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Rigorous statistical analysis of the data from many different cell types has thus
far supported the existence of a common intersection of the G vs. f relationships
measured after treatment with a large panel of cytoskeletally active drugs (Laudadio,
Millet et al., 2005). This holds true regardless of the receptor-ligand pathway that
was used to probe cell rheology (Puig-de-Morales, Millet et al., 2004). The same
statistical analysis, however, hints that the crossover frequency 0 may not be the
same for all cell types and receptor-ligand pathways. Unfortunately, so far 0 cannot
be measured with high-enough accuracy to resolve such differences. Thus for all
practical purposes we can regard 0 as being the same for all cell types and for all
receptor-ligand combinations (Fabry, Maksym et al., 2003; Puig-de-Morales, Millet
et al., 2004).
The structural damping relationship has long been applied to describe the rheolog-
ical data for a variety of biological tissues (Weber, 1835; Kohlrausch, 1847; Fung,
62 J. Fredberg and B. Fabry

1967; Hildebrandt, 1969; Fredberg and Stamenovic, 1989; Suki, Peslin et al., 1989;
Hantos, Daroczy et al., 1990; Navajas, Mijailovich et al., 1992; Fredberg, Bunk et al.,
1993). Thus, it may seem natural (but still intriguing) that living cells, too, exhibit
structural damping behavior, and in this regard the collapse of the · vs. x data from
different cell types and drug treatments onto the same relationship (Fig. 3-8b) is a
necessary consequence of such behavior. But it is utterly mystifying why the G n vs.
x data from different drug treatments should form any relationship at all and why,
moreover, the data from different cell types and different receptor-ligand pathways
should collapse onto the very same relationship.


Theory of soft glassy rheology

What are soft glassy materials
The master relationships shown in Figs. 3-8 and 3-9 demonstrate that when the me-
chanical properties of the cell change, they do so along a special trajectory. This
trajectory is found to be identical in a large variety of cell types that are probed via dif-
ferent receptor-ligand pathways and over many frequency decades. In all those cases,
changes of stiffness and friction induced by pharmacological interventions could be
accounted for solely by changes in x. This parameter x appears to play a central
organizing role leading to the collapse of all data onto master curves. But what is x?
A possible answer may come “ surprisingly “ from a theory of soft glassy materials
that was developed by Sollich and colleagues (Sollich, Lequeux et al., 1997). In the
remainder of this chapter a brief introduction is given to some fundamental principles
and ideas about soft glassy rheology. Parallels between living cells and soft glassy
materials are shown, and a discussion ensues on what insights may be gained from
this into the mechanisms involved.
The class of soft glassy materials (SGM) comprises what would at ¬rst glance
seem to be a remarkably diverse group of substances that includes foams, pastes,
colloids, emulsions, and slurries. Yet the mechanical behavior of each of these sub-
stances is surprisingly alike. The common empirical criteria that de¬ne this class of
materials are that they are very soft (in the range of Pa to kPa), that both G and G
increase with the same weak power-law dependencies on frequency, and that the loss
tangent · is frequency insensitive and of the order 0.1 (Sollich, Lequeux et al., 1997;
Sollich, 1998). The data presented so far establish that the cytoskeleton of living cells
satis¬es all of these criteria. Accordingly, we propose the working hypothesis that the
cytoskeleton of the living cell can be added to the list of soft glassy materials.
Sollich reasoned that because the materials comprising this class are so diverse, the
common rheological features must be not so much a re¬‚ection of speci¬c molecules
or molecular mechanisms as they are a re¬‚ection of generic system properties that
play out at some higher level of structural organization (Sollich, 1998). The generic
features that all soft glassy materials share are that each is composed of elements
that are discrete, numerous, and aggregated with one another via weak interactions.
In addition, these materials exist far away from thermodynamic equilibrium and are
arrayed in a microstructural geometry that is inherently disordered and metastable.
Note that the cytoskeleton of living cells shares all of these features.
The cytoskeleton as a soft glassy material 63


Sollich™s theory of SGMs
To describe the interaction between the elements within the matrix, Sollich developed
a theory of soft glassy rheology (SGR) using earlier work by Bouchaud as a point of
departure (Bouchaud, 1992). SGR theory considers that each individual element of the
matrix exists within an energy landscape containing many wells, or traps, of differing-
depth E. These traps are formed by interactions of the element with neighboring
elements. In the case of living cells those traps might be plausibly thought to be
formed by binding energies between neighboring cytoskeletal elements including but
not limited to cross-links between actin ¬laments, cross-bridges between actin and
myosin, hydrophilic interactions between various proteins, charge effects, or simple
steric constraints.
In Bouchaud™s theory of glasses, an element can escape its energy well and fall into
another nearby well; such hopping events are activated by thermally driven random
¬‚uctuations. As distinct from Bouchaud™s theory, in Sollich™s theory of soft glassy
materials each energy well is regarded as being so deep that the elements are unlikely
to escape the well by thermal ¬‚uctuations alone. Instead, elements are imagined to be
agitated, or jostled, by their mutual interactions with neighboring elements (Sollich,
1998). A clear notion of the source of the nonthermal agitation remains to be identi¬ed,
but this agitation can be represented nonetheless by an effective temperature, or noise
level, x.
Sollich™s SGR theory follows from a conservation law for probability of an element
being trapped in an energy well of depth E and local displacement (strain) l, at time
t, denoted P(E, l, t). Dynamics is then governed by a conservation equation for this
probability, given by

‚ P(E, l, t)/‚t + γ ‚ P/‚l = ’g(E, l)P(E, l, t) + f (E) (t)δ(l) (3.5)

where (t) = d Edl g(E, l)P(E, l, t) (required for conservation of probability),
and δ(l) is the Dirac delta function. Here γ = dl/dt and f (E) is the distribution of
energy-well depths. Eq. 3.5 states that the material rate of change of P is given by
the sum of two terms. The ¬rst term is depletion, equal to the probability of resident
elements hopping out, given by the product of the probability of occupancy P and a
transition rate g(E, l). The second term is the accumulation rate, equal to the product
of the total number of available transitions (t) and the delta function constraint
forcing elements to hop into wells at zero local strain.
Sollich takes g(E, l) = 0 exp(’(E + kl 2 /2)/x) and f (E) = 0 exp (’E), re-
spectively. Note that the transition rate g(E, l) for hopping out of wells is distributed
over E, which, in the nonlinear regime, is also a function of strain. Note further that
the transition rate into wells of depth E only depends on strain through the constraint
that l = 0 following a hop.
When x > 1, there is suf¬cient agitation in the matrix that the element can hop
randomly between wells and, as a result, the system as a whole can ¬‚ow and become
disordered. When x approaches 1, however, the elements become trapped in deeper
and deeper wells from which they are unable to escape: the system exhibits a glass
transition and becomes a simple elastic solid with stiffness G 0 .
64 J. Fredberg and B. Fabry


Soft glassy rheology and structural damping
Remarkably, in the limit that the frequency is small (compared to 0 ) and the imposed
deformations are small (such that the rate for hopping out of wells is dominated by
x), Sollich™s theory leads directly to the structural damping equation (Eq. 3.2).
The data reported above establish ¬rmly that the mechanical behavior of cells
conforms well to Eq. 3.2. If Sollich™s theory and underlying ideas are assumed to
apply to the data reported here, then the parameters in Eq. 3.2 (x, G 0 , 0 ) can be
identi¬ed as follows.
The parameter x is identi¬ed as being the noise temperature of the cytoskeletal
matrix. The measured values of x in cells lie between 1.15 and 1.35, indicating that
cells exist close to a glass transition.
G 0 is identi¬ed as being the stiffness of the cytoskeleton at the glass transition
(x = 1). In this connection, Satcher and Dewey (1996) developed a static model of
cell stiffness based on consideration of cell actin content and matrix geometry. All
dynamic interactions were neglected in their model, as would be the case in SGR
theory in the limit that x approaches 1, when all hopping ceases. As such, it might
be expected that their model would predict this limiting value of the stiffness, G 0 ,
as de¬ned in Eq. 3.2. Indeed, we have found a remarkably good correspondence
between their prediction (order of 10 kPa) and our estimate for G 0 (41 kPa in HASM
cells).
Finally, 0 is identi¬ed in Sollich™s theory as being the maximum rate at which
cytoskeletal elements can escape their traps. However, for soft glassy materials in
general, and the case of living cells in particular, the factors that determine 0 re-
main unclear. Statistical analysis of our data suggests that 0 did not vary with drug
treatments and possibly not even across cell type (Fig. 3-7). But why 0 is invariant
is not at all clear, and is not explained by SGR theory.


Open questions
Crucial aspects of soft glassy rheology (SGR) theory remain incomplete, however.
First, the effective noise temperature x is a temperature to the extent that the rate at
which elements can hop out of a trap assumes the form exp(’E/x), where x takes
the usual position of a thermal energy k B T in the familiar Boltzmann exponential.
By analogy, x has been interpreted by Sollich as re¬‚ecting jostling of elements by
an unidenti¬ed but nonthermal origin. It appears an interesting question whether the
ambiguity surrounding x might be resolved in the case of living cells (as opposed to
the inert materials for which SGR theory was originally devised) by an obvious and
ready source of nonthermal energy injection, namely those proteins that go through
cyclic conformational changes and thus agitate the matrix by mechanisms that are
ATP dependent.
Second, Sollich interprets E as an energy-well depth, but there are dif¬culties with
this interpretation. In SGR theory the total energy is not a conserved quantity even in
the zero-strain case, and the energy landscape has no spatial dimension, precluding
explicit computation of microstructural rearrangements. Such microstructural rear-
rangements in the form of cytoskeletal reorganization during cell division, crawling,
The cytoskeleton as a soft glassy material 65

or intracellular transport processes are of fundamental interest in cell biology. The
¬‚uctuation-dissipation theorem implies a profound connection between dissipative
phenomena as re¬‚ected in the measured values of G during oscillatory forcing, and
the temporal evolution of the mean square displacement (MSD), or ¬‚uctuations, of
free unforced particles in the medium. Experiments from our lab and others revealed
a behavior of MSD that lies somewhere between that of simple diffusion in a homoge-
neous medium and ballistic behavior characteristic of short time displacements (An,
Fabry et al., 2004). To what extent this is consistent with SGR theory also remains an
open question (Lau, Hoffman et al., 2003).
Despite these questions, however, it is clear from Sollich™s theory that for a soft
glass to elastically deform, its elements must remain in energy wells; in order to ¬‚ow,
the elements must hop out of these wells. In the case of cells, these processes depend
mainly on a putative energy level in the cytoskeletal lattice, where that energy is
representative of the amount of molecular agitation, or jostling, present in the lattice
relative to the depth of energy wells that constrain molecular motions. This energy
level can be expressed as an effective lattice temperature (x) “ as distinct from the
familiar thermodynamic temperature. Even while the thermodynamic temperature is
held ¬xed, this effective temperature can change, can be manipulated, and can be
measured. The higher the effective temperature, the more frequently do elemental
structures trapped in one energy well manage to hop out of that well only to fall
into another. The hop, therefore, can be thought of as the fundamental molecular
remodeling event.
It is interesting that, from a mechanistic point of view, the parameter x plays a
central role in the theory of soft glassy materials. At the same time, from a purely
empirical point of view, the parameter x is found to play a central organizing role
leading to the collapse of all data onto master curves (Fig. 3-8). Whether or not the
measured value of x might ultimately be shown to correspond to an effective lattice
temperature, this empirical analysis would appear to provide a unifying framework
for studying protein interactions within the complex integrative microenvironment of
the cell body.
In the next section, the concepts developed so far are employed to tie together within
such a uni¬ed framework diverse behaviors of cell physiology that were previously
unexplained or regarded as unrelated.


Biological insights from SGR theory

Malleability of airway smooth muscle
The function of smooth muscle is to maintain shape and/or tone of hollow organs
(Murphy, 1988). Typically, smooth muscle must do so over an extremely wide range
of working lengths. Two unique features enable smooth muscle to do this. First,
smooth muscle can develop its contractile forces almost independently of muscle
length (Wang, Pare et al., 2001). To achieve this, the cytoskeletal lattice and associated
contractile machinery of smooth muscle is disordered and highly malleable, quite
unlike the ordered and ¬xed structure of striated muscle and the rather narrow range
of lengths over which striated muscle can generate appreciable tension. Second, at the
66 J. Fredberg and B. Fabry

height of force development, smooth muscle can “latch” its contractile machinery,
that is to say, down-regulate the rate of acto-myosin cycling, thereby leading to tone
maintenance very economically in terms of energy metabolism (Hai and Murphy,
1989). To produce the same steady-state isometric force, for example, striated muscle
hydrolyzes more ATP at a rate 300 times higher than does smooth muscle (Murphy,
1988).
Soft glassy rheology theory helps to piece together such information into an in-
tegrative context. Below, we present some earlier data from our laboratory on the
mechanical and contractile properties of smooth muscle tissue; these data contained
some previously unexplained loose ends. The glass hypothesis now offers a new and
consistent explanation of these data.
Our earlier work focused on the contractile states of smooth muscle that were
inferred from the responses to sinusoidal length or force perturbations. Fig. 3-9 sum-
marizes a typical result obtained from sinusoidal length perturbations.
The stiffness E in those experiments is a measure of the number of force-generating
acto-myosin bridges, while the hysteresivity · is a measure of internal mechanical
friction and is closely coupled to the rate of cross-bridge cycling as re¬‚ected both in
the unloaded shortening velocity and the rate of ATP utilization measured by NADH
¬‚uorimetry (Fredberg, Jones et al., 1996). The dramatic increase in force and stiffness
after contractile stimulation (Fig. 3-10) therefore re¬‚ects an increase in the number
of acto-myosin bridges. The progressive fall of · after contractile stimulus onset
(Fig. 3-9) has been interpreted as re¬‚ecting rapidly cycling cross-bridges early in the
contractile event converting to slowly cycling latch-bridges later in the contractile
event (Fredberg, Jones et al., 1996). This molecular picture ¬ts exceptionally well
with computational analysis based on ¬rst principles of myosin-binding dynamics.
According to this picture, imposed sinusoidal length oscillations (between 400 s and
900 s in Fig. 3-9) around a constant mean length lead to a disruption of acto-myosin
cross-bridges and latch-bridges. This shifts the binding equilibrium of myosin toward
a faster cycling rate such that with increasing oscillation amplitude hysteresivity
increases, and muscle force and stiffness fall (Fredberg, 2000; Mijailovich, Butler
et al., 2000). However, this picture is unable to explain why force and stiffness re-
mained suppressed even after the length oscillations had stopped (Fig. 3-9).
Much the same behavior was found in a similar experiment in which force oscilla-
tions were imposed around a constant mean force (Fredberg, 2000): With increasing
amplitude of the force oscillations, stiffness decreased and muscle length and hystere-
sivity increased (Fig. 3-10). Again, this behavior was exceptionally well explained by
acto-myosin binding dynamics (Fredberg, 2000), but when the amplitude of the force
oscillations was reduced, length and stiffness inexplicably did not return (Fig. 3-10).
A rather different perspective on these observations (Figs. 3-9 and 3-10) arises when
they are viewed instead through the lens of glassy behavior. Accordingly, the relaxed
smooth muscle cell is in a relatively “cold” state, with a noise temperature close to
unity, but with the onset of contractile stimulation the cell very rapidly becomes “hot.”
After this hot initial transient, the cell then begins to gradually “cool” in the process
of sustained contractile stimulation until, eventually, it approaches a steady-state that
approximates a “frozen” state not only mechanically (high stiffness and low noise
temperature) but also biochemically and metabolically (Gunst and Fredberg, 2003).
The cytoskeleton as a soft glassy material 67




}quench
(A)
F




µ

}quench
(B)
E




1.4
1.3
1.2
·




x
(C)
1.1

1.0




warm
cold hot cold cold
cold

Fig. 3-9. Time courses of force (A), stiffness (B), and hysteresivity (C) in a bovine tracheal smooth
muscle strip. To continuously track changes in stiffness and hysteresivity, sinusoidal length oscilla-
tions at a frequency of 0.2 Hz and with amplitude µ were superimposed throughout the measurement
period. The tangent of the phase angle between the sinusoidal length oscillations and the resulting
force oscillations (that is, the hysteresivity ·) de¬nes the noise temperature x (panel (C), right side)
by a simple relationship (Eq. 3.5). The strip was stimulated at 100 s with acetylcholine (10’4 M). The
solid trace in each panel corresponds to cyclic strain (µ) maintained throughout at 0.25%. Broken
lines correspond to runs in the same muscle subjected to graded increments of µ from 0.25 to either
0.5, 1, 2, 4, or 8% over the time interval from 400 to 900 s. Adapted from Fredberg, Inouye et al.,
1997.



The progressive decrease of stiffness and increase of hysteresivity with increas-
ing amplitude of the imposed cyclic strain (Figs. 3-9 and 3-10) is consistent with a
¬‚uidization of the CSK matrix due to the application of a shear stress. SGR theory
predicts that shear stress imposed at the macroscale adds to the agitation already
present at the microscale, and thereby increases the noise temperature in the matrix
(Sollich, 1998). This in turn allows elements to escape their cages more easily, such
that friction and hysteresivity increases while stiffness decreases (Sollich, 1998; Fabry
and Fredberg, 2003).
SGR theory now goes beyond the theory of perturbed myosin binding by predicting
that the stretch-induced increase in noise temperature speeds up all internal molecular
events, including accelerated plastic restructuring events within the CSK. When the
68 J. Fredberg and B. Fabry


δF


}
LL quench (a)
L SE

} quench (b)
E




1.2
·




(c)




x
1.1

1.0
cold warm cold

Fig. 3-10. Evolution of mechanical properties of bovine tracheal smooth muscle during contraction
against a constant mean force on which force ¬‚uctuations (0.2 Hz) of graded amplitude δ F were
superimposed. (a) Mean muscle length L (relative to optimal length L 0 ). (b) Loop stiffness (percent-
age of maximum isometric value). (c) hysteresivity · and noise temperature x. L SE is the statically
equilibrated length of the muscle after 120 min. of unperturbed contraction against a constant load
of 32% of maximum force (F0 ). Adapted from Fredberg, 2000.


tidal stretches are terminated (Figs. 3-9 and 3-10), however, the noise temperature
is suddenly lowered, and all plastic changes might become trapped, or quenched, so
that the muscle is unable to return to maximum force and stiffness (Fig. 3.10), or
maximum shortening (Fig. 3-10) (Fabry and Fredberg, 2003).
The glass hypothesis predicts, therefore, that the cell ought to be able to adapt
faster to step-length changes imposed while the cell is transiently ˜hot™ (that is, early
in activation), and far less so after it has cooled in the process of sustained activation
(Fabry and Fredberg, 2003; Gunst and Fredberg, 2003). Indeed, Gunst and colleagues
showed that a step change of muscle length alters the level of the subsequent force
plateau to a degree that depends mostly on the timing of the length change with respect
to stimulus onset (Gunst, Meiss et al., 1995; Gunst and Fredberg, 2003).

Conclusion
The behavior of soft glasses, and the underlying notion of the noise temperature, might
provide a unifying explanation of the ability of the cytoskeletal lattice to deform, to
¬‚ow, and to remodel. Such a view does not point to speci¬c molecular processes that
occur, but instead derives the mechanical properties from generic features: structural
elements that are discrete, numerous, aggregated with one another via weak interac-
tions, and arrayed in a geometry that is structurally disordered and metastable. We
have proposed here that these features may comprise the basis of CSK rheology and
remodeling.

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4 Continuum elastic or viscoelastic models for the cell
Mohammad R. K. Mofrad, Helene Karcher, and Roger D. Kamm




Cells can be modeled as continuum media if the smallest operative length scale
ABSTRACT:
of interest is much larger than the distance over which cellular structure or properties may
vary. Continuum description uses a coarse-graining approach that replaces the contributions of
the cytoskeleton™s discrete stress ¬bers to the local microscopic stress-strain relationship with
averaged constitutive laws that apply at macroscopic scale. This in turn leads to continuous
stress-strain relationships and deformation descriptions that are applicable to the whole cell
or cellular compartments. Depending on the dynamic time scale of interest, such continuum
description can be elastic or viscoelastic with appropriate complexity. This chapter presents
the elastic and viscoelastic continuum multicompartment descriptions of the cell and shows a
successful representation of such an approach by implementing ¬nite element-based two- and
three-dimensional models of the cell comprising separate compartments for cellular membrane
and actin cortex, cytoskeleton, and nucleus. To the extent that such continuum models can
capture stress and strain patterns within the cell, it can help relate biological in¬‚uences of
various types of force application and dynamics under different geometrical con¬gurations of
the cell.


Introduction
Cells can be modeled as continuum media if the smallest length scale of interest
is signi¬cantly larger than the dimensions of the microstructure. For example when
whole-cell deformations are considered, the length scale of interest is at least one or
two orders of magnitude larger than the distance between the cell™s microstructural el-
ements (namely, the cytoskeletal ¬laments), and as such a continuum description may
be appropriate. In the case of erythrocytes or neutrophils in micropipette aspiration,
the macroscopic mechanical behavior has been successfully captured by continuum
viscoelastic models. Another example is the cell deformation in magnetocytometry,
the application of a controlled force or torque via magnetic microbeads tethered to
a single cell. Because the bead size and the resulting deformation in such experi-
ments are much larger than the mesh size of the cytoskeletal network, a continuum
viscoelastic model has been successfully applied without the need to worry about the
heterogeneous distribution of ¬lamentous proteins in the cytoskeleton. It should be
noted that in using a continuum model, there are no constraints in terms of isotropy or
homogeneity of properties, as these can easily be incorporated to the extent they are
71
72 M.R.K. Mofrad, H. Karcher, and R. Kamm

known. Predictions of the continuum model, however, are only as good as the consti-
tutive law “ stress-strain relation “ on which they are based. This could range from a
simple linear elasticity model to a description that captures the viscoelastic behavior
of a soft glassy material (see, for example Chapter 3). Accordingly, the continuum
model tells us nothing about the microstructure, other than what might be indirectly
inferred based on the ability of one constitutive law or another to capture the observed
cellular strains. It is important that modelers recognize this limitation.
In essence, continuum mechanics is a coarse-graining approach that replaces the
contributions of the cytoskeleton™s discrete stress ¬bers to the local microscopic stress-
strain relationship with averaged constitutive laws that apply at macroscopic scale.
This in turn leads to continuous stress-strain relationships and deformation descrip-
tions that are applicable to the whole cell or cellular compartments. Depending on
the dynamic time scale of interest, such continuum descriptions can be elastic or
viscoelastic with appropriate complexity.
This chapter presents elastic and viscoelastic continuum multicompartment de-
scriptions of the cell and shows a successful representation of such approaches by
implementing ¬nite element-based two- and three-dimensional models of the cell
comprising separate compartments for cellular membrane and actin cortex, cytoskele-
ton, and the nucleus. To the extent that such continuum models can capture stress and
strain patterns within the cell, they can help us relate biological in¬‚uences of various
types of force application and dynamics under different geometrical con¬gurations
of the cell.
By contrasting the computational results against experimental data obtained using
various techniques probing single cells “ such as micropipette aspiration (Discher
et al., 1998; Drury and Dembo, 2001), microindentation (Bathe et al., 2002), atomic
force microscopy (AFM) (Charras et al., 2001), or magnetocytometry (Figs. 4-7, 4-8,
Karcher et al., 2003; Mack et al., 2004) “ the validity and limits of such continuum
mechanics models will be assessed. In addition, different aspects of the model will be
characterized by examining, for instance, the mechanical role of the membrane and
actin cortex in the overall cell behavior. Lastly, the applicability of different elastic
and viscoelastic models in the form of various constitutive laws to describe the cell
under different loading conditions will be addressed.


Purpose of continuum models
Continuum models of the cell are developed toward two main purposes: analyzing ex-
periments probing single cell mechanics, and evaluating the level of forces sensed by
various parts of the cell in vivo or in vitro. In the latter case, a continuum model eval-
uates the stress and strain patterns induced in the cell by the experimental technique.
Comparison of theoretical and computational predictions proposed by the continuum
model against the experimental observations then allows for deduction of the cell™s
mechanical properties. In magnetocytometry, for example, the same torque or tan-
gential force applied experimentally to a microbead attached atop a cell is imposed
in continuum models of the cell. Material properties introduced in the model that
reproduce the observed bead displacement yield possible mechanical properties of
the probed cell (see Mijailovich et al., 2002, and Fig. 4-7 for torque application,
Continuum elastic or viscoelastic models for the cell 73




Fig. 4-1. Simulation of a small erythrocyte under aspiration. The micropipette, indicated by the solid
gray shading, has an inside diameter of 0.9 µm. The surface of the cell is triangulated with 6110
vertex nodes that represent the spectrin-actin junction complexes of the erythrocyte cytoskeleton.
The volume of the cell is 0.6 times the fully in¬‚ated volume, and the simulation is drawn from the
stress-free model in the free shape ensemble. From Discher et al., 1998.


and Karcher et al., 2003, and Fig. 4-8 for tangential force application). Continuum
models have also shed light on mechanical effects of other techniques probing single
cells, such as micropipette aspiration (Figs. 4-1, 4-6, and for example, Theret et al.,
1988; Yeung and Evans, 1989; Dong and Skalak, 1992; Sato et al., 1996; Guilak
et al., 2000; Drury and Dembo, 2001), microindentation (for example, Bathe et al.,
2002, probing neutrophils, Fig. 4-2 left), atomic force microscopy (AFM) (for ex-
ample, Charras et al., 2001 and Charras and Horton, 2002, deducing mechanical




Fig. 4-2. Microindentation of a neutrophil (left) and passage through a capillary (right) (¬nite element
model). From Bathe et al., 2002.
74 M.R.K. Mofrad, H. Karcher, and R. Kamm

(b)
(a)




(c) (d)




Fig. 4-3. Strain distributions elicited by AFM indentation. All of the scales are in strains. The
numerical values chosen for this simulation were: E = 10 kPa, ν = 0.3, R = 15 µm, F = 1 nN.
(a) Radial strain distribution. The largest radial strains are found on the cell surface. A large strain
gradient is present at the boundary between the region where the sphere is in contact with the
cell surface and the region where it is not. (b) Tangential strain distribution. The largest tangential
strains occurred at the cell surface in the area of indentation. (c) Vertical strain distribution. The
largest vertical strains were located directly under the area of indentation within the cell thickness.
(d) Deformations elicited by AFM indentation. The deformations have been ampli¬ed 15-fold in the
z-direction. From Charras et al., 2001.


properties of osteoblasts, Figs. 4-3, 4-4), magnetocytometry (Figs. 4-7, 4-8, Karcher
et al., 2003; Mack et al., 2004; Mijailovich et al., 2002), or optical tweezers (for
example, Mills et al., 2004 stretching erythrocytes, Fig. 4-5). Finally, comparison of
continuum models with corresponding experiments could help to distinguish active
biological responses of the cell (such as remodeling and formation of pseudopods)
from passive mechanical deformations, the only deformations captured by the model.
This capability has not been exploited yet to the best of our knowledge.
In addition to helping interpret experiments, continuum models are also used to
evaluate strains and stresses under biological conditions (for example, Fung and Liu,
1993, for endothelium of blood vessels). One example is found in the microcircu-
lation where studies have examined the passage of blood cells through a narrow
capillary (for example, Bathe et al., 2002, for neutrophils (Fig. 4-2 left), Barthes-
Biesel, 1996, for erythrocytes) where ¬nite element models have been used to predict
the changes in cell shape and the cell™s transit time through capillaries. In the case
Continuum elastic or viscoelastic models for the cell 75




Fig. 4-4. The effect of ¬‚uid shear. (a) The shear stress resultant in the z-direction („z ) for a nominal
5 Pa shear stress on a ¬‚at substrate. The shear stresses are tensile and lower upstream and higher
downstream. The imposed parabolic ¬‚ow pro¬le is shown at the entry and the boundary conditions
are indicated on the graph. (b) The vertical strain distribution (µzz ) for a cell submitted to ¬‚uid shear
stresses. Black triangles indicate where the substrate was fully constrained. The cellular strains are
maximal downstream from the cell apex and in the cellular region. In (a) and (b), the arrow indicates
the direction of ¬‚ow. From Charras and Horton, 2002.


of neutrophils, these inputs are crucial in understanding their high concentration in
capillaries, neutrophil margination, and in understanding individual neutrophil activa-
tion preceding their leaving the blood circulation to reach infection sites. Neutrophil
concentration depends indeed on transit time, and activation has recently been shown
experimentally to depend on the time scale of shape changes (Yap and Kamm, 2005).
Similarly, continuum models can shed light on blood cells™ dysfunctional microrhe-
ology arising from changes in cell shape or mechanical properties (for example, time-
dependent stiffening of erythrocytes infected by malaria parasites in Mills et al., 2004
(Fig. 4-5)).
Other examples include the prediction of forces exerted on a migrating cell in
a three-dimensional scaffold gel (Zaman et al., 2005), prediction of single cell at-
tachment and motility on a substrate, for example the model for ¬broblasts or the
unicellular organism Ameboid (Gracheva and Othmer, 2004), or individual protopod
dynamics based on actin polymerization (Schmid-Sch¨ nbein, 1984).
o


Principles of continuum models
A continuum cell model provides the displacement, strain, and stress ¬elds induced in
the cell, given its initial geometry and material properties, and the boundary conditions
it is subjected to (such as displacements or forces applied on the cell surface). Laws
of continuum mechanics are used to solve for the distribution of mechanical stress
and deformation in the cell. Continuum cell models of interest lead to equations
that are generally not tractable analytically. In practice, the solution is often obtained
numerically via discretization of the cell volume into smaller computational cells
using (for example) ¬nite element techniques.
A typical continuum model relies on linear momentum conservation (applicable to
the whole cell volume). Because body forces within the cell are typically small, and,
76 M.R.K. Mofrad, H. Karcher, and R. Kamm

maximum principal strain


0% 20% 40% 60% 80% 100% 120%


simulations
experiment


0 pN




67 pN




130 pN




193 pN

(a) (b) (c)
Fig. 4-5. Images of erythrocytes being stretched using optical tweezer at various pulling forces. The
images in the left column are obtained from experimental video photography whereas the images
in the center column (top view) and in the right column (half model 3D view) correspond to large
deformation computational simulation of the biconcave red cell. The middle column shows a plan
view of the stretched biconcave cell undergoing large deformation at the forces indicated on the left.
The predicted shape changes are in reasonable agreement with observations. The contours in the
middle column represent spatial variation of constant maximum principal strain. The right column
shows one half of the full 3D shape of the cell at different imposed forces. Here, the membrane is
assumed to contain a ¬‚uid with preserved the internal volume. From Mills et al., 2004.

at the scale of a cell, inertial effects are negligible in comparison to stress magnitudes
the conservation equation simply reads:
∇· σ =0
with σ = Cauchy™s stress tensor.

Boundary conditions
For the solution to uniquely exist, either a surface force or a displacement (possibly
equal to zero) should be imposed on each point of the cell boundary. Continuity of
normal surface forces and of displacement imposes necessary conditions to ensure
uniqueness of the solution.

Mechanical and material characteristics
Mechanical properties of the cell must be introduced in the model to link strain
and stress ¬elds. Because a cell is composed of various parts with vastly different
mechanical properties, the model ideally should distinguish between the main parts
Continuum elastic or viscoelastic models for the cell 77

(a)




Rc




(b)
cex




zax


fin


cin
z
fex


Fig. 4-6. Geometry of a typical computational domain at two stages. (a) The domain in its initial,
round state. (b) The domain has been partially aspirated into the pipet. Here, the interior, exterior,
and nozzle of the pipet are indicated. ¬n , free-interior; fex , free-exterior; cin , constrained-interior;
and cex , constrained-exterior boundaries. There is a ¬fth, purely logical boundary, zax , which is
the axis of symmetry. From Drury and Dembo, 2001.

of the cell, namely the plasma membrane, the nucleus, the cytoplasm, and organelles,
which are all assigned different mechanical properties. This often leads to the in-
troduction of many poorly known parameters. A compromise must then be found
between the number of cellular compartments modeled and the number of parame-
ters introduced.
The cytoskeleton is dif¬cult to model, both because of its intricate structure and
because it typically exhibits both solid- and ¬‚uid-like characteristics, both active
and passive. Indeed, a purely solid passive model would not capture functions like
crawling, spreading, extravasion, invasion, or division. Similarly, a purely ¬‚uid model
would fail in describing the ability to maintain the structural integrity of cells, unless
the membrane is suf¬ciently stiff.
The nucleus has generally been found to be stiffer and more viscous than the
cytoskeleton. Probing isolated chondrocyte nuclei with micropipette aspiration Guilak
et al. (2000) found nuclei to be three to four times stiffer and nearly twice as viscous
as the cytoplasm. Its higher viscosity results in a slower time scale of response, so that
the nucleus can often be considered as elastic, even when the rest of the cell requires
viscoelastic modeling. Nonetheless, the available data on nuclear stiffness seem to be
rather divergent, with values ranging from 18 Pa to nearly 10 kPa (Tseng et al., 2004;
Dahl et al., 2005), due perhaps to factors such as differences in cell type, measurement
technique, length scale of measurement, and also method of interpretation.
The cellular membrane has very different mechanical properties from the rest of
the cell, and hence, despite its thinness, often requires separate modeling. It is more
78 M.R.K. Mofrad, H. Karcher, and R. Kamm

(a) (b)




(d)
(c)




Fig. 4-7. Deformed shapes and strain ¬elds in a cell 5 µm in height for bead embedded 10% of
its diameter. Shown are strain ¬elds of the components of strain: µzz (a), µ yy (b), µ yz (c), and the
effective strain µe f f (d). The effective strain is de¬ned as: µe f f = 2 µi j ’ µi j , where µi j are strain
3
components in Cartesian system xi (x, y, z). From Mijailovitch et al., 2002.


¬‚uid-like (Evans, 1989; Evans and Yeung, 1989) and should be modeled as a vis-
coelastic material with time constants of the order of tens of µs.
The cortex, that is, the shell of cytoskeleton that is just beneath the membrane,
is in most cell types stiffer than the rest of the cytoskeleton. Bending stiffness
of the membrane and cortex has been measured in red blood cells (Hwang and
Waugh, 1997; Zhelev et al., 1994). A cortical tension when the cell is at its (un-
stimulated) resting state has also been observed in endothelial cells and leukocytes
(Schmid-Sch¨ nbein et al., 1995).
o

Example of studied cell types

Blood cells: leukocytes and erythrocytes
Blood cells are subjected to intense mechanical stimulation from both blood ¬‚ow and
vessel walls, and their rheological properties are important to their effectiveness in per-
forming their biological functions in the microcirculation. Modeling of neutrophils™
viscoelastic large deformations in narrow capillaries or in micropipette experiments
has shed light on their deformation and their passage time through a capillary or
entrance time in a pipette. Examples of such studies are Dong et al. (1988), Dong
and Skalak (1992), Bathe et al. (2002), and Drury and Dembo (2001) (see Fig. 4.6),
who used ¬nite element techniques and/or analytical methods to model the large de-
formations in neutrophils. Shape recovery after micropipette aspiration “ a measure
Continuum elastic or viscoelastic models for the cell 79




Fig. 4-8. Computational ¬nite element models of a cell monolayer being pulled at 500 pN using
magnetic cytometry experiment. Top panels show the pressure and effective stress ¬elds induced in
the cell after 2 s. (effective stress is a scalar invariant of the stress tensor excluding the compressive
part). Lower left panel shows the membrane xx-stretch (in the direction of the applied force), while
the lower right panel shows the induced deformation in the cytoskeleton in the direction of the applied
force. From Karcher et al., 2003.


of the neutrophil™s viscoelastic properties and its active remodeling “ was for ex-
ample investigated with a theoretical continuum model consisting of two compart-
ments: a cytoplasm modeled as a Newtonian liquid, and a membrane modeled with
a Maxwell viscoelastic ¬‚uid in the ¬rst time of recovery and a constant surface ten-
sion for the later times (Tran-Son-Tay et al., 1991). Erythrocytes have typically been
modeled as viscoelastic membranes ¬lled with viscous ¬‚uids, mostly to understand
microcirculation phenomena, but also to explain the formation of “spikes” or crena-
tions on their surface (Landman, 1984).


Adherent cells: ¬brobasts, epithelial cells, and endothelial cells
Many types of cell, anchored to a basal substrate and sensitive to mechanical stimuli “
like ¬brobasts and epithelial and endothelial cells “ have been probed by magnetocy-
tometry, the forcing of a µm-sized bead attached atop a single cell through a certain
type of membrane receptor (such as integrins).
Continuum modeling of this experiment was successfully developed to analyze
the detailed strain/stress ¬elds induced in the cell by various types of bead forcing
(oscillatory or ramp forces of various magnitudes) (Mijailovitch et al., 2002; Fig. 4-7).
80 M.R.K. Mofrad, H. Karcher, and R. Kamm

y


x


F

Eff. Stress
27.0 Pa
0.13
µm 18.0
0.11 9.0
0.09 0.0
0.07 -9.0
-18.0
0.05
-27.0
0.03




Fig. 4-9. A continuum, viscoelastic ¬nite element simulation representing experimental cell contact
sites on the basal cell surface estimated the focal adhesion shear stress distribution during magneto-
cytometry. Left panel shows merged experimental ¬‚uorescent images depicting the focal adhesion
sites. Middle and right panels show displacement and shear stress in the basal membrane of the cell.
Zero displacement and elevated shear stresses are evident in the focal adhesion regions. From Mack
et al., 2004.



(Karcher et al., 2003; Fig. 4-8). Modeling the cell with two Maxwell viscoelastic
compartments representing, respectively, the cytoskeleton and the membrane/cortex,
the authors found that the membrane/cortex contributed a negligible mechanical effect
on the bead displacement at the time scales corresponding to magnetocytometry.
Comparison with experiments on NIH 3T3 ¬broblasts led to a predicted viscoelastic
time scale of ∼1 s and a shear modulus of ∼1000 Pa for these cells. In addition, the
model showed that the degree to which the bead is embedded in the cell, a parameter
dif¬cult to control and measure in experiments (Laurent et al., 2002; Ohayon et al.,
2004), dramatically changes the magnitude of stress and strain, although it in¬‚uences
their pattern very little. Continuum modeling also allowed for modulation of cell
height and material properties to investigate the behavior of different adherent cell
types. It also demonstrated that the response of the cell when forced with the microbead
was consistent with that of a linear elastic model, quite surprising in view of the locally
large strains.
The cell attachment to its substrate by the basal membrane was later modi¬ed to
investigate force transmission from the bead to the basal membrane (Mack et al.,
2004) (Fig. 4-9). Only experimentally observed points of attachments, that is, focal
adhesion sites, were ¬xed in the model, allowing for the rest of the cell substrate
to move freely. Forcing of the bead on the apical surface of NIH 3T3 ¬broblasts
preferentially displaced focal adhesion sites closer to the bead and induced a larger
shear on the corresponding ¬xed locations in the model, implying that focal adhesion
translation correlates with the local level of force they sense.
An alternative experiment to probe cell deformation and adhesion consists of plat-
ing them on a compliant substrate. Finite element modeling of cells probed by this
Continuum elastic or viscoelastic models for the cell 81

technique was recently used to evaluate the stress and strain experienced at the nuclear
envelope, thereby investigating the mechanical interplay between the cytoskeleton
and the nucleus. The ultimate goal of this study was to identify potential sources of
mechanical dysfunction in ¬broblasts de¬cient in speci¬c structural nuclear mem-
brane proteins (Hsiao, 2004). The model showed that the effect of nuclear shape,
relative material properties of the nucleus and cytoskeleton, and focal adhesion size
were important parameters in determining the magnitude of stress and strain at the
nucleus/cytoskeleton interface.

Limitations of continuum model
Continuum models of the cell aim at capturing its passive dynamics. In addition
to the limitations mentioned above, current models do not yet typically account for
active biology: deformations and stresses experienced as a direct consequence of
biochemical responses of the cell to mechanical load cannot be predicted by current
continuum models. However, by contrasting the predicted purely mechanical cell
response to experimental observations, one could isolate phenomena involving active
biology, such as cell contraction or migration, from the passive mechanical response
of the cell. Alternatively, continuum models might be envisioned that account for
active processes through time-dependent properties or residual strains that are linked
to biological processes. (See also Chapter 10.)
Another limitation of continuum models stems from lack of description of cy-
toskeletal ¬bers. As such, they are not applicable for micromanipulations of the cell
with a probe of the same size or smaller than the cytoskeletal mesh (∼0.1“1.0 µm).
This includes most AFM experiments. In addition, the continuum models exclude
small Brownian motions due to thermal ¬‚uctuations of the cytoskeleton, which would
correspond to ¬‚uctuations of the network nodes in a continuum model and have been
shown to play a key role in cell motility (Mogilner and Oster, 1996).
Finally, continuum models have so far employed a limited number of time constants
to characterize the cell™s behavior. However, cells have recently been shown to exhibit
behaviors with power-law rheology implying a continuous spectrum of time scales
(Fabry et al., 2001; Desprat et al., 2004, and Chapter 3). Modeling the cell with no
intrinsic time constant has successfully captured this behavior (for example, Djord-
jevi¯ et al., 2003), though this type of model cannot and does not aim at predicting or
c
describing force or strain distribution within the cell. One of the challenges, therefore,
to the use of continuum models for the prediction of intracellular stress and strain
patterns is to develop cell material models that capture this complex behavior. In the
meantime, models involving a ¬nite number of time constants consistent with the
time scale of the experimental technique can be used, recognizing their limitations.


Conclusion
Continuum mechanical models have proven useful in exploiting and interpreting re-
sults of a number of experimental techniques probing single cells or cell monolayers.
They can help identify the stress and strain patterns induced within the cell by ex-
perimental perturpations, or the material properties of various cell compartments. In
82 M.R.K. Mofrad, H. Karcher, and R. Kamm

addition, continuum models enable us to predict the forces experienced within cells
in vivo, and to then form hypotheses on how cells might sense and transduce forces
into behavior such as changes in shape or gene expression.
The time scale of cell stimulation in experiments in vivo often requires that we take
into account the time-dependent response of the cell, that is, to model it or some of its
components as viscous or viscoelastic. Likewise, it is often necessary to model cell
compartments with different materials, as their composition gives them very distinct
mechanical properties.
Such continuum models have proven useful in the past, and will continue to play a
role in cell modeling. As we gain more accurate experimental data on cellular rheology,
these results can be incorporated into continuum models of improved accuracy of
representation. As such, they are useful “receptacles” of experimental data with the
capability to then predict the cellular response to mechanical stimulus, provided one
accepts the limitations, and recognizes that they provide little by way of insight into
the microstructural basis for macroscopic rheology.



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5 Multiphasic models of cell mechanics
Farshid Guilak, Mansoor A. Haider, Lori A. Setton,
Tod A. Laursen, and Frank P.T. Baaijens




Cells are highly complex structures whose physiology and biomechanical proper-
ABSTRACT:
ties depend on the interactions among the varying concentrations of water, charged or uncharged
macromolecules, ions, and other molecular components contained within the cytoplasm. To
further investigate the mechanistic basis of the mechanical behaviors of cells, recent studies
have developed models of single cells and cell“matrix interactions that use multiphasic consti-
tutive laws to represent the interactions among solid, ¬‚uid, and in some cases, ionic phases of
cells. The goals of such studies have been to characterize the relative contributions of different
physical mechanisms responsible for empirically observed phenomena such as cell viscoelas-
ticity or volume change under mechanical or osmotic loading, and to account for the coupling
of mechanical, chemical, and electrical events within living cells. This chapter describes sev-
eral two-phase (¬‚uid-solid) or three-phase (¬‚uid-solid-ion) models, originally developed for
studying soft hydrated tissues, that have been extended to describe the biomechanical behavior
of individual cells or cell“matrix interactions in various tissue systems. The application of such
“biphasic” or “triphasic” continuum-based approaches can be combined with other structurally
based models to study the interactions of the different constitutive phases in governing cell
mechanical behavior.


Introduction
Cells of the human body are regularly subjected to a complex mechanical environment,
consisting of temporally and spatially varying stresses, strains, ¬‚uid ¬‚ow, osmotic
pressure, and other biophysical factors. In many cases, the mechanical properties and
the rheology of cells play a critical role in their ability to withstand mechanical loading
while performing their physiologic functions. In other cases, mechanical factors serve
as important signals that in¬‚uence, and potentially regulate, cell phenotype in both
health and disease. An important goal in the ¬eld of cell mechanics thus has been the
study of the mechanical properties of the cell and its biomechanical interactions with
the extracellular matrix. Accordingly, such approaches have required the development
of constitutive models based on realistic cellular structure and composition to better
describe cell behavior.
Based on empirical studies of cell mechanical behavior, continuum models of
cell mechanics generally have assumed either ¬‚uid or solid composition and cell
properties, potentially including cortical tension at the membrane (Evans and Yeung,
84
Multiphasic models of cell mechanics 85

1989; Dong et al., 1991; Needham and Hochmuth, 1992; Karcher et al., 2003). In other
approaches, the elastic behavior of the cell has been described using structural models
such as the “tensegrity” approach (Ingber, 2003). Most such models have employed
constitutive models that assume cells consist of a single-phase material (that is, ¬‚uid
or solid), as detailed in Chapters 4, and 6. However, a number of recent studies have
developed models of single cells and cell“matrix interactions that use multiphasic
constitutive laws to account for interactions among solid, ¬‚uid, and in some cases,
ionic phases of cells. The goals of such studies have been to characterize the relative
importance of the mechanisms accounting for empirically observed phenomena such
as cell volume change under mechanical or osmotic loading, the mechanistic basis
responsible for cell viscoelasticity, and the coupling of various mechanical, chemical,
and electrical events within living cells. The presence of these behaviors, which arise
from interactions among different phases, often cannot be described by single-phase
models.
The cell cytoplasm may consist of varying concentrations of water, charged or un-
charged macromolecules, ions, and other molecular components. Furthermore, due
to the highly charged and hydrated nature of its various components (Maughan and
Godt, 1989; Cantiello et al., 1991), the cytoplasm™s gel-like properties have been
described under several different contexts see for example Chapter 7 and Pollack
(2001). Much of the supporting data for the application of multiphasic models of
cells has come from the study of volumetric and morphologic changes of cells in re-
sponse to mechanical or osmotic loading. The majority of work in this area has been
performed on cells of articular cartilage (chondrocytes), likely due to the fact that
these cells are embedded within a highly charged and hydrated extracellular matrix
that has been modeled extensively using multiphasic descriptions. For example (see
Fig. 5-1), chondrocytes in articular cartilage exhibit signi¬cant changes in shape and
volume that occur in coordination with the deformation and dilatation of the extra-
cellular matrix (Guilak, 1995; Guilak et al., 1995; Buschmann et al., 1996). By using
generalized continuum models of cells and tissue, the essential characteristics of cell
and tissue mechanics and their mechanical interactions can be better understood. In
this chapter, we describe several experimental and theoretical approaches for studying
the multiphasic behavior of living cells.


Biphasic (solid“¬‚uid) models of cell mechanics
Viscoelastic behavior in cells can arise from both ¬‚ow-dependent (¬‚uid“solid inter-
actions and ¬‚uid viscosity) and ¬‚ow-independent mechanisms (for example, intrinsic
viscoelasticity of the cytoskeleton). Previous studies have described the cytoplasm of
“solid-like” cells as a gel or as a porous-permeable, ¬‚uid-saturated meshwork (Oster,
1984; Oster, 1989; Pollack, 2001) such that the forces within the cell exhibit a balance
of stresses arising from hydrostatic and osmotic pressures and the elastic properties
of the cytoskeleton. This representation of cell mechanical behavior is consistent with
the fundamental concepts of the biphasic theory, which has been used to represent the
mechanical behavior of soft hydrated tissues as being that of a two-phase material.
This continuum mixture theory approach has been adopted in several studies to model
volumetric and viscoelastic cell behaviors and to investigate potential mechanisms
86 F. Guilak et al.




Fig. 5-1. Three-dimensional reconstructions of viable chondrocytes within the extracellular matrix
before (left) and after (right) compression of the tissue to 15% surface-to-surface tissue strain. Sig-
ni¬cant changes in chondrocyte height and volume were observed, showing that cellular deformation
was coordinated with deformation of the tissue extracellular matrix.


responsible for cell mechanical behavior (Bachrach et al., 1995; Shin and Athanasiou,
1999; Guilak and Mow, 2000; Baaijens et al., 2005; Trickey et al., 2006).
Modern mixture theories (Truesdell and Toupin, 1960; Bowen, 1980) provide a
foundation for multiphasic modeling of cell mechanics as well as of soft hydrated
tissues. The biphasic model (Mow et al., 1980; Mow et al., 1984), based on Bowen™s
theory of incompressible mixtures (Bowen, 1980), has been widely employed in mod-
eling the mechanics of articular cartilage and other musculoskeletal tissues, such as
intervertebral disc (Iatridis et al., 1998), bone (Mak et al., 1997), or meniscus. (Spilker
et al., 1992). In such models, the cell or tissue is idealized as a porous and permeable
solid material that is saturated by a second phase consisting of interstitial ¬‚uid (water
with dissolved ions). Viscoelastic behavior can arise from intrinsic viscoelasticity of
the solid phase, or from diffusive drag between the solid and ¬‚uid phases.
In the biphasic theory, originaly developed to describe the mechanical behavior of
soft, hydrated tissues (Mow et al., 1980), the momentum balance laws for the solid
and ¬‚uid phases, respectively, are written as:

∇ · σ s + Π = 0, ∇ ·σf ’Π = 0 (5.1)

where σ s and σ f are partial Cauchy stress tensors that measure the force per unit
mixture area on each phase. The symbol Π denotes a momentum exchange vector
that accounts for the interphase drag force as ¬‚uid ¬‚ows past solid in the mixture.
Note that, in biphasic models of cells or cartilage, the contribution of inertial terms to
the momentum balance equations is negligible, as the motion is dominated by elastic
deformation and diffusive drag and occurs at relatively low frequencies. The mixture
is assumed to be intrinsically incompressible and saturated, so that:

∇ · (φ s us + φ f v f ) = 0, where φ s + φ f = 1,
™ (5.2)
Multiphasic models of cell mechanics 87

us is the solid displacement, v f is the ¬‚uid velocity, and φ s is the solid volume fraction.
For example, under the assumption of in¬nitesimal strain, with isotropic solid phase
and inviscid ¬‚uid phase, while the momentum exchange is described by Darcy™s Law,
the resulting constitutive laws are:

σ s = ’φ s pI + »s tr (e)I + 2µs e, σ f = ’φ f pI, Π = K (v f ’ us )
™ (5.3)

where I is the identity tensor, p is a pore pressure used to enforce the incompressibility
constraint, e = 1/2[∇us + (∇us )T ] is the in¬nitesimal strain tensor, »s, µs are Lam´ e
coef¬cients for the solid phase, and K is a diffusive drag coef¬cient. The Lam´ e
coef¬cients » , µ are associated with “drained” elastic equilibrium states that occur
s s

under static loading when all ¬‚uid ¬‚ow has ceased in the mixture. An alternate set of
elastic moduli are the Young™s modulus E s and Poisson ratio ν s (0 ¤ ν s < 0.5) where
ν
Es ss
µs = 2(1+ν s ) is the solid phase shear modulus and »s = (1+ν sE)(1’2ν s ) .
For this linear biphasic model, by substituting Eq. 5.3 into Eq. 5.1, the governing
equations Eq. 5.1 and Eq. 5.2 constitute a system of seven equations in the seven
unknowns us , v f, p. The ¬‚uid velocity v f is commonly eliminated to yield a “u-p
formulation” consisting of the four equations:
1
‚t (∇ · us ) = k∇ 2 p, µs ∇(∇ · us ) + ∇ 2 us = ∇ p (5.4)
1 ’ 2ν s


where k = (φ f )2 /K is the permeability. The ¬‚uid velocity is then given by:
φf
v = ‚t u ’ ∇p
f s
(5.5)
K
The governing equations Eqs. 5.4“5.5 illustrate a common formulation of the linear
isotropic biphasic model. Within the framework of Eqs. 5.1“5.2, this fundamen-
tal model can be extended to account for additional mechanisms via modi¬cation
of the constitutive relations in Eq. 5.3. Such mechanisms have included transverse
isotropy of the solid phase, large deformation, solid matrix viscoelasticity, nonlin-
ear strain-dependent permeability, intrinsic ¬‚uid viscosity, and tension-compression
nonlinearity (Lai et al., 1981; Holmes et al., 1985; Cohen et al., 1998).


Biphasic poroviscoelastic models of cell mechanics
In other approaches, both the ¬‚ow-dependent and ¬‚ow-independent viscoelastic be-
haviors have been taken into account to describe transient cell response to loading.
This type of approach has been used previously to separate the in¬‚uence of “intrin-
sic” viscoelastic behavior of the solid extracellular matrix of tissues such as articular
cartilage from the time- and rate-dependent effects due to ¬‚uid“solid interactions in
the tissue (Mak, 1986a; Mak, 1986b; Setton et al., 1993; DiSilvestro and Suh, 2002).
For example, in modeling the creep response of chondrocytes during both full and
partial micropipette aspiration (Baaijens et al., 2005; Trickey et al., in press), it was
found that an elastic biphasic model cannot capture the time-dependent response of
chondrocytes accurately (Baaijens et al., 2005). To examine the relative contributions
of intrinsic solid viscoelasticity (solid“solid interactions) as compared to biphasic
viscoelastic behavior (¬‚uid“solid interactions), a large strain, ¬nite element simulation
88 F. Guilak et al.

of the micropipette aspiration experiment was developed to model the cell using ¬nite
strain incompressible and compressible elastic models, a two-mode compressible
viscoelastic model, a biphasic elastic, or a biphasic viscoelastic model.
Assuming isotropic and constant permeability, the governing equations Eq. 5.1 and
5.2 may be rewritten (Mow et al., 1980; Sengers et al., 2004) as:

∇ ·σ ’∇p = 0 (5.6)
∇ · v ’ ∇ · k∇ p = 0

where v denotes the solid velocity. If a viscoelastic model is used to investigate the
time-dependent behavior, a two-mode model may be used. The stress tensor is split
into an elastic part and a viscoelastic part:

σ = σe + „ . (5.7)

If a ¬nite strain formulation is used, a suitable constitutive model for the compressible
elastic contribution can be given by
G
σ e = κ(J ’ 1)I + (B ’ J 2/3 I), (5.8)
J
where, using the deformation tensor F = (∇0 x)T with ∇0 the gradient operator with
respect to the reference con¬guration, the volume ratio is given by J = det(F), and the
right Cauchy-Green tensor by B = F ·FT . The material parameters κ and G denote
the compressibility modulus and the shear modulus, respectively. The viscoelastic
response is modeled using a compressible Upper Convected Maxwell model:
1

„ + „ = 2G v Dd (5.9)
»
where the operator ∇ denotes the upper-convected time derivative (Baaijens, 1998),
and Dd is the deviatoric part of the rate of deformation tensor, de¬ned by:
1 1 ™ ’1
(F · F + F’T · FT )
Dd = D ’ tr (D) I, where D = ™ (5.10)
3 2
G v is the modulus and » is the relaxation time of the viscoelastic mode.


Multiphasic and triphasic models (solid“¬‚uid“ion)
In response to alterations in their osmotic environment, cells passively swell or shrink.
The capability of the biphasic model to describe this osmotic response is limited to
the determination of effective biphasic material parameters that vary with extracel-
lular osmolality. The triphasic continuum mixture model (Lai et al., 1991) provides
a framework that has the capability to more completely describe mechanochemical
coupling via both mechanical and chemical material parameters in the governing
equations. This model has been successfully employed in quantitative descriptions of
mechanochemical coupling in articular cartilage, where the aggrecan of the extracel-
lular matrix gives rise to a net negative ¬xed-charge density within the tissue. Similar
approaches have been used to describe other charged hydrated soft tissues (Huyghe
et al., 2003).
Multiphasic models of cell mechanics 89

To date, there has been only limited application of the triphasic model to cell
mechanics (Gu et al., 1997). At mechanochemical equilibrium in vitro, many cells are
known to exhibit a passive volumetric response corresponding to an ideal osmometer.
For an ideal osmometer, cell volume and inverse osmolality (normalized to their values
in the iso-osmotic state) are linearly related via the Boyle van™t Hoff relation. The
resulting states of mechanochemical equilibrium of the cell exhibit an internal balance
between mechanical and chemical stresses. The triphasic model, in the absence of
electrical ¬elds, gives rise to the mixture momentum equation:
∇ · σ = 0, where: σ = ’ pI + σ E (5.11)
where σ is the mixture stress and σ E is the extra stress in the solid phase. The balance
of electrochemical potentials for intracellular and extracellular water gives rise to the
Donnan osmotic pressure relation:
p = RT (•c ’ • — c— ) (5.12)
where T is the temperature, • and • — are the intracellular and extracellular osmotic
activity coef¬cients, and c and c— are the intracellular and extracellular ion concentra-
tions, respectively (R is the universal gas constant). To close the system of governing
equations (Eqs. 5.11“5.12), the intracellular extra stress and the intracellular ion con-
centration need to be characterized. While the former may be postulated via a con-
stitutive description for the subcellular components (such as nucleus, cytoskeleton,
membrane), the latter necessitates a detailed analysis of electrochemical ion poten-
tials inside a cell that accounts for intracellular ionic composition and biophysical
mechanisms such as the selective permeability of the bilayer lipid membrane, and the
nontransient activity of ion pumps and ion channels.
The strength of this type of mixture theory approach was recently illustrated in a
study modeling the transient swelling and recovery behavior of a single cell subjected
to an osmotic stress with neutrally charged solutes (Ateshian et al., 2006). A general-
ized “triphasic” formulation and notation (Gu et al., 1998) were used to account for
multiple solute species and incorporated partition coef¬cients for the solutes in the
cytoplasm relative to the external solution. Numerical simulations demonstrate that
the volume response of the cell to osmotic loading is very sensitive to the partition
coef¬cient of the solute in the cytoplasm, which controls the magnitude of cell volume
recovery. Furthermore, incorporation of tension in the cell membrane signi¬cantly
affected the mechanical response of the cell to an osmotic stress. Of particular interest
was the fact that the resulting equations could be reduced to the classical equations
of Kedem and Katchalsky (1958) in the limit when the membrane tension is equal to
zero and the solute partition coef¬cient in the cytoplasm is equal to unity. These ¬nd-
ings emphasize the strength of using more generalized mixture approaches that can
be selectively simpli¬ed in their representation of various aspects of cell mechanical
behavior.


Analysis of cell mechanical tests
Similar to other experiments of cell mechanics, the analysis of cell multiphasic
properties has involved the comparison and matching of different experimental
90 F. Guilak et al.

con¬gurations to the theoretical response as predicted by analytical or numerical
models of each individual experimental con¬guration. Given the complexity of the
governing equations for multiphasic or poroelastic models, analytical solutions have
only been possible for simpli¬ed geometries that approximate various testing con-
¬gurations. In most cases, numerical methods such as ¬nite element techniques are
used in combination with optimization methods to best ¬t the predicted behavior to
the actual cellular response in order to determine the intrinsic material properties of
the cell.


Micropipette aspiration
Cells
The micropipette aspiration technique has been used extensively to study the me-
chanical properties of both ¬‚uid-like and solid-like cells (Hochmuth, 2000), including
circulating cells such as red blood cells (Evans, 1989) and neutrophils (Sung et al.,
1982; Dong et al., 1991; Ting-Beall et al., 1993), or adhesion-dependent cells such as
¬broblasts (Thoumine and Ott, 1997), endothelial cells (Theret et al., 1988; Sato et al.,
1990), or chondrocytes (Jones et al., 1999; Trickey et al., 2000; Guilak et al., 2002).
This technique involves the use of a small glass pipette to apply controlled suction
pressures to the cell surface while measuring the ensuing transient deformation via
video microscopy. The analysis of such experiments has required the development
of a variety of theoretical models that assume cells behave as viscous liquid drops
(Yeung and Evans, 1989), potentially possessing cortical tension (Evans and Yeung,
1989), or as elastic or viscoelastic solids (Theret et al., 1988; Sato et al., 1990; Haider
and Guilak, 2000; Haider and Guilak, 2002) and speci¬cally, to model the solid-like
response of cells to micropipette aspiration, Theret et al. (1988) developed an elegant
analytical solution of an associated contact problem to calculate the Young™s modulus
(E) of an incompressible cell. This elastic model was subsequently extended to a
standard linear solid (Kelvin) model, thus incorporating viscoelastic cell properties
(Sato et al., 1990). These models idealized the cell as an elastic or viscoelastic incom-
pressible and homogeneous half-space. Experimentally, the length of cell aspiration
was measured at several pressure increments, and the Young™s modulus (E) was deter-
mined as a function of the applied pressure ( p), the length of aspiration of the cell into
the micropipette (L), and the radius of the micropipette (a) as E = 3a p/(2π L),
where is a function of the ratio of the micropipette thickness to its inner radius
(Fig. 5-2).
In recent studies, the micropipette aspiration test has been modeled assuming that
cells exhibit biphasic behavior. The cell was modeled using ¬nite strain incompress-
ible and compressible elastic models, a two-mode compressible viscoelastic model,
or a biphasic elastic or viscoelastic model. Comparison of the model to the experi-
mentally measured response of chondrocytes to a step increase in aspiration pressure
showed that a two-mode compressible viscoelastic formulation could predict the creep
response of chondrocytes during micropipette aspiration (Fig. 5-3). Similarly, a bipha-
sic two-mode viscoelastic analysis could predict all aspects of the cell™s creep response
to a step aspiration. In contrast, a purely biphasic elastic formulation was not capable
Multiphasic models of cell mechanics 91




Fig. 5-2. Viscoelastic creep response of a chondrocyte to a step increase in pressure in the
micropipette aspiration test. Images show the chondrocyte in the resting state under a tare pres-
sure at time zero, followed by increasing cell displacement over time after step application of the
test pressure. From Haider and Guilak, 2000.


(a) (b)
Normalized Cell Displacement




e
Normalized Time

Fig. 5-3. (a) Biphasic, viscoelastic ¬nite element mesh used to model the micropipette aspiration
test. The mesh contains 906 elements and has 3243 nodes, with biquadratic interpolation for the
displacement and a bilinear, continuous interpolation for the pressure. Along the left boundary,
symmetry conditions are applied, while along the portion of the cell boundary within the micropipette,
suction pressure is applied. Sliding contact conditions along the interface with the micropipette are
enforced through the use of a Lagrange multiplier formulation. (b) Normalized aspiration length
of a chondrocyte as a function of the normalized time. The solid dots indicate the experimental
behavior of an average chondrocyte. The solid line corresponds to the two-mode viscoelastic model.
The dashed line corresponds to the biphasic, two-mode viscoelastic model. For the latter model, the
triangle marks the end of the pressure application and the start of the creep response. The creep
response of the cell was well described by both the two-mode viscoelastic model and the biphasic
viscoelastic model. From Baaijens et al., 2005.
92 F. Guilak et al.




Fig. 5-4. Micropipette aspiration test to examine the volumetric response of cells to mechanical
deformation. Video images of a chondrocyte and micropipette before (a) and after (b) complete
aspiration of the cell. Cells show a signi¬cant decrease in volume, which when matched to a theoretical
model can be used to determine Poisson™s ratio as one measure of compressibility. From Trickey
et al., 2006.



of predicting the complete creep response, suggesting that the viscoelastic response
of the chondrocytes under micropipette aspiration is predominantly due to intrinsic
viscoelastic phenomena and is not due to the biphasic behavior.
Other studies have also used the micropipette technique to determine the volume
change of chondrocytes after complete aspiration into a micropipette (Jones et al.,
1999). While many cells are assumed to be incompressible with a Poisson ratio
of 0.5, these studies demonstrated that certain cells, such as chondrocytes, in fact
exhibit a certain level of compressibility (Fig. 5-4), presumably due to the expulsion of
intracellular ¬‚uid. Isolated cells were fully aspirated into a micropipette and allowed to
reach mechanical equilibrium. Cells were then extruded from the micropipette and cell
volume and morphology were measured over time. By simulating this experimental
procedure with a ¬nite element analysis modeling the cell as either a biphasic or
viscoelastic material, the Poisson ratio and viscoelastic recovery properties of the cell
were determined. The Poisson ratio of chondrocytes was found to be ∼0.38, suggesting
that cells may in fact show volumetric changes in response to mechanical compression.
The ¬nding of cell compressibility in response to mechanical loading is consistent with
previous studies showing signi¬cant loss of cell volume in chondrocytes embedded
within the extracellular matrix (Guilak et al., 1995). Taken together with micropipette
studies, these studies suggest that cell volume changes are due to biphasic mechanical
effects resulting in ¬‚uid exudation from the cell, while cellular viscoelasticity is
more likely due to intrinsic behavior of the cytoplasm and not to ¬‚ow-dependent
effects.


Biphasic properties of the pericellular matrix
In vitro experimental analysis of the mechanics of isolated cells provides a simpli¬ed
and controlled environment in which theoretical models, and associated numerical
solutions, can be employed to measure and compare cell properties via material param-
eters. Ultimately, however, most biophysical analyses of cell mechanics are motivated
by a need to extrapolate the in vitro ¬ndings to a characterization of the physiological
Multiphasic models of cell mechanics 93

(in vivo) environment of the cell. For cells such as articular chondrocytes, it is pos-
sible to isolate a functional cell“matrix unit and analyze its mechanical properties in
vitro, thus providing a link to the physiological setting (Poole, 1992). This modeling
approach is brie¬‚y described here in the context of the articular chondrocyte.
Chondrocytes in articular cartilage are completely surrounded by a narrow region
of tissue, termed the pericellular matrix (PCM). The PCM is characterized by the
presence of type VI collagen (Poole, 1992), which is not found elsewhere in cartilage
under normal circumstances, and a higher concentration of aggrecan relative to the
extracellular matrix (ECM), as well as smaller amounts of other collagen types and
proteins. The chondrocyte together with the pericellular matrix and the surround-

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