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INTEGRATED
NANO-
BIOMECHANICS
INTEGRATED
NANO-
BIOMECHANICS

Edited by
TAKAMI YAMAGUCHI
TAKUJI ISHIKAWA
YOHSUKE IMAI
Elsevier
Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands
The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom
50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States

# 2018 Elsevier Inc. All rights reserved.

No part of this publication may be reproduced or transmitted in any form or by any means, electronic or
mechanical, including photocopying, recording, or any information storage and retrieval system, without
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This book and the individual contributions contained in it are protected under copyright by the Publisher
(other than as may be noted herein).

Notices
Knowledge and best practice in this field are constantly changing. As new research and experience broaden
our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using
any information, methods, compounds, or experiments described herein. In using such information or methods
they should be mindful of their own safety and the safety of others, including parties for whom they have a professional
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liability for any injury and/or damage to persons or property as a matter of products liability, negligence or
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CONTRIBUTORS

Hamid Hamed Department of Chemical & Petroleum Engineering,

Sharif University of Technology, Tehran, Iran
Yohsuke Imai School of Engineering, Tohoku University, Sendai, Japan
Takuji Ishikawa Department of Finemechanics; Department of
Bioengineering and Robotics, Tohoku University, Sendai, Japan
Hiroki Kamada Department of Diagnostic Radiology, Tohoku University
Hospital, Sendai, Japan
Azusa Kage Department of Bioengineering and Robotics, Tohoku
University, Sendai, Japan
Kenji Kikuchi Department of Finemechanics, Tohoku University,
Sendai, Japan
Rui Lima MEtRiS, Department of Mechanical Engineering, Minho
University, Guimarães, Portugal; Transport Phenomena Research
Center (CEFT), Engineering Faculty, University of Porto, Porto,
Portugal
Noriaki Matsuki Department of Biomedical Engineering, Graduate
School of Engineering, Okayama University of Science, Okayama,
Japan
Fumio Mizuno Department of Electronics and Intelligent Systems,
Tohoku Institute of Technology, Sendai, Japan
Sephanie Nix Department of Machine Intelligence and Systems
Engineering, Akita Prefectural University, Akita, Japan
Toshihiro Omori Department of Bioengineering and Robotics,
Tohoku University, Sendai, Japan
Maryam Saadatmand Department of Chemical & Petroleum
Engineering, Sharif University of Technology, Tehran, Iran
Yuji Shimogonya Frontier Research Institute for Interdisciplinary
Sciences, Tohoku University, Sendai, Japan
Hironori Ueno Molecular Function & Life Sciences, Aichi University
of Education, Kariya, Japan
Takami Yamaguchi Biomedical Engineering, Tohoku University, Sendai,
Japan

xiii
PREFACE

This is a book on integrated nanobiomechanics that is a com-

pilation of the studies that underwent in the research group sup-
ported by the Grant-in-Aid for Specially Promoted Research by the
Ministry of Education, Culture, Sports, Science, and Technology
(MEXT) No. 25000008 (2013–17) under the same title of “inte-
grated nanobiomechanics.” When we started the research, we
aimed to establish this field of study, based on mechanical basis,
to understand life itself to predict the progression of many dis-
eases quantitatively. We claimed that this would provide a novel
tool for clinical treatment, and the reliability of treatment should
be considerably improved compared with existing empirics-
based treatments. Integrated nanobiomechanics will also acceler-
ate, we also claimed, the development of new medicine, since one
would become able to discuss the effects of medicine by perform-
ing, for example, computational simulations instead of time-
consuming experiments. In order to approach this goal, we tried
to establish integrated nanobiomechanics by modeling multiscale
physical and biomedical phenomena, from the molecular level, to
the cellular, tissue, and organ levels, to the whole-body level. Stud-
ies were extended to a wide variety of fields, and we believe that
understandings of the nanoscale biological phenomena greatly
progressed.
It is our greatest pleasure to report the results of the collabora-
tive studies carried out by the grant, and we would like to hear
from wide audience about the study. We also would like to extend
our sincere gratitude to the Japanese Ministry of Education, Cul-
ture, Sports, Science, and Technology (MEXT) for their generous
support and to the colleagues and supporters of the study at
Tohoku University, Japan.

Takami Yamaguchi
Takuji Ishikawa
Yohsuke Imai

xv
INTRODUCTION
1
Takami Yamaguchi
Biomedical Engineering, Tohoku University, Sendai, Japan

Biomechanics contains a wide variety of research fields related

to biology and mechanics. It was conventionally regarded to deal
with rather mechanical aspects of biology such as motion, defor-
mation, and flow of biological existence. However, nowadays, bio-
mechanics extends its realm to almost all essential parts of studies
of life. Embryology is a good example of the extension of biome-
chanics to the deepest secret of life. The very first most important
event in the embryonic development of the fetus, the gastrulation,
is now one of the most actively studied phenomena of cellular
biomechanics. Through gastrulation, we were transformed from
two-dimensional object to truly three-dimensional existence.
Biomechanics of solid structure is mandatory to understand the
process. Moreover, the determination of the laterality of the body
is now known as being governed by the flow produced by the
flagella motion in the fetus.
Migration, attachment, and invasion of white blood cells
(WBCs) are the key phenomenon in the inflammatory response
of the living system against infection. To construct a comprehen-
sive view of the recruit, compilation, and immune activity of the
WBCs, we find the biomechanics mandatory. Some key phenom-
ena should be clarified by mechanics, such as how the WBCs
come close to the vascular inner wall in the blood flow streamline,
how it is attached to the endothelium, and how it goes across the
intraendothelial junction to work in the extracellular inflamma-
tory region. These questions can never be fully answered without
the consideration of blood flow and cellular mechanics. Chemical
attraction, which is commonly recalled by biologists, cannot
explain the phenomena because the blood flow and the concen-
tration boundary layer produced by the flow interfere the trans-
mission of chemical substances between the vascular wall and
flowing cells.

Integrated Nano-Biomechanics. https://doi.org/10.1016/B978-0-323-38944-0.00001-2

# 2018 Elsevier Inc. All rights reserved. 1
2 Chapter 1 INTRODUCTION

Above all, the discovery of the mechanical transduction of the

cell opened a wide gate to a new discipline of mechanobiology
that can help in the understanding of integrated biological prob-
lem. Living system lives in the mechanical environment from its
first stage of germs to the senescent degradation. It is now widely
recognized that there is no complete nor fruitful study on the liv-
ing system without considering its mechanical condition and the
response and adaptation to it.
Needless to say, progress of biology in the 20th and 21st
century was remarkable. Led by the development of molecular
biology, understanding of function and structure of living system
has been incredibly advanced in these centuries. To be notewor-
thy, from the late 20th century, mechanical responses of living sys-
tem drew strong attention of biologists who had never been
interested in or even neglected mechanical environment of living
cells. First, it was recognized by the biomechanists that endothe-
lial cells of relatively large vessels have a capability of sensing fluid
mechanical stress and adapt to the shear stress exerted by the
blood flow. Before that time, these mechanical responses were
interesting to a limited group of biomechanists and were esti-
mated as rather a peripheral phenomenon to the mainstream of
biology. Most of biologists tend to regard the mechanical phe-
nomena as a toy of mechanists who do not understand the main-
stream of biology. However, studies have revealed that many
biological processes that seem to be purely biological not
mechanical are caused and governed by the mechanical events
and conditions. If we include, for example, the diffusion that is
naturally a physical phenomenon, all the signal transduction in
the living system can be said to be mechanically driven.
The reason why we are particularly concerned in the nanoscale
study is that the nanometer-scale interactions of the component of
the living organism should be regarded as the site of life itself.
Apparently, most fundamental substances comprising life, for
example, protein, nucleic acids, lipids, and carbohydrate mole-
cules, are not living. They are just chemical substances. Life lies
under the interactions of these components, that is, chemical
and physical reactions, of the nanometer scale. If we proceed more
into picometer- or femtometer-scale phenomena, we see how those
molecules are composed but not the life phenomenon as it is clearly
distinguished from merely chemical reactions. Though we do not
want to revive the archaic vitalism, needless to say, nanoscale inter-
actions among components of the cell, if they are fully understood,
would answer our most fundamental question, what is life.
Life emerged in the cosmos of purely inorganic materials
under physical or mechanical conditions and evolved in the
 Chapter 1 INTRODUCTION 3

terrestrial environment. It forged itself a distinguished existence

that lives. Throughout its evolution, it was immersed in the grav-
itational influences whence its structure and function were devel-
oped inevitably under mechanical constraints. It is natural,
therefore, that mechanical construction and behavior are crucial
to understand and comprehend its secret. It is obvious that our
most fundamental question, that is, “what is life?,” will never be
solved without the wide and profound use of mechanics of the
utmost wide, from molecular to ecological length and timescales.
Living organisms are extremely complex system, of which multi-
ple levels of interactions take place and regulated by the physico-
chemical control system. It is now widely accepted even by
traditional biologists that mechanical behavior is the key to eluci-
date the mechanism of life even when it seems to be pure chem-
ical or biological. Mechanics always play important roles in the
living beings and the contribution of mechanical structures and
the functions of living organisms. Of many levels of mechanical
interactions that govern life, cellular and subcellular mechanical
interactions are most significant.
So far, we have mainly concentrated into healthy or normal life
phenomena. However, boundaries of normal condition are delin-
eated by abnormal or disease phenomena. Those boundaries
must be carefully determined by studying various pathological
degradations by diseases. It is noteworthy that the boundary
between healthy and diseased condition is not usually clearly
divided. They are in a sense continuous. Therefore, we now have
to study the transition and difference between normal healthy
mechanical condition and disease pathological escape, as well
as the purely normal healthy conditions. By this study, we will
more deeply understand life. Research of the pathological state
is of course potentially useful to find measures of diagnosis and
treatment of diseases. Though our principal purpose of the cur-
rent series of studies is to understand the mechanics and mech-
anism of life, we believe that we will be able to contribute to
medicine through such approaches.
As earlier discussed, it is certain that the cell is the entity that is
alive. Because there is no concise definition available, we usually
use some descriptive definition of life. Though there are some var-
iations of the descriptions, metabolism, reactivity, compartment,
and reproduction are major necessities by which we distinguish
the living and nonliving system. Cell undoubtedly fulfills the
requirement. However, none of subcellular components do. Mito-
chondrion is an exception, but it cannot survive when it is taken
out from the cell after a very long cohabitation with eukaryotic
cells. It is also undoubtedly clear that any molecules, even very
4 Chapter 1 INTRODUCTION

large and complex protein, are not alive. If we accept the fact, we
will be obliged to consider that the relationship or interactions
between these components of the cell are life, and this is the rea-
son why the nanoscale biomechanics is necessary. After the estab-
lishment of molecular biology, particularly so-called omics
studies, complex nature of the biological existence is exceedingly
pronounced. However, not the appreciation of complex nature
nor its massive statistical analysis without guiding principle leads
to truly comprehensive understanding of life. In this sense, nano-
scale interactions between subnanoscale substances in the cell,
which our nanoscale study is aimed, are the key issue of the
biology.
Biology of the cell revealed that every cell has an ability to sense
and quantitate the mechanical environments surrounding it. This
understanding is now widely possessed by researchers, and thus,
the field of the study is called mechanobiology. Not only mechan-
ics is important in the analysis of mechanical response of special
cells such as endothelium of the vascular wall, but also the study
of many different kinds of cells should be carried out. Mechanical
receptor proteins or mechanoreceptor channels are widely stud-
ied in sensing cells. These receptor-mediated mechanisms may be
a part of mechanobiological processes. However, we need to con-
sider the whole structure of the cell that responds to mechanical
stimuli and detailed mechanism. Again, this is the level of nano-
biomechanics, and further study, not merely molecular research,
is now necessary.
In the multicellular organism including us, no cells move and
work independently, except blood cells and germ cells, such as
sperm and egg. Of those exceptions, egg is literally single, and
others are collective. In order to analyze and understand the col-
lective motion of those cells, single-cell motion should be exam-
ined first. Of those cells of collective motion, erythrocyte or red
blood cells (RBCs) and platelets are thought of as purely passive
because they have no motive element. Leucocytes or WBCs and
sperms are active in their motion. However, their motility has
quite different timescale. The timescale of spermatic motion is
an order of 1–10 3 s, while motion of leucocytes is almost passive
when they are in the blood flow and become motile after they stick
to the vascular wall. Its motion on and through the vascular wall
could be an order of 1–10 3 s or longer. In the case of sperm, their
interaction to each other may not be important. However, it is
shown that sperms move collectively. Interaction is very impor-
tant in the case of blood cells. RBCs, particularly due its concen-
tration in the blood, must be eventually analyzed through their
multiple-body interactions. WBC motion must be analyzed with
 Chapter 1 INTRODUCTION 5

their interactions not with each other but with their interactions
with the erythrocytes in the blood flow and with endothelial cells
when they reside adjacent to the wall. Platelets are smaller than
those blood cells, and their motion is strongly influenced by the
RBC. Formation of thrombosis and hemostasis should be under-
stood under this condition. Consequently, motion and character-
istics of the multicellular fluid such as the blood must be
examined through the single-cell mechanics, and their collective
nature should be reconstructed by the collective motion of the
unit component. This is an attempt to reorganize the rheology
of the physiological fluids and is now possible to discuss this
thanks to huge-scale computation.
Traditionally, particularly in the field of engineering, mechan-
ics studies are divided into some subfields, typically fluid
mechanics, solid mechanics, thermodynamics and transport phe-
nomena, mechanical dynamics, and electromagnetics. However,
we have to place a stress on the fact that none of the biological
phenomenon can be analyzed by only one of those subspecific
measures. For example, production, transmission, and transduc-
tion of biological signals are governed by the interaction of the
electric field and electromagnetic transformation of ion channel
protein molecules and the diffusion of molecules in the synaptic
gap. This typically ends up in the contraction of muscle to pro-
duce macroscopic motion of the body. Conventionally, this pro-
cess is not studied from the mechanical viewpoint. However,
whole process is based on mechanics. Electric excitement and
transmission of the pulse are sequential processes of the electro-
magnetic deformation of the channel protein, and final transmis-
sion of the excitement is based on the axonal transport of the
material from the neuronal cell body to the synapse. Nanoscale
biomechanics should be involved in the study of whole physiology
of the living system as such.
Fluid-solid interaction is another problem of intermechanics
or combined studies. As shown later in this book, a number of bio-
logical systems are composed of solid components and fluid com-
ponents. Simplest cell in the whole body, the RBC, is a good
example. It has no intracellular solid components such as the
nucleus and the mitochondria, but just a thick solution of hemo-
globin occupies the cell. Our analysis revealed, nevertheless, that
the stiffness of the cell membrane and the viscosity of the intra-
cellular fluid relative to that of surrounding fluid, which is the
plasma, greatly affect the deformation and motion of the RBC
and therefore total blood flow in the smaller vasculature. This
is a typical example showing the necessity of the fluid-solid
interaction studies, though the theoretical difficulty and the
6 Chapter 1 INTRODUCTION

computational load are incredibly high. Existence of the cell-free

layer in the vasculature is another example of the study that needs
the extensive and rigorous fluid-solid interactions analysis of very
large number of RBCs.
Diffusion of gases such as oxygen and carbon dioxide gener-
ated from metabolic processes, nutrients of a wide range of
molecular weight, and messenger molecules from small-molecu-
lar-weight gas to large-molecular-weight polypeptides and pro-
teins play very important roles in physiology. Particularly, the
diffusion of these particles in the thick solution of cells, such as
RBC, should be carefully analyzed because the interactions of
the large (i.e., cellular)-scale motion of the medium strongly affect
the diffusion and distribution of the smaller solutes in the blood.
Solid mechanics is now not only for the analysis of hard tissue
such as bones and cartilages. The smaller the scale of biological
motion analysis is analyzed, the smaller the source of force gen-
eration should be considered. In the fetal development, earlier
mentioned, the flow of fluid produced by flagella motion is now
focused in the study of the origin of laterality. This is another
fluid-solid interaction problem, and the solid components
actively contribute. In the utmost large scale, in terms of biolog-
ical process, cardiac muscle contraction and the macroscopic
blood flow are also such kind of complex fluid-solid interaction
problem, although this is out of our scope of nanobiomechanics.
Traditionally, two measures have been recognized in the field
of mechanics research; they are experimental and theoretical
studies. The third category, computational studies, became feasi-
ble after the late 20th century, and it occupies almost equal part as
other two measures nowadays. Because we have been involved in
computational studies for a long period, let us first depict the
advancement of computational studies.
It is not very long ago when the third way of study, the compu-
tation, was recognized and accepted as the potentially important
measure. When we started to construct the computational biome-
chanics, some 40 years ago, there were many controversies against
it. At that moment, power of computer was so poor that it could
not allow us to deal with complicated phenomena, such as fluid-
solid interactions. It is well known that the first supercomputer,
the Cray-1, whose number of installation was very limited in
the world, had an ability of peak floating point operations of
108 FLOPS (100 MFLOPS) that is now much slower than that
of the portable telephones. In the highest range of scientific com-
puting, as is well known, computational power is discussed by an
order of 1016 FLOPS. In early days, available computer power to
ordinary research was much less than that of the highest level
 Chapter 1 INTRODUCTION 7

at that time, and we had to conduct studies under such restriction.

However, owing to very rapid development of computer technol-
ogy, we can now utilize incredibly higher (than that were available
then) computer power and extend studies on what was thought to
be impossible at that time. Of particular interest is the interven-
tion of so-called graphics processing unit (GPU) microchips.
Though this is an offspring of popular microcomputer technology,
especially that of game machines, its speedup of floating point
multiply-accumulate operations drastically affects the scientific
computing. If we carefully choose the computing algorithm, we
can now build a laboratory level supercomputer with affordable
price as is discussed in the later part of this book.
This amazing increase of computer power actualized two
important aspects of computational studies. First, it became pos-
sible to apply computation to problems of very large scale. In this
book, the reader will find some examples of the advancement
such as the flow analysis of RBCs in even small arterioles of the
radius of an order 10 4 m and number of cellular components
involved approaches to the order of 10 4–10 6. Second, advance-
ment is the refinement of methods and accuracy of computation.
Orders of accuracy have been steadily improved for about
10 3–10 6 or even smaller. Thus, we can now try to challenge
every complex problem with extremely large scale in terms of
numbers of elements in the computational domain with very high
accuracy, so that we can now discuss the computational results
with high confidence as theoretical and experimental results in
many field of interest. We will see number of our results of com-
plex computational studies in the following sections of this book.
Experimental studies are also advanced by the introduction of
many sophisticated instruments and powerful experimental tools.
From the mechanical viewpoints, development of various kinds of
microscope significantly improved our understandings of nano-
scale biomechanical phenomena. Of many microscopic instru-
ments currently available, we find that the cryoelectron
microscopy, particularly its tomographic extension applied to
motor proteins in the cell, is of great significance. As later dis-
cussed in this book, it can help our understanding of the mecha-
nism of intracellular transport phenomena. So, many transport
mechanisms contribute to the cellular functions. Forces also
emerged through the transport mechanism. The cryoelectron
microscopy visualizes the molecular mechanism of those trans-
port phenomena in a form that computational studies can be used
to quantitatively follow the course of reactions. Another example
of newly introduced microscopic method is the confocal laser
scanning microscope. When it is applied to relatively large-scale
8 Chapter 1 INTRODUCTION

RBC flows, it proves to be a powerful tool to analyze various

types of diffusion processes in the cellular motion, again which
is shown in a form that modern ultralarge-scale computa-
tional analysis results can be combined to build truly integrated
nanobiomechanics.
Nanobiomechanics so far discussed should be said a compre-
hensive endeavor to understand life itself. It also bridges the
molecular biology and mesoscale, that is, cellular and tissue level,
biology. As previously discussed, molecule itself does not live, but
interactions of molecules are where the life phenomenon rises up.
Consequently, all the macroscopic biological phenomena, how-
ever complex its appearance seems, can be understood on the
basis of nanobiomechanics. For example, our studies on the
fluid-solid interactions between RBC and plasma fluid, started
from analyses of single-cell motion, now reached to the level that
the macroscopic rheological properties are reproduced by mas-
sive computation. Though, needless to say, the living organism
is extremely complex and studies of biology are explosively
advancing, we believe that the nanobiomechanics approach is
one of most important keys to understand the whole life phe-
nomena and to develop methods to cope with diseases or
malfunction of life.
BIOMECHANICS OF
2
MICROCIRCULATION

CHAPTER OUTLINE
2.1 Behavior of Capsules in Flow 10
2.1.1 Governing Equations and Numerical Method 10
2.1.2 Capsules in an Unbounded Simple Shear Flow 14
2.1.3 Lateral Migration of Capsules in Shear Flow 17
2.1.4 Capsules in an Oscillatory Shear Flow 22
2.2 Behavior of Red Blood Cells 26
2.2.1 Mechanical Modeling of RBCs 26
2.2.2 An RBC in Simple Shear Flow; Tumbling-Swinging Transition 28
2.2.3 An RBC Flowing Thorough a Micropore 32
2.3 Cell Adhesion in Microvessels 34
2.3.1 Margination in Microvessels 35
2.3.2 Cell Adhesion in Capillaries 41
2.3.3 Malaria Infection 45
2.4 Formation and Destruction of the Primary Thrombus 50
2.4.1 Physiology and Pathology of Primary Thrombus
in Blood Flow 50
2.4.2 Particle Method Simulation 52
2.4.3 Simulation of Primary Thrombus Formation and Influence of
Vessel Geometry 54
2.4.4 Influence of Platelet Glycoprotein Receptors and SIPA 55
2.4.5 Interaction Between Platelets and RBCs on the Primary
Thrombus Formation and Mechanical Hemolysis 58
2.4.6 Summary 62
References 63

Integrated Nano-Biomechanics. https://doi.org/10.1016/B978-0-323-38944-0.00002-4

# 2018 Elsevier Inc. All rights reserved. 9
10 Chapter 2 BIOMECHANICS OF MICROCIRCULATION

2.1
BEHAVIOR OF CAPSULES IN FLOW
Stephanie Nix
Department of Machine Intelligence and Systems Engineering, Akita Prefectural
University, Akita, Japan

A capsule consists of a liquid enclosed by some kind of elastic

membrane, often consisting of cross-linked polymers or proteins.
The study of the fluid dynamics of capsules has a number of bio-
logical, biomedical, and industrial applications; capsules are now
being used in applications as diverse as the food industry
(Madene et al., 2006), agricultural industry (Friedman and
Mualem, 1994), cosmetics (Kromidas et al., 2006), and drug deliv-
ery and synthetic biology (Sta €dler et al., 2009; van Dongen et al.,
2009). In many of these cases, capsules are used for transport
within some additional fluid, so it is essential to understand their
behavior in flows, in addition to their mechanical properties in
stasis.
In addition to the clarification of the physical properties of arti-
ficial capsules, the study of capsule properties is also useful in
understanding objects found in nature. For example, capsules
are often used as a simple model for blood cells. In particular,
red blood cells, which consist of an incompressible viscoelastic
membrane and inner Newtonian-like hemoglobin solution, are
often modeled as capsules in fluid simulations.
In this section, we start by introducing the assumptions and
modeling used in the simulation of capsules. Then, we explore
the behavior of initially spherical capsules in several flavors of
shear flow: first, an infinite shear flow; then, a shear flow bounded
by a single infinite planar wall; and finally, an infinite oscillatory
shear flow.

2.1.1 Governing Equations and

Numerical Method
Consider an initially spherical capsule of radius a suspended in
a fluid with density ρ and viscosity μ undergoing an applied flow
with velocity field u∞, as shown in Fig. 2.1.1. The capsule consists
of an inner fluid with viscosity λμ and a two-dimensional hyper-
elastic membrane with surface shear modulus Gs and negligible
 Chapter 2 BIOMECHANICS OF MICROCIRCULATION 11

Fig. 2.1.1 Schematic of a capsule suspended in a flow with applied velocity

field u∞.

bending resistance. Assuming that the internal and external fluids

are incompressible, the velocities of the fluids inside and outside
the capsule are each described by the continuity equation and the
incompressible Navier-Stokes equation, respectively:
ru¼0 (2.1.1)
∂u
ρ + ρðu  rÞu ¼ rp + μr2 u + f (2.1.2)
∂t
where u is a velocity field existing either inside or outside the cap-
sule, p is the associated pressure field, and f is some externally
applied force density, such as a gravitational or electromagnetic
force. The nondimensionalized form of Eq. (2.1.2) is

0 
∂u
+ ðu  r Þu ¼ r0 p0 + r0 u0 + f 0
0 0 0 2
Re (2.1.3)
∂t 0
where primes represent nondimensionalized quantities: the
velocity u is nondimensionalized by a characteristic value of the
strength of the applied velocity field, which we denote as j u∞j,
lengths by a, time by a/j u∞j, and the pressure by the quantity
μ ju∞j/a. By this nondimensionalization, the Reynolds number
ρju∞ ja
Re ¼ (2.1.4)
μ
is derived, which represents the ratio of the inertial forces to the
viscous forces in the fluid. If the Reynolds number is much smaller
than unity, then the effects of inertia can be neglected, as the left-
hand side of Eq. (2.1.3) approaches zero, and the flow can be trea-
ted as a Stokes flow.
Here, we evaluate the validity of the Stokes limit approxima-
tion. The density and viscosity of water are ρ  1  103 kg/m3
and μ  1  103 Pa s, respectively, and these values reflect the
12 Chapter 2 BIOMECHANICS OF MICROCIRCULATION

leading-order conditions that capsules experience. Artificial cap-

sules and cells have radii in the range 106–104 m. In the micro-
circulation, the characteristic velocity of blood flow takes values of
j u∞j  102 m/s (Lipowsky, 2005), leading to a maximum Reyn-
olds number of Re < 102 for a capsule size of a few micrometers.
The governing equations of Stokes flow are
ru¼0 (2.1.5)
rp + μr2 u + f ¼ 0 (2.1.6)
for the flows both outside and inside the capsule. The assumption
that the motion of the capsule is governed by the Stokes equations
allows us to ignore the inertial terms on the left-hand side of
Eq. (2.1.2) and makes it possible to solve the velocity field exactly.
The exact solution for Stokes flow can be expressed using
Green’s functions for the velocity, pressure, and stress tensor.
The free-space Green’s functions are determined by setting a point
force f ¼ gδ(x  y) at y and substituting into Eq. (2.1.6); then, the
resulting partial differential equation is solved given the addi-
tional constraint for the pressure
r2 p ¼ 0 (2.1.7)
and the definition of the stress tensor


∂ui ∂uk
σ ik ¼ pδik + μ + (2.1.8)
∂xk ∂xi
so that there are three equations for the three unknown functions.
The velocity, pressure, stress tensor, and the corresponding
Green’s functions are given by
1 δij ri rj
u i ðx Þ ¼ Jij ðrÞgj , Jij ðrÞ ¼ + 3 (2.1.9)
8πμ r r
1 ri
pðx Þ ¼ Pj ðrÞgj , Pi ðrÞ ¼ 2 3 (2.1.10)
8π r
1 ri rj rk
σ ik ðx Þ ¼ Tijk ðrÞgj , Tijk ðrÞ ¼ 6 5 (2.1.11)
8π r
where r ¼ x 2 y and r ¼ jrj. Details of the derivation of the above
Green’s functions are given in Pozrikidis (1992).
The boundary integral equation giving the velocity at an arbi-
trary point x in a Stokes flow
ð
1
u j ðx Þ ¼  qi ðy Þ  Jij ðy  x ÞdAðy Þ
8πμ
ð A
1
+ ui ðy Þ  Tijk ðy  x Þ  nk ðy ÞdAðy Þ (2.1.12)
8π A
 Chapter 2 BIOMECHANICS OF MICROCIRCULATION 13

results from the generalization from a single point force to a force

density q on a surface A with unit normal n through the use of the
Lorentz reciprocal identity. In the particular case of a capsule with
viscosity ratio λ, the velocity at a point x on the capsule surface
takes on the form (Pozrikidis, 1992; Foessel et al., 2011)
ð
1
u j ð x Þ ¼ u∞
j ð x Þ  qi ðy Þ  Jij ðy  x ÞdAðy Þ
ð 8πμ A
1λ
+ ½ui ðy Þ  ui ðx Þ  Tijk ðy  x Þ  nk ðy ÞdAðy Þ
8π A
(2.1.13)
where u∞ is an externally applied flow and q is the membrane load
at a point y on the capsule surface due to the capsule elasticity.
The membrane load q is given as a function of the membrane
tension via a weak formulation of the finite element method
(Walter et al., 2010):
ð ð
^  qdA ¼ ^ε : TdA
u (2.1.14)
A A

where u^ is a virtual displacement that gives a virtual strain ^ε and T

is the membrane tension. The membrane tension is given as a
function of the membrane strain energy ws by the relation
2 ∂ws αβ ∂ws αβ
T αβ ¼ G + 2Js g (2.1.15)
Js ∂I1 ∂I2
Here, Gαβ and gαβ are the contravariant metric tensors of the unde-
formed and deformed states, respectively. I1 and I2 are the strain
invariants, given by
I1 ¼ gαβ G αβ  2 (2.1.16)
  αβ 
I2 ¼ gαβ G   1 ¼ Js2  1 (2.1.17)
where gαβ is the covariant metric tensor. Several membrane con-
stitutive laws are examined in the upcoming work. One such law is
the neoHookean law, with strain energy function:


Gs 1
ws ¼ I1  1 + (2.1.18)
2 I2 + 1
The neoHookean constitutive law is strain softening and is often
used to model rubberlike materials. The red blood cell membrane
is described by the strain-hardening Skalak constitutive law
(Skalak et al., 1973):
Gs  2 
ws ¼ I + 2I1  2I2 + CI 22 (2.1.19)
4 1
14 Chapter 2 BIOMECHANICS OF MICROCIRCULATION

where C is a constant that describes the degree of area incompres-

sibility of the membrane, such that the area dilation modulus is
equal to Ks ¼ Gs(1 + 2C). Assuming that the red blood cell mem-
brane is well described by the Skalak law, C has been estimated
to be in the order of C ¼ 1000 (Skalak et al., 1973).
The relative effect of the viscous forces induced by the external
flow to the elastic forces in the membrane is described by the cap-
illary number Ca:

μ_γ a
Ca ¼ (2.1.20)
Gs

when the capsule is suspended in a simple shear flow of the form

u∞ ¼ γ_ x3 e1 .
In the next subsections, we will examine the dynamics of cap-
sules in a variety of external flows. First, we will consider the sim-
plest case of a capsule suspended in an unbounded simple shear
flow. Then, we will consider the effects of boundaries by consid-
ering the motion of a capsule in a simple shear flow near an
infinite planar wall. Finally, we will consider the effects of time-
varying flows by considering the motion of a capsule in a periodic
sinusoidal flow.

2.1.2 Capsules in an Unbounded Simple

Shear Flow
The behavior of a solitary capsule in a simple shear flow has
been investigated extensively in the past 30–40 years. The defor-
mation of a capsule in a simple shear flow in the limit of small
deformation was first worked out analytically by Barthès-Biesel
and coworkers (Barthès-Biesel, 1980; Barthès-Biesel and
Rallison, 1981). In recent years, increases in computing power
have led to an explosion in numerical analyses of various param-
eters that affect the flow behavior of capsules.
First, we examine the motion of a capsule in a simple shear
flow in the limit of an infinite flow, as shown in Fig. 2.1.2. The cen-
ter of the capsule is placed in the center of the flow at x ¼ (0, 0, 0),
such that the capsule does not undergo translation within the
flow. A shear flow of the form u∞ ¼ γ_ x3 e1 is applied instanta-
neously at t ¼ 0, and the capsule deforms freely in response to
the flow.
 Chapter 2 BIOMECHANICS OF MICROCIRCULATION 15

Fig. 2.1.2 Schematic of computational setup.

The general behavior of the capsule can be explained through

decomposing the shear flow into extensional and rotational
components:
0 10 1
0 0 γ_ x1
u∞ ¼ @ 0 0 0 A@ x 2 A
00 0 0 x13 0 1 0 10 1
0 0 γ_ =2 x1 0 0 γ_ =2 x1
¼ @ 0 0 0 A@ x2 A + @ 0 0 0 A@ x2 A (2.1.21)
γ_ =2 0 0 x3 _γ =2 0 0 x3
where the first term on the right-hand side is the extensional com-
ponent of the shear flow, which is oriented 45° in the +e3 direction
from the + e1 axis, and the second term is the rotational compo-
nent of the shear flow. Due to the extensional component, the cap-
sule deforms from a spherical to an ellipsoidal shape initially
oriented in the direction of the extensional flow. Simultaneously,
the rotational component of the flow acts to rotate the membrane
clockwise and, in a flow with a large γ_ , leads to the capsule aligning
in the flow direction. Then, due to membrane elasticity, which
acts as a restoring force to return the capsule to its initially spher-
ical shape, an extensional flow is induced in the direction opposite
to the extensional component of the applied shear flow, and the
shape of the capsule converges when the forces due to these
two extensional flows are balanced. The rotational component
of the shear flow, however, has no restoring force acting against
it, so the membrane continuously rotates around the inner fluid
in what is known as tank-treading motion.
Since the capsule takes on an approximately ellipsoidal shape,
the extent of deformation can be quantified using the lengths of
16 Chapter 2 BIOMECHANICS OF MICROCIRCULATION

the major and minor axes in the deformed state. Here, we use the
Taylor parameter (Taylor, 1934)
Ll  Ls
D¼ (2.1.22)
Ll + Ls
where Ll and Ls are the long and short axes of the deformed cap-
sule, respectively. The extent of the capsule deformation is deter-
mined by the membrane constitutive law, the capillary number
Ca, and the viscosity ratio λ.
First, we examine the effect of the constitutive law on the
deformation of the capsule at constant Ca and λ; examples are
shown in Fig. 2.1.3 that illustrate changes in the capsule deforma-
tion under the same flow conditions. In particular, the capsule
with the neoHookean membrane undergoes the largest deforma-
tion, as shown on the left-hand side of Fig. 2.1.3. The capsules with
a membrane described by the Skalak constitutive law undergo less
deformation, with the extent of deformation decreasing with an
increase in the area incompressibility constant C. This behavior
occurs because the neoHookean membrane is strain softening,
while the Skalak membrane is strain hardening.
Next, we examine the effect of Ca at constant λ and constitutive
law, with examples shown in Fig. 2.1.4. When Ca is small, the cap-
sule undergoes little deformation and so takes on a nearly spher-
ical shape. In addition, when Ca is smaller than a threshold value
CaL, membrane buckling occurs in the equatorial region, as can be
seen on the left-hand side of Fig. 2.1.4. As Ca is increased, the
buckling in the equatorial region disappears, as can be seen in
the middle capsule in Fig. 2.1.4. Then, as Ca is increased larger
than a threshold value CaH, membrane buckling again occurs
but this time at the tips of the capsule.

Fig. 2.1.3 Snapshots of converged shapes of capsules at Ca ¼ 0.4 for varying

constitutive law. Gradation represents the magnitude of the nondimensionalized
normal membrane load.
 Chapter 2 BIOMECHANICS OF MICROCIRCULATION 17

Fig. 2.1.4 Snapshots of converged shapes of the Skalak capsules for different Ca.
Gradation represents the magnitude of the nondimensionalized normal membrane
load.

Fig. 2.1.5 Capsule deformation with varying Ca and constitutive law.

Results for the deformation of capsules with varying constitu-

tive law and Ca at λ ¼ 1 are summarized in Fig. 2.1.5. In the small-
deformation limit (Ca ≪ 1), capsule deformation is independent
of the membrane constitutive law and varies linearly with Ca.
Simulations are carried out until the point at which buckling
appears at high Ca, which varies for each constitutive law. Thus,
from these results, it can be inferred that a capsule with a Skalak
membrane at C ¼ 10 is stable in flows five times as strong as a cap-
sule with a neoHookean membrane.

2.1.3 Lateral Migration of Capsules in

Shear Flow
Next, we examine the motion of a capsule in a shear flow near a
boundary. When a solid sphere is placed in shear flow near a wall
in the limit of Stokes flow, the distance between the sphere and the
18 Chapter 2 BIOMECHANICS OF MICROCIRCULATION

wall remains constant due to the reversibility of Stokes flow. How-

ever, when a spherical capsule is placed near a wall, the deform-
ability of the capsule breaks the symmetry of the flow, and the
capsule moves away from the wall in a process called lateral
migration. Similarly, when a capsule is placed in a parabolic flow,
such as Poiseuille flow, the deformability of the capsule leads to
lateral migration in the direction of lower shear rate. In this sec-
tion, we explore the lateral migration of capsules due to a wall
or a nonzero gradient in the shear rate.
A capsule with an inner viscosity of μ is placed at an initial dis-
tance of h from an infinitely planar wall placed at x3 ¼ 0, as shown
in Fig. 2.1.6. A simple shear flow of the form u∞ ¼ γ_ x3 e1 is started
instantaneously at t ¼ 0, after which the capsule is allowed to
move freely from the wall.
To implement an infinite planar wall using the boundary inte-
gral method, a modified Green’s function for the velocity is used
(Blake, 1971), of the form

Jijw ðrÞ ¼ Jij0 ðrÞ  Jij0 ðRÞ + 2h2 JijD ðRÞ  2hJ SD
i3j ðRÞ (2.1.23)
where Ri ¼ (1  2δi3) ri; the following definitions modified from
Pozrikidis (1992) are used:


  δij 3Ri Rj
JijD ðRÞ ¼ 1  2δj3  (2.1.24)
R3 R5


  δij R3  δi3 Rj + δj3 Ri 3Ri Rj R3
JijSD ðRÞ ¼ 1  2δj3  (2.1.25)
R3 R5
and δij is the Kronecker delta. This modified Green’s function rep-
resents the image system for a point force generated by placing
several singularities at the point reflected across the wall.
For a nonunity viscosity ratio, a modified Green’s function for
the stress tensor is also necessary. This modified Green’s function

Fig. 2.1.6 Schematic of computational domain.

 Chapter 2 BIOMECHANICS OF MICROCIRCULATION 19

for the stress tensor, like the stress tensor itself, must satisfy
Eq. (2.1.8); thus, it can be derived if the Green’s functions for
the velocity and pressure are known. The modified Green’s func-
tion for the pressure was previously derived by Blake and Chwang
(1974):



rj Rj Rj 3Rj R3 R3 δj3 3Rj R3
Pj ðrÞ ¼ 2 3 + 3  2 3 
w
+ 2h 3  5 (2.1.26)
r R R R5 R R
and the portion within the brackets can be shown to be equivalent
to the modified Green’s function for the point source. Using the
same logic, we use the modified Green’s function for the stresslet
singularity (Nix et al., 2016):

w 3ri rj rk δik rj 3Ri Rj Rk δik Rj D D

Kijk ðrÞ ¼  3  + 3 + δk3 Gijk ðRÞ + δi3 Gkji ðRÞ
r5 r R5 R
D D
δk3 ð1  2δi3 Þ 2hDij ðRÞ + Gi3j ðRÞ  δi3 ð1  2δk3 Þ 2hDjk ðRÞ + Gj3k ð RÞ
Q
2hð1  2δi3 Þð1  2δk3 Þ hQijk ðRÞ + Gi3jk ð RÞ
(2.1.27)
to form the stress tensor
w
Tijk ðrÞ ¼ δik Pjw + Kijk
w w
+ Kkji (2.1.28)
where the singularities used in Eq. (2.1.27) are defined as
δij Rk  δjk Ri + δik Rj 3Ri Rj Rk
D
Gijk ðRÞ ¼  (2.1.29)
R3 R5
δij 3Ri Rj
Dij ðRÞ ¼   5 (2.1.30)
R3 R
3δij Rk  3δik Rj  3δjk Ri 15Ri Rj Rk
Qijk ðRÞ ¼  (2.1.31)
R5 R7
Q δjk δil  δjl δik  δij δkl 15Ri Rj Rk Rl
Gijkl ð RÞ ¼  +
R5 R7
3δkl Ri Rj + 3δjl Ri Rk + 3δjk Ri Rl
 (2.1.32)
R5
3δil Rj Rk + 3δik Rj Rl  3δij Rk Rl

R5
Snapshots of simulations of a capsule initially placed h/a ¼ 1.1
from a wall with Ca ¼ 0.4 are shown in Figs. 2.1.7 and 2.1.8.
Fig. 2.1.7 shows a capsule with a neoHookean membrane, while
Fig. 2.1.8 shows a capsule with a Skalak membrane, with area
incompressibility constant C ¼ 1. As seen in the figures, under
the same flow conditions, variations in the capsule deformation
are observed.
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*** END OF THE PROJECT GUTENBERG EBOOK MUTTER
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