. , and Karniadakis, G. , 2010, “ Dissipative Particle Dynamics Modeling of Red Blood Cells,” Computational Hydrodynamics of Capsules and Biological Cells, C. Pozrikidis , ed., CRC Press, Boca Raton, FL, pp. 183–218. [CrossRef]
Li, X. , Vlahovska, P. M. , and Karniadakis, G. E. , 2013, “ Continuum- and Particle-Based Modeling of Shapes and Dynamics of Red Blood Cells in Health and Disease,” Soft Matter, 9(1), pp. 28–37. [CrossRef] [PubMed]
Pivkin, I. , and Karniadakis, G. , 2008, “ Accurate Coarse-Grained Modeling of Red Blood Cells,” Phys. Rev. Lett., 101(11), pp. 1–4. [CrossRef]
Alizadehrad, D. , Imai, Y. , Nakaaki, K. , Ishikawa, T. , and Yamaguchi, T. , 2012, “ Quantification of Red Blood Cell Deformation at High-Hematocrit Blood Flow in Microvessels,” J. Biomech., 45(15), pp. 2684–2689. [CrossRef] [PubMed]
Lim, H. W. G. , Wortis, M. , and Mukhopadhyay, R. , 2008, “ Red Blood Cell Shapes and Shape Transformations: Newtonian Mechanics of a Composite Membrane,” Soft Matter, Wiley-VCH Verlag GmbH, Weinheim, Germany, pp. 139–204.
Evans, E. A. , and Fung, Y.-C. , 1972, “ Improved Measurements of the Erythrocyte Geometry,” Microvasc. Res., 4(4), pp. 335–347. [CrossRef] [PubMed]
Boey, S. K. , Boal, D. H. , and Discher, D. E. , 1998, “ Simulations of the Erythrocyte Cytoskeleton at Large Deformation—I: Microscopic Models.,” Biophys. J., 75(3), pp. 1573–1583. [CrossRef] [PubMed]
Li, J. , Dao, M. , Lim, C. T. , and Suresh, S. , 2005, “ Spectrin-Level Modeling of the Cytoskeleton and Optical Tweezers Stretching of the Erythrocyte,” Biophys. J., 88(5), pp. 3707–3719. [CrossRef] [PubMed]
Gompper, G. , and Kroll, D. M. , 1996, “ Random Surface Discretizations and the Renormalization of the Bending Rigidity,” J. Phys. I, 6(10), pp. 1305–1320.
Chen, M. , and Boyle, F. J. , 2014, “ Investigation of Membrane Mechanics Using Spring Networks: Application to Red-Blood-Cell Modelling,” Mater. Sci. Eng. C, 43, pp. 506–516. [CrossRef]
Boal, D. , 2012, “ Introduction to the Cell,” Mechanics of the Cell, Cambridge University Press, Cambridge, UK, pp. 1–24.
Housner, G. W. , and Hudson, D. E. , 1980, “ Dynamics of a Particle,” Applied Mechanics Dynamics, California Institute of Technology, Pasadena, CA, pp. 26–47.
Svetina, S. , and Zeks, B. , 2002, “ Shape Behavior of Lipid Vesicles as the Basis of Some Cellular Processes,” Anat. Rec., 268(3), pp. 215–225. [CrossRef] [PubMed]
Käs, J. , and Sackmann, E. , 1991, “ Shape Transitions and Shape Stability of Giant Phospholipid Vesicles in Pure Water Induced by Area-to-Volume Changes,” Biophys. J., 60(4), pp. 825–844. [CrossRef] [PubMed]
Käs, J. , Sackmann, E. , Podgornik, R. , Svetina, S. , and Žekš, B. , 1993, “ Thermally Induced Budding of Phospholipid Vesicles—A Discontinuous Process,” J. Phys. II, 3(5), pp. 631–645.
Seifert, U. , Berndl, K. , and Lipowsky, R. , 1991, “ Shape Transformations of Vesicles: Phase Diagram for Spontaneous-Curvature and Bilayer-Coupling Models,” Phys. Rev. A, 44(2), pp. 1182–1202. [CrossRef] [PubMed]
Bessis, M. , 1973, “ Red Cell Shapes: An Illustrated Classification and Its Rationale,” Red Cell Shape, M. Bessis , R. I. Weed , and P. Leblond , eds., Springer, Berlin, pp. 1–26. [CrossRef]
Brailsford, J. D. , Korpman, R. A. , and Bull, B. S. , 1980, “ Crenation and Cupping of the Red Cell: A New Theoretical Approach—Part II: Cupping,” J. Theor. Biol., 86(3), pp. 531–546. [CrossRef] [PubMed]
Jay, A. W. , 1975, “ Geometry of the Human Erythrocyte—I: Effect of Albumin on Cell Geometry,” Biophys. J., 15(3), pp. 205–222. [CrossRef] [PubMed]
Pan, W. , Fedosov, D. A. , Caswell, B. , and Karniadakis, G. E. , 2011, “ Predicting Dynamics and Rheology of Blood Flow: A Comparative Study of Multiscale and Low-Dimensional Models of Red Blood Cells.,” Microvasc. Res., 82(2), pp. 163–170. [CrossRef] [PubMed]
Helfrich, W. , 1973, “ Elastic Properties of Lipid Bilayers: Theory and Possible Experiments,” Z. Naturforsch. C, 28(11–12), pp. 693–703. [PubMed]


Grahic Jump Location
Fig. 1

Schematic diagrams of (a) a healthy RBC and (b) the RBC membrane. The PM consists mainly of a lipid bilayer, while the cytoskeleton is a hyper-elastic-behaving network attached to the PM via anchoring proteins.

Grahic Jump Location
Fig. 2

Illustration of the curvature of particle i. The shaded area is the area occupied by particle i, i.e., Ai, Lj is the length of edge j, n1 and n2 are the normal vectors to the two triangular elements which share edge j, and θj is the angle formed by these vectors, i.e., the included angle.

Grahic Jump Location
Fig. 3

Comparison of the geometry of (a) the initial discocyte with (b) the ellipsoid which is used to define the reference spring lengths, Lo,j, for the WLC springs. The magnified sections show the equivalent spring in both meshes.

Grahic Jump Location
Fig. 4

The numerically predicted Δa–v phase diagram of vesicle transformation. The predicted transformed shapes show contour maps of curvature. The boundaries shown are reproduced with permission from Seifert et al. [36].

Grahic Jump Location
Fig. 5

Predicted echinocytes III with a reduced volume v ≈ 0.645 and a reduced MAD Δa ≈ 1.66. These transformed shapes were predicted using meshes with (a) 2664, (b) 4636, (c) 8610, and (d) 12,930 triangular elements.

Grahic Jump Location
Fig. 6

The numerically predicted Δa–v phase diagram of the SDE transformation from discocytes to echinocytes I, II, and III

Grahic Jump Location
Fig. 7

The numerically predicted Δa–v phase diagram of the SDE transformation from discocytes to stomatocytes I, II, and III

Grahic Jump Location
Fig. 9

Calculation of the model bending energy using curvature. i is the particle index. For an edge jξj and ζj are the surface-normal vectors to the neighboring elements, Lj is the edge length, and θj is the subtended angle.

Grahic Jump Location
Fig. 8

A thin membrane element of area A. R1 and R2 are the principle radii of curvature at the neutral plane, and H is the element thickness.

Grahic Jump Location
Fig. 10

Calculation of the model bending energy using the subtended angle. ξ and ζ are the surface normal vectors, L is the common-edge length, and θ is the subtended angle.



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