Procedia IUTAM 10 ( 2014 ) 193 – 200

2210-9838 © 2013 Published by Elsevier Ltd. Open access under CC BY-NC-ND license.

Selection and/or peer-review under responsibility of the Organizing Committee of The 23rd International Congress of Theoretical and Applied Mechanics, ICTAM2012doi: 10.1016/j.piutam.2014.01.018

Available online at www.sciencedirect.com

ScienceDirect

23rd International Congress of Theoretical and Applied Mechanics

Toward the multi-scale simulation for a human body using thenext-generation supercomputer

Yoichiro Matsumotoa,∗ , Satoshi Iib , Seiji Shiozakic ,

Kazuyasu Sugiyamaa,c , Shu Takagia,c

aSchool of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, JapanbDepartment of Mechanical Science and Bioengineering, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan

cComputational Science Research Program, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan

Abstract

We have developed novel numerical methods for fluid-structure and fluid-membrane interaction problems. The basic

equation set is formulated in a full Eulerian framework. The method is based on the finite difference volume-of-fluid

scheme with fractional step algorithm. It is validated through a numerical solution to a deformable vesicle problem, and

applied to blood flows including red blood cells (RBCs) and platelets. Further, to gain insight into the mechanism of

thrombus formation, a stochastic Monte Carlo model to describe the platelet-vessel wall interaction is incorporated into

the Eulerian method. The effect of the RBCs on the platelet motion is discussed.

c© 2013 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Yukio kaneda and Tomasz A.

Kowalewski.

Keywords: Eulerian method, fluid-structure interaction, blood flow, stochastic Monte Carlo method

1. Introduction

A 10-Peta-flops class supercomputer (the K computer) has been developed in the Next-Generation Su-

percomputer Project in Japan, and it will go into full-scale operation at the end of September 2012 [1].

The software development as well as the hardware development have been highly expected. Especially, the

development of human body simulator has been assigned as a grand challenge program for the effective use

of the K computer, in which the multi-scale and multi-physics natures of the living matter are emphasized.

Under this concept, we have been developing the multi-scale simulator for a living human body. Basic strat-

egy of the simulator is to utilize the medical image data taken by MRI, CT, or ultrasound for the prediction

of disease and planning of therapy. For this purpose, we have developed full Eulerian Fluid-Structure Inter-

action (FSI) solver [2–8] without mesh generation procedure, which enables us to conduct the simulations

directly from medical images. In this paper, we shall focus on the application to microcirculation systems

including deformable blood cells.

Blood flow plays important roles in several functions to sustain life, and exhibits phenomenologically

rich behavior owing to its multi-physics nature, the complex geometry, and the suspension of blood cells.

∗Corresponding author. E-mail address: ymats@fel.t.u-tokyo.ac.jp.

© 2013 Published by Elsevier Ltd. Open access under CC BY-NC-ND license.

Selection and/or peer-review under responsibility of the Organizing Committee of The 23rd International Congress of Theoretical and Applied Mechanics, ICTAM2012

194 Yoichiro Matsumoto et al. / Procedia IUTAM 10 ( 2014 ) 193 – 200

In a microcirculation system with a sub-millimetric scale vessel, a red blood cell (RBC) subjected to a

shearing fluid motion undergoes large deformation. Such a distinctive deformability and a dense particulate

flow nature characterize rheological properties, which are relevant to transport phenomena and hemostatic

processes [9]. Recent advances in high-performance computing and numerical methods have promoted

interest in hemodynamic simulations [10–12]. In order to obtain practically important knowledge of the

integrated dynamics, it is important to perform a large-scale computation close to the realistic scale since

the system is hydrodynamically and geometrically nonlinear in size. An efficient and scalable numerical

method is desirable for the large-scale computation.

Conventionally, the computational fluid dynamics is commonly described in an Eulerian way, while the

computational structure dynamics is more straightforward to be described in an Lagrangian way. The cou-

pling of the fluid and structure dynamics is a nontrivial task due to such a difference in the numerical frame-

work. There are currently several approaches classified with respect to the computational treatment how the

kinematic and dynamic interactions are coupled on the moving interface. When addressing moving inter-

face problems, one has preferably employed the Lagrangian method using a moving finite element mesh.

The numerical methods include Arbitrary Lagrangian Eulerian [13], Deforming-Spatial-Domain/Stabilized

Space-Time [14] and Immersed Boundary [15] approaches. Once the body-fitted mesh is provided, the state-

of-the-art Lagrangian approaches are satisfactory for achieving accurate predictions. However, for a system

involving complex geometry of solid and/or a large number of bodies, it requires not only a high computa-

tional cost but also great efforts to generate the high quality mesh and to reconstruct the mesh topology when

the mesh element is added/deleted. Moreover, in a large parallel computation, one encounters a nontrivial

issue how the respective quantities on the Eulerian and Lagrangian meshes are adequately communicated to

keep the load balance of each compute node.

To release the FSI simulation from the mesh generation/reconstruction procedure, and to extend the

applicability to certain additional classes of problems, full Eulerian (fixed-mesh) methods have been devel-

oped [16–19]. Following an idea of a one-fluid formulation [20], in which one set of governing equations

is written over the whole field, as frequently used in multiphase flow simulations, the authors have for-

mulated the basic equations for general FSI problems suited to the finite difference method [4, 7]. The

developed method has revealed practical advantages of geometrical flexibility [2, 3, 8] since it can directly

access voxel data and avoid a breakdown in a large deformation owing to the absence of the mesh distortion

problem. The monolithic formulation discretized on regular grids systematically offers numerical stability

conditions [5]. Further, in view of computational efficiency, it readily exploits vector processing and reduces

a computational-load imbalance in parallel computing. The full Eulerian method allows us to utilize effi-

cient computational techniques cultivated in the field of computational fluid dynamics, that is an advantage

in the realization of massively parallel computing [21].

In this paper, firstly, the basic equations of the system consisting of Newtonian fluid and hyperelastic

structure/membrane will be outlined. Secondly, the validity of the advocated numerical method will be

demonstrated. Thirdly, the method will be applied to three-dimensional blood flows including blood cells

in a capillary vessel and in a microchannel, and the relevance of the RBC to the thrombus formation will be

discussed.

2. Basic equations

We consider a system comprised of Newtonian fluid and visco-hyperelastic material regions, and also

membrane surfaces, which enclose portions of fluid. The fluid and solid phases are incompressible (namely,

∇ · v = 0, here, v is the velocity vector) with the identical density ρ as in many analyses for biological

systems. We assume that the membrane has no thickness, but the elastic tension thereon causes the traction

jump and thus the discontinuity of the fluid stress. Here, we shall briefly explain the Eulerian formulation

[4, 7]. The momentum equation for the whole field is written monolithically as

ρ(∂t + v · ∇)v = ∇ · σ + |∇φ| f , (1)

where σ denotes the Cauchy stress tensor, φ the volume fraction of a dispersed phase (namely, the VOF

function [22]), and f the local force density vector to describe the traction jump.

195 Yoichiro Matsumoto et al. / Procedia IUTAM 10 ( 2014 ) 193 – 200

The kinematics of structure/membrane is described by φ, a reference mean curvature κR(= −∇ · nR/2),

a left Cauchy–Green deformation tensor B(= F · FT), and a tensor Gs(= F · PR · FT) (here n(= ∇φ/|∇φ|)is the unit normal vector, F the deformation gradient tensor, P(= I − nn) the surface projection tensor, and

the suffix R indicates the reference configuration), which are temporally updated by solving the transport

equations

(∂t + v · ∇)φ = 0, (2)

(∂t + v · ∇)κR = 0, (3)

(∂t + v · ∇)B = L · B + B · LT, (4)

(∂t + v · ∇)Gs = L · Gs + Gs · LT, (5)

where L(= ∇vT) is the velocity gradient tensor.

We write the Cauchy stress in a fluid-structure mixture form

σ = −pI + 2(1 − φ)μf D′ + φσ′s, (6)

where p denotes the pressure, I the unit tensor, μ the viscosity, and D(= (L + LT)/2) the strain rate tensor.

The prime on the second-order tensor stands for the deviatoric tensor, and the suffices f and s indicate the

fluid and solid phases, respectively. For an incompressible visco-hyperelastic material obeying Newtonian

and neo-Hookean constitutive laws, the deviatoric stress tensor of solid is given by

σ′s = 2μs D′ +GB′, (7)

where G is the modulus of transverse elasticity [4]. In consideration of the surface elasticity and the bending

resistance on the membrane, the local force density is given by

f = (P · ∇) · (τs + {(P · ∇) · m} · Pn), (8)

where τs denotes the in-plane Cauchy stress tensor, and m the bending moment tensor. For a hyperelastic

membrane, the in-plane stress is expressed as

τs =2√

IIs + 1

[∂Ws

∂Is

Bs + (IIs + 1)∂Ws

∂IIs

P], (9)

where Bs(= P · F · PR · FT · PT) denotes the surface left Cauchy-Green deformation tensor, Ws the surface

strain energy potential, and Is(= tr(Bs) − 2) and IIs(= {tr(Bs)2 − tr(B2

s )}/2 − 1) are the invariants of Bs

[23, 24]. Note that Bs can be determined on the fixed mesh (instead of incorporating a Lagrangian mesh)

from a relationship Bs = P · Gs · PT. The bending moment is written as

m = Eb(κ − κR P), (10)

where Eb denotes the bending stiffness [25], and κ(= −∇n) the curvature tensor.

In numerical simulations, the basic equations are solved by means of a finite-difference method using

a regular Cartesian mesh. We follow a conventional staggered grid arrangement. We employ the Simpli-

fied MAC algorithm [26] to temporally update the incompressible field. For detailed descriptions on the

numerical schemes and algorithms, and the validation tests, we refer the readers to [4, 5, 7].

3. Results and discussion

3.1. A deformable spherical membrane in a shear flow

To demonstrate the validity of the numerical method, a deformed motion of a spherical vesicle subjected

to an unbounded shear flow is addressed [6]. To make comparisons with available numerical works [27, 28],

196 Yoichiro Matsumoto et al. / Procedia IUTAM 10 ( 2014 ) 193 – 200

we neglect the bending force, and consider the neo-Hookean membrane, of which the surface strain energy

potential is given by

Ws =Es

6

(Is +

1

IIs + 1− 1

), (11)

where Es is the surface elastic modulus. The viscosity inside the membrane is the same as that outside it.

The system is characterized by two dimensionless parameters: the Reynolds number Re = ργd2/(4μ), and

the capillary number Ca = μγd/(2Es), here d denotes the vesicle diameter and γ the shear rate. In the

present study, the Reynolds number is fixed at Re = 0.01, while the capillary number is varied.

The simulated results at the fully developed state are shown in Fig. 1 a, b, c. The vesicle deformation

is confirmed to be higher with increasing the capillary number Ca. To quantify the deformation level, the

Taylor deformation parameter D(= (l − s)/(l + s)) (here l and s are the semi-major and semi-minor axes of

an ellipse, respectively) is introduced. Here, l and s are determined through fitting the obtained interface

on a shear plane. The temporal evolutions of D for various capillary numbers are reported in Fig. 1 d. The

present results are in better agreement with the well-validated results [27, 28] with decreasing the grid width

Δx, indicating the convergence behavior and the validity of the present approach.

0

0.1

0.2

0.3

0.4

0.5

(d) 0.6

0 0.5 1 1.5 2

D

t

Ca=0.0125

Ca=0.025

Ca=0.05

Ca=0.1

Ca=0.2

Eggleton et al.Pozrikidis et al.Full Eluerian (Δx=d/16)Full Eluerian (Δx=d/32)Full Eluerian (Δx=d/64)

Fig. 1. The left panels show the shape of the vesicle subjected to the linear shear flow with a spacial resolution of Δx = d/32 at the

capillary numbers of (a) Ca = 0.0125, (b) Ca = 0.05 and (c) Ca = 0.2. The right panel d shows the the Taylor deformation parameter

D versus time t. The open-triangles to the results in [27] using the boundary element method and the filled-circles correspond to the

results in [28] using the IB method. The curves correspond to the present results for various spatial resolutions.

3.2. Blood flow in a capillary vessel

The Eulerian method is applied to a blood flow [8]. The system considered is enclosed by a vessel tube,

and includes RBCs and platelets. For the RBC, the Evans–Skalak model [23] is employed to describe the

surface strain energy potential

Ws =Es

6

[(Is + 1)2 + αII2

s − 2IIs − 1], (12)

where α is the resistance to the surface dilation. The platelet and the vessel wall are treated as the neo-

Hookean solid. Initially, the RBCs and platelets are randomly seeded. The volume fraction of RBC is

set to approximately Ht = 20% (here, Ht denotes the hematocrit). Initial shapes of the RBC and platelet

are of biconcave discoid with a diameter of 7.8 μm and spheroid with diameters of 2 μm × 2 μm × 1.2 μm,

respectively. The size of the computational domain is 44 μm×22 μm×22 μm, and the number of grid points

197 Yoichiro Matsumoto et al. / Procedia IUTAM 10 ( 2014 ) 193 – 200

is 320 × 160 × 160. The lumen diameter is 20 μm. To pump the fluid and the blood cells, the uniform

pressure gradient of 400 kPa/m is applied to the system. The periodic boundary condition is imposed in

the streamwise (x) direction, and wall boundary conditions are employed for both y and z directions. A

numerical simulation is run up to t = 75 ms. To advect the VOF function, the MTHINC method [29] is

employed.

Figure 2 visualizes the shape of the RBCs, platelets and vessel wall for three time instants. As the time

goes on, the RBCs rotate and tend to be more mixed, giving rise to a rich behavior. Each of RBCs has a

nonunique shape such as a parachute shape or slipper one. As well-known, the deformed RBCs are likely to

migrate in a direction from the vessel wall to the vessel core due to the hydrodynamic lift force. In Fig. 2,

the cell-free layer (in which the number density of the RBC is very low) is confirmed to form near the wall

and to be thicker over time.

(a) t = 7.5ms (b) t = 30ms (c) t = 75ms

Fig. 2. Snapshots of the RBCs (in red) and platelets (in yellow).

To clarify the effect of the RBCs on the platelet motion, Fig. 3 shows the trajectories of the respective

platelets in a radial direction for cases with and without RBCs. It is clearly shown that in the absence of

the RBC (Fig. 3 a), the platelets almost straightly drift in the flow with small radial fluctuation, whereas

in the presence of the RBC (Fig. 3 b), the level of the radial fluctuation is much higher, indicating that the

RBCs agitate the surrounding fluid flow and strongly affect the platelet motion. As shown in Fig. 3 b, the

level of the platelet fluctuation is preferentially high around the axis (r < 4 μm). It is explained from that

in the RBC-rich core region, the platelets are pushed out by the RBCs and thus much dispersed, whereas

in the cell-free layer, the radial motion is restricted. Further, a specific separation of the radial positions is

observed between the axial center (r < 4 μm) and near the wall (6 μm < r < 8 μm). Such a two-forked

tendency implies that once the platelets come closer to the cell-free layer, they hardly move back toward the

vessel core due to the barrier effect of the RBCs [30], and thus are likely to be trapped therein.

0

1

2

3

4

5

6

7

8

9

10

0 10 20 30 40 50 60 70

r c [

μm]

t [ms]

(a) Ht = 0%

0

1

2

3

4

5

6

7

8

9

10

0 10 20 30 40 50 60 70

r c [

μm]

t [ms]

(b) Ht = 20%

Fig. 3. Trajectories of the respective platelets in a radial direction.

3.3. Thrombosis simulatorWe have been developing a numerical model of an initial stage of thrombus formation, in which platelets

adhere to an injured vessel wall. Although the platelet aggregation is essential for a hemostatic process to

198 Yoichiro Matsumoto et al. / Procedia IUTAM 10 ( 2014 ) 193 – 200

arrest the bleeding, an exaggerated aggregation occludes the vessel leading to thrombosis, corresponding

to one of the most crucial diseases to possibly cause myocardial and cerebral infarctions [31, 32]. Fluid-

structure/membrane coupling simulations are expected to much improve the understanding of the dynamic

mechanism in the aggregation process beyond the epidemiological knowledge. It is nontrivial to numerically

reproduce the platelet adhesion since it involves a wide variety of phenomena ranging from molecular to

continuum scales. A major challenge is the modeling of the ligand-receptor type interaction between the

von Willebrand Factor (vWF) attaching to the injured wall and the Glycoprotein Ibα (GPIbα) to the platelet

membrane [33, 34] in a form to be incorporated into the Eulerian FSI model [4, 7]. The basic concept of the

multi-scale simulator is schematically illustrated in Fig. 4. The molecular scale interaction is reflected on

the continuum scale force in a gap between the platelet and the injured vessel in a stochastic manner. To

this end, a stochastic Monte Carlo simulation is performed to determine the formation and breakage of the

GPIbα-vWF bond, and also to evaluate the bond force [35].

Fig. 4. Multi-scale modeling on the platelets adhesion.

Here, we demonstrate preliminary results of blood flow simulations coupled with the stochastic mod-

el. We examine the relevance of the RBC-induced fluctuation (as mentioned in Sect. 3.2) to the platelet

adhesion in a channel flow. Initially, the platelet is positioned in the vicinity of the channel wall. Figure 5

visualizes the shape of the RBCs and the platelet at the fully developed state. Notably, in the absence of

the RBC (Fig. 5a), the platelet moves to the channel core, whereas in the presence of the RBC (Fig. 5a),

the platelet adheres to the wall due to the growth of the GPIbα-vWF bond as clearly shown in the bottom

panel. The platelet with an almost ellipsoidal shape undergoes the hydrodynamic lift force, and is likely to

migrate away from the wall if there exists no additional mechanism to transport the platelet to the wall. In

the presence of the RBC, however, the RBCs migrate to the channel core more rapidly than the platelet, and

thus push out the platelet toward the wall. Further, the RBCs induce the wall-normal fluctuation of the fluid

flow. Therefore, the RBC plays a role to increase the chance for the platelet to approach to the wall. It should

be noticed that the present results support the fact that the thrombus does not grow when the hematocrit Htis too small.

4. Conclusion and perspectives

We have developed the Eulerian numerical approach applicable for both fluid-structure and fluid-membrane

interaction problems [4, 6, 7]. The method facilitates the treatment of a time-dependent complex geometry.

199 Yoichiro Matsumoto et al. / Procedia IUTAM 10 ( 2014 ) 193 – 200

(a) without RBCs (no adhesion) (b) with RBCs

Fig. 5. Effect of the presence of the RBC (in red) on the platelet adhesion in the channel flow. The vWF-rich region on the channel

wall is in dark yellow. The color on the platelet surface indicates the number density of the GPIbα.

We applied the method to the vesicle deformation subjected to the shearing fluid motion, and the blood flow

in the capillary vessel. Further, we demonstrated the preliminary results using the thrombosis simulator, in

which the stochastic Monte Carlo method in consideration of the ligand-receptor interaction between the

proteins was incorporated [35]. The results illustrate that platelets are much easier to aggregate on the wall

in the presence of red blood cells.

It should be noticed that the Eulerian method makes it easily possible to realize a massively parallel

computation and to attain excellent scalability on state-of-the-art scalar-type supercomputers [21]. The in-

tegrated dynamics in e.g. a brain arteriole of 100 μm or longer diameter is expected to be covered using

thousands compute nodes of the K computer. Challenging tasks that should be tackled are improvements of

the thrombosis simulator to capture the later stages of the thrombus formation involving more complicated

phenomena such as the platelet activation, the change in the hemorheology, and the fibrin network forma-

tion/breakage. The modeling of the bio-chemical reactions and the substance diffusions caused after the

activation of platelet triggered by the GPIbα-vWF binding (Fig. 4), and the implementation of a robust time

advancement during the coupling are the ongoing subjects of the authors.

Acknowledgments

This research was supported by Research and Development of the Next-Generation Integrated Simula-

tion of Living Matter, a part of the Development and Use of the Next-Generation Supercomputer Project

of the Ministry of Education, Culture, Sports, Science and Technology (MEXT). Part of the results was

obtained by using the RIKEN Integrated Cluster of Clusters (RICC) and by early access to the K computer

at the RIKEN Advanced Institute for Computational Science.

References

[1] Yonezawa A, Watanabe T, Yokokawa M, Sato M, Hirao K, Advanced Institute for Computational Science (AICS) – Japanese

national high-performance computing research institute and its 10-Petaflops supercomputer “K”. Proc of SC ’11 State of thePractice Reports 2011; doi: 10.1145/2063348.2063366.

[2] Sugiyama K, Ii S, Takeuchi S, Takagi S, Matsumoto Y, Full Eulerian simulations of biconcave neo-Hookean particles in a

Poiseuille flow. Comput Mech 2010; 46: 147–157.

[3] Nagano N, Sugiyama K, Takeuchi S, Ii S, Takagi S, Matsumoto Y, Full Eulerian finite-difference simulation of fluid flow in

hyperelastic wavy channel. J Fluid Sci Tech 2010; 5: 475–490.

[4] Sugiyama K, Ii S, Takeuchi S, Takagi S, Matsumoto Y, A full Eulerian finite difference approach for solving fluid-structure

coupling problems. J Comput Phys 2011; 230: 596–627.

200 Yoichiro Matsumoto et al. / Procedia IUTAM 10 ( 2014 ) 193 – 200

[5] Ii S, Sugiyama K, Takeuchi S, Takagi S, Matsumoto Y, An implicit full Eulerian method for the fluid-structure interaction

problem. Int J Numer Meth Fluids 2011; 65: 150–165.

[6] Takagi S, Sugiyama K, Ii S, Matsumoto Y, A review of full Eulerian methods for fluid structure interaction problems. J ApplMech 2012; 79: 010911.

[7] Ii S, Gong X, Sugiyama K, Wu J, Huang H, Takagi S, A full Eulerian fluid-membrane coupling method with a smoothed volume-

of-fluid approach. Comm Comput Phys 2012; 12: 544–576.

[8] Ii S, Sugiyama K, Takagi S, Matsumoto Y, A computational blood flow analysis in a capillary vessel including multiple red blood

cells and platelets. J Biomech Sci Engrg 2012; 7: 72–83.

[9] Popel AS, Johnson PC, Microcirculation and and hemorheology. Annu Rev Fluid Mech 2005; 37: 43–69.

[10] Pozrikidis S (Ed.), Modeling and simulation of capsules and biological cells. Boca Raton: Chapman & Hall/CRC; 2003.

[11] Bungartz HJ, Schafer M (Eds.), Fluid-structure interaction – modelling, simulation, optimisation. Lecture Notes Comput SciEngrg 2006; 53.

[12] Pozrikidis C (Ed.), Computational hydrodynamics of capsules and biological cells. Boca Raton: CRC Press; 2011.

[13] Hirt CW, Amsden AA, Cook JL, An arbitrary Lagrangian-Eulerian computing method for all flow speeds. J Comput Phys 1974;

14: 227–253.

[14] Tezduyar TE, Behr M, Liou J, A new strategy for finite element computations involving moving boundaries and interfaces - The

deforming-spatial-domain/space-time procedure: I. The concept and the preliminary numerical tests. Comput Meth Appl MechEngrg 1992; 94: 339–351.

[15] Peskin CS, Flow patterns around heart valves: a numerical method. J Comput Phys 1972; 10: 252–271.

[16] Van Hoogstraten PAA, Slaats PMA, Baaijens FPT, An Eulerian approach to the finite element modelling of neo-Hookean rubber

material. Appl Sci Res 1991; 48: 193–210.

[17] Liu C, Walkington NJ, An Eulerian description of fluids containing visco-elastic particles. Arch Ration Mech Anal 2001; 159:

229–252.

[18] Dunne T, An Eulerian approach to fluid-structure interaction and goal-oriented mesh adaptation. Int J Numer Meth Fluids 2006;

51: 1017–1039.

[19] Cottet GH, Maitre E, Milcent T, Eulerian formulation and level set models for incompressible fluid-structure interaction. MathModelling Numer Anal 2008; 42: 471–492.

[20] Tryggvason G, Sussman M, Hussaini MY, Immersed boundary methods for fluid interfaces. in Prosperetti A, Tryggvason G

(Eds.), Computational Methods for Multiphase Flow, Cambridge: Cambridge University Press; 2007, Chap 3.

[21] Sugiyama K, Kawashima Y, Koyama H, Noda S, Ii S, Takagi S, et al., Development of explicit Eulerian finite difference solver

for large-scale fluid-structure interaction systems. Proc High Peformance Computing Symp 2012 2012; IPSJ- HPCS2012056.

[22] Hirt CW, Nichols BD, Volume of fluid (VOF) method for the dynamics of free boundaries. J Comput Phys 1981; 38: 201–225.

[23] Evans EA, Skalak A, Mechanics and Thermodynamics of Biomembranes. Boca Raton: CRC; 1980.

[24] Barthes-Biesel D, Rallison JM, The time-dependent deformation of a capsule freely suspended in a linear shear flow. J FluidMech 1981; 113: 251–267.

[25] Pozrikidis C, Effect of membrane bending stiffness on the deformation of capsules in simple shear flow. J Fluid Mech 2001; 440:

269–291.

[26] Amsden AA, Harlow FH, A simplified MAC technique for incompressible fluid flow calculations. J Comput Phys 1970; 6:

322–325.

[27] Pozrikidis C, Finite deformation of liquid capsules enclosed by elastic membranes in simple shear flow. J Fluid Mech 1995; 297:

123–152.

[28] Eggleton CD, Popel AS, Large deformation of red blood cell ghosts in a simple shear flow. Phys Fluids 1998; 10: 1834–1845.

[29] Ii S, Sugiyama K, Takeuchi S, Takagi S, Matsumoto Y, Xiao F, An interface capturing method with a continuous function: the

THINC method with multi-dimensional reconstruction. J Comput Phys 2012; 231: 2328–2358.

[30] Crowl LM, Fogelson AL, Analysis of mechanisms for platelet near-wall excess under arterial blood flow conditions. J FluidMech 2011; 676: 348–375.

[31] Stoltz JF, Singh M, Riha P, Hemorheology in Practice. Amsterdam: IOS; 1999.

[32] Wootton DM, Ku DN, Fluid mechanics on vascular systems, diseases, and thrombosis. Annu Rev Biomed Engrg 1999; 1: 299–

329.

[33] Coller BS, Peerschke EI, Scudder LE, Sullivan CA, Studies with a murine monoclonal antibody that abolishes ristocetin-induced

binding of von Willebrand factor to platelets – additional evidence in support of GPIb as a platelet receptor for von Willebrand

factor. Blood 1983; 61: 99–110.

[34] Goto S, Salomon DR, Ikeda Y, Ruggeri ZM, Characterization of the unique mechanism mediating the shear-dependent binding

of soluble von Willebrand factor to platelets. J Biol Chem 1995; 270: 23352–23361.

[35] Shiozaki S, Ishikawa KL, Takagi S, Numerical study on platelet adhesion to vessel walls using the kinetic Monte Carlo method.

J Biomech Sci Engrg 2012; 7: 275–283.