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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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Figures

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. 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. 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. 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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