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Mechanical Aspects of Tumour Growth
Multiphase Modelling, Adhesion, and Evolving
Natural Configurations
L.Preziosi
Dipartimento di Matematica, Politecnico di Torino
Corso Duca degli Abruzzi 24, I-10129, Torino, Italy.
G.Vitale
Dipartimento di Matematica, Politecnico di Torino
Corso Duca degli Abruzzi 24, I-10129, Torino, Italy.
1
Contents
1 Mechanical Aspects of Tumour Growth: Multiphase Mod-
elling, Adhesion, and Evolving Natural Configurations 1
1.1 Introduction 1
1.2 Mass Balance for a Multicomponent System 2
1.3 Force Balance for a Multicomponent System 5
1.4 Liquid-Solid Interactions in a Saturated Mixture 7
1.5 Modelling Adhesion Forces 10
1.6 Modelling Cell-ECM Interaction Force 16
1.7 Tumour Cell Constituent as a Liquid 18
1.8 The Tumour Mass as a Solid: Evolving Natural Configurations 27
1.9 Response to Shear Tests 33
1.10 Uniaxial Compression Tests 37
References 43
1
Mechanical Aspects of Tumour
Growth: Multiphase Modelling,
Adhesion, and Evolving Natural
Configurations
1.1 Introduction
In the last ten years mixture theory has been applied with success to describe several
problems related to tumour growth, as reviewed in (Ambrosi and Preziosi, 2002),
(Araujo and McElwain, 2005a), (Graziano and Preziosi, 2007), (Lowengrub et al.,
2010), (Tracqui, 2009). This chapter will give the basics to deduce multiphase models
for few but essential constituents (cells, extracellular matrix, extracellular liquid, and
possibly blood and limphatic vasculature). All the steps of the modelling procedure
are explained in detail, special attention being paid to the meaning of all the different
terms involved in the model. We will also show how to take into account of the presence
of several sub-populations of cells, of the different components of the extracellular
matrix (ECM), and possibly of a homogenised vascular network.
Though the general model can be quite complicated, there are two basic assump-
tions that allow to obtain more manageable models. The first one consists in assuming
that the components of the extracellular matrix and possibly of the vasculature form
such an intricate network that they all move together so that the same deformation
and velocity describe their evolution. The second one consists in assuming that the
pressure gradient and the interaction forces between the liquid and the other con-
stituents are very small if compared for instance to the internal stresses of the solid
constituents or, say, the adhesion forces acting between cells and extracellular matrix.
We will also focus on other mechanical aspects related to tumour growth, related to
the activation of mechanotransduction pathways inside the cell, to contact inhibition
phenomena and to the formation of stiffer fibrotic tissues.
In order to focus on the aspects above, we will instead forget about the role that
nutrients and growth factors still have on growth. This would require the introduction
of reaction-diffusion equations in a multicomponent model. The reader interested in
such extensions can look at (Astanin and Preziosi, 2008), (Bear and Bachmat, 1990).
Specifically, Sections 2 and 3 will respectively describe how to derive for multicom-
ponent systems the balances of mass and momentum. Sections 4 through 6 will focus
on the interaction forces acting between different constituents, respectively, on the
Mechanical Aspects of Tumour Growth: Multiphase Modelling, Adhesion, and Evolving Natural Configurations
relative importance of the interaction with the extracellular liquid with respect to the
interaction between solid-like constituents, on how to transfer microscopic information
on the forces exerted by the adhesion molecules up to the continuum level and on how
to model the interaction force between cells and the extracellular matrix. Section 7
will specialize the model to the case in which the ensemble of cells is treated as a liquid
with two applications having to do with two different mechanical aspects related to
tumour growth: contact inhibition and the formation of fibrotic tissues. Section 8 will
describe how to model the ensemble of cells as a solid. Finally, Sections 9 and 10 will
give a comparison with several experimental tests on cell aggregates and multicellular
spheroids.
1.2 Mass Balance for a Multicomponent System
Tumours as their sorrounding tissues are mainly made of cells, extracellular matrix
(ECM), blood and lymphatic vessels, and physiological liquids. The growth and be-
haviour of cells depends on many chemical compounds, e.g., nutrients, growth factors,
chemotactic factors. Of course, the typical size of molecules is extremely small com-
pared to cells, vascular structures and ECM fibers, so that the space they occupy can
be neglected and they can be considered as part of the other constituents, i.e., dis-
solved in the extracellular liquid, within the vasculature, attached to the extracellular
matrix, within the cells.
For all the other constituents of the mixture it is useful to refer to their volume
ratios which can be defined as follows. Given a point in the mixture consider a sequence
of spheres centered in that point. Measuring the ratio of the volume occupied by
the constituent of interest inside the sphere to the volume of the sphere one may
observe the dependency shown in Fig. 1.1. For sample volumes having the size of the
microscopic constituent the ratio is likely to oscillate very strongly due to microscopic
inhomogenity. At the other extreme, macroscopic inhomogenities may affect the ratio
for large sample volumes. However, for sample volumes in between, that in this case
means at scales larger than the cell size and smaller than the typical tissue scale,
it is nearly constant and allows us to define a quantity called volume ratio of the
constituent.
Let us then denote by φc, φm, φℓ, φv∈[0,1] the volume ratios respectively occupied
by cells, extracellular matrix, extracellular liquid, and vascular structures. Of course,
X
α=c,m,ℓ,v
φα≤1,(1.1)
because the above structures can at most occupy the entire space and the model
must be such that the solution satisfies this unilateral constraint. If, instead, one has
considered all the constituents occupying the region of interest, then the mixture is
said to be saturated and X
α=c,m,ℓ,v
φα= 1 .(1.2)
Actually, in some cases the upper constraint on the volume ratio (i.e., 1 in (1.2)) is
replaced by a constant value φ < 1, possibly space dependent, allowing for some fixed
Mass Balance for a Multicomponent System
Fig. 1.1 Plot of the volume ratio of the constituent as a function of sample volume size.
portion of space to be occupied by other constituents not considered in the mixture.
To be specific, if we are fixing the volume occupied by vessels or by a general stroma
Eq.(1.2) can be replaced by
φc+φm+φℓ=φ≤1.(1.3)
From the mathematical viewpoint this geometrical constraint still plays the same role
as the saturation assumption (1.2).
The mass balance equations for the different constituents can be obtained starting
from a global mass balance of that constituent over a general control volume V fixed
in space with boundary ∂V and external normal to the boundary n. As an example,
let us focus on the cellular constituent of the tissue. Denoting by ρthe density of the
single cells, the mass of the constituent contained in V is
M=ZV
ρφcdV
and can change due to the flux caused by the motion of the constituent through the
boundary ∂V and to growth or death of cells. One then has
dM
dt =−Z∂V
ρφcvc·ndΣ + ZV
ρΓcdV ,
where vcis the cell velocity and Γcis the growth/death rate for the cellular mass.
Using the divergence theorem, one can write
ZV∂
∂t (ρφc) + ∇ · (ρφcvc)−ρΓcdV = 0 .
Mechanical Aspects of Tumour Growth: Multiphase Modelling, Adhesion, and Evolving Natural Configurations
If the integrands are smooth, the arbitrariety of the volume of integration V assures
that the integrand must vanish, i.e.,
∂
∂t (ρφc) + ∇ · (ρφcvc) = ρΓc.(1.4)
where ρcan be taken constant and equal to the density of water.
The same procedure can be reproposed for all the volume ratios so that one can
write the mass balance equations
∂φα
∂t +∇ · (φαvα) = Γα,for α=c, m, ℓ, v . (1.5)
In the case of a saturated mixture, summing up the above equations gives
∇ · X
α=c,m,ℓ,v
(φαvα) = X
α=c,m,ℓ,v
Γα,(1.6)
If vessels are not taken into account, as for instance happens in the avascular
phase of growth, and mass exchange only occurs between the constituents taken into
consideration the mixture is said to be closed and, if for sake of simplicity one takes
all constituents to have the same density ρ, one can write
Γc+ Γm+ Γℓ= 0 ,(1.7)
(otherwise it is the sum of ραΓαthat need to vanish).
On the other hand, the presence of a homogenised vascular or lymphatic structure
makes the system open, because blood and lymphatic fluids are flowing in the ves-
sels and perfusing through their walls. This allows the sum of the growth rates to be
non vanishing. In some models, for sake of simplicity, the presence of blood and lym-
phatic vessels and even of the extracellular liquid is neglected, though their influence is
reflected in the presence of external mass source/sink terms not satisfying (1.7). How-
ever, also in this case one needs to assure that during its evolution the solution is such
that the geometrical condition (1.1) is never violated. This approach is, for instance,
used in (Franks et al., 2003a), (Franks et al., 2003b), (Franks and King, 2003).
According to the details needed to describe the phenomenon of interest, modelling
the cellular population as a single constituent might not be enough. It may be necessary
to distinguish different cell populations, e.g., tumour cells, endothelial cells, epithelial
cells, fibroblasts, macrophages, lymphocytes, or to distinguish different clones within
the same population characterized by relevant differences in their behaviour, or to
distinguish the cells according to their phase in the cell cycle, because, for instance,
the response to therapies depends on whether the cell is resting, growing, or undergoing
mitosis.
If this is the case, the same procedure can be reproposed for each sub-population,
so that actually Eq.(1.4) is split in Iequations
∂φci
∂t +∇ · (φcivci) = Γci, i = 1,...,I, (1.8)
where Iis the number of subpopulations, φciis the volume ratio of the subpopulation
i,vciis its velocity and Γciis its growth/death rate, that in this case can also take
Force Balance for a Multicomponent System
into account of trasition phenomena, e.g., between clones or between phases of the cell
cycle. Of course,
I
X
i=1
φci=φc,
I
X
i=1
φcivci=φcvc,
I
X
i=1
Γci= Γc,(1.9)
and therefore summing all (1.8) over igives Eq. (1.4) back.
In a similar way, it might be necessary to distinguish the ECM in its main con-
stituents, e.g., collagen, elastin, fibronectin, vitronectin, proteoglycans, because of the
different mechanical behaviour and chemical properties, or because of the different
production/degradation phenomena. One then has
∂φmj
∂t +∇ · φmjvm= Γmj, j = 1,...,J, (1.10)
where mjis the volume ratio of the j-th component and Γmjis its remodelling rate. We
stress that in (1.10) the ECM velocity is taken to be the same for all ECM components,
which means describing them as an intricate network of fibres that have to move all
together. This is called a constrained sub-mixture assumption, that, as we shall see in
the following, will call for a single force balance equation for the ECM. As before
J
X
j=1
mj=m ,
J
X
j=1
Γmj= Γm,(1.11)
and summing (1.10) over jgives
∂φm
∂t +∇ · (φmvm) = Γm,(1.12)
back. The same thing could be repeated for the vascular networks because lymphatic,
arterial flows and venous flows play different roles. For these constituents is however
plausible to assume that the vessel move together with the ECM structure being
entangled in their network. In this case vv=vm.
1.3 Force Balance for a Multicomponent System
In order to determine the velocity fields appearing in the mass balance equations, in the
theory of mixture one can write a momentum balance equation for each constituent.
As we shall see in the following, luckily in dealing with growth phenomena inertial
terms can be neglected, so that the equations simplify to force balance equations.
A further simplification is related to the fact that the ensemble of cells and ECM
constituents make a porous material in which the extracellular liquid flows, so it is
possible to use tools and results that are classical in the theory of porous media. On the
other hand, the behaviour of the porous material is complicated by its deformability
and by the presence of adhesion bonds between the constituents and within the single
constituent that because of their weakness are easily breakable, as occurs during growth
or remodelling.
Mechanical Aspects of Tumour Growth: Multiphase Modelling, Adhesion, and Evolving Natural Configurations
In order to clarify the origin of all terms appearing in the momentum equations,
we here focus on the cellular constituent of the tissue.
Starting again with the integral formulation, the variation of momentum of the
cellular constituent in the fixed volume V (term on the l.h.s. of Eq.(1.13) below) is
due to
•the momentum flux caused by the motion of the cells through the boundary ∂V
(first term on the r.h.s. of Eq.(1.13));
•contact forces with other cells through the boundary ∂V, which are codirected
with n(second term on the r.h.s. of Eq.(1.13));
•contact forces due to the interaction with the other constituents within the do-
main through the interface separating the constituents, say the part of the cell
membrane wet by the extracellular liquid or in contact with the extracellular
matrix through the adhesion sites (third term on the r.h.s. of Eq.(1.13));
•momentum supply related to mass exchange among the constituents, e.g., fluid
absorbed by a growing cell, or ECM production (fourth term on the r.h.s. of
Eq.(1.13));
•body forces, e.g., chemotaxis or haptotaxis can be modelled in this way, though
they actually involve the activation of sub-cellular mechanisms rather than an
external action. (see (Chauviere and Preziosi, 2010)) (fifth term on the r.h.s. of
Eq.(1.13)).
One then has
d
dtZV
ρφcvcdV =−Z∂V
ρφcvc(vc·n)dΣ + Z∂Ve
TT
cndΣ
+ZVe
mcdV +ZV
ρΓcvcdV +ZV
ρφcbcdV ,
(1.13)
where e
Tcis called partial stress and e
mcis called interaction force.
So using the divergence theorem we can write
ZV∂
∂t (ρφcvc) + ∇ · (ρφcvc⊗vc−e
Tc)−ρφcbc−e
mc−ρΓcvcdV = 0 .(1.14)
If the integrands are smooth, the arbitrariety of the volume of integration V assures
that the integrand must vanish, i.e.,
∂
∂t (ρφcvc) + ∇ · (ρφcvc⊗vc) = ∇ · e
Tc+ρφcbc+e
mc+ρΓcvc.(1.15)
Actually, using the mass balance equation (1.5)1, Eq. (1.15) can be simplified as
ρφc∂vc
∂t +vc· ∇vc=∇ · e
Tc+ρφcbc+e
mc.(1.16)
As already stated, in growth phenomena the inertial term on the left hand side can
be neglected, so that (1.16) simplifies even further in
Liquid-Solid Interactions in a Saturated Mixture
∇ · e
Tc+ρφcbc+e
mc= 0 .(1.17)
Proceeding in a similar way for the other constituents, one can write
∇ · e
Tm+e
mm=0,(1.18)
∇ · e
Tℓ+e
mℓ=0,(1.19)
where we dropped the body force term because, apart from gravity that plays no role
in the phenomena of interest for this chapter, it is difficult to imagine relevant body
forces acting on the ECM and on the liquid.
We observe explicitly, that the equation for the cellular constituent can be special-
ized for any sub-population. On the other hand, even if it is found necessary to specify
different ECM components, the constrained sub-mixture hypothesis assures that a sin-
gle force balance equation (1.26) is enough to determine the common velocity vm. Of
course, all the ECM components will contribute to the constitutive equation for the
stress tensor according to their relative proportions.
If vascular structures are considered and move together with the ECM, Eq.(1.18)
will still hold including in Tmthe contribution due to vessels and in mmthe interaction
of the vessels with the other constituents.
If the mixture is closed and then momentum is exchanged only between the con-
stituents taken into consideration, then one can enforce (Bowen, 1976), (Rajagopal
and Tao, 1995) X
α=c,m,ℓ
(e
mα+ρΓαvα) = 0,(1.20)
or, coherently with the fact that inertia can be neglected, the contribution due to mass
exchange can be dropped with respect to that due to the interaction forces (Preziosi
and Farina, 2001), so that X
α=c,m,ℓ e
mα=0.(1.21)
Hence, one can say that the interaction forces sum up to zero, as it might be expected
since they act as internal forces among the constituents of the whole mixture.
Actually, each term mαcontains all forces acting on the constituent αdue to its
interaction with the other constituents β. We here assume that an action-reaction
principle holds for each interaction pair, i.e., mαβ =−mβ α, where mαβ is the force
acting on the constituent αdue to β. This, however, is an approximation, for instance,
for the presence of exchanges of mass and of other effects, that however, as stated
above, can be considered negligible in growth phenomena.
1.4 Liquid-Solid Interactions in a Saturated Mixture
In a saturated and closed mixture the geometrical constraint (1.2) describing satura-
tion implies that not all motions are possible, but only those satisfying (1.6) with a
vanishing r.h.s., or,
Mechanical Aspects of Tumour Growth: Multiphase Modelling, Adhesion, and Evolving Natural Configurations
0 = X
α=c,m,ℓ,v ∇ · (φαvα) = X
α=c,m,ℓ,v
(φαI:Lα+vα· ∇φα),(1.22)
where Iis the identity tensor and the colon indicates tensor saturation. Without going
into technical details, this means that it is possible to test the stress response only for
those deformation gradients satisfying the above constraint, or that the validity of the
second principle of thermodynamics is checked only for all those processes satisfying
the saturation constraint.
A similar reasoning is classically used when studying the flow of incompressible
fluids or the deformation of composite materials that can be assumed to be inestensi-
ble in a given direction. Enforcing the above conditions imply that one can only study
those deformations satisfying the constraints and only for these classes of processes
the second principle of thermodynamics should hold. The consequence is that in the
former case the isotropic part of the stress tensor for an incompressible fluid can not be
determined constitutively but is a reaction that adjusts so that the incompressibility
constraint is satisfied. Similarly, in the latter case in which a material is inextensi-
ble along the direction Nin the reference configuration one can not determine the
component n⊗n(with n=FN and Fis the deformation gradient) of the stress
tensor.
In the case of mixtures, the saturation constraint implies that the constitutive
equations for the partial stresses and for the interaction forces are characterized by
the presence of a Lagrange multiplier classically identified with the interstitial pressure
of the extracellular liquid (Bowen, 1976; Rajagopal and Tao, 1995), that for the cellular
constituent writes as
e
Tc=−φcPI+φcTc,(1.23)
e
mc=P∇φc+mc,(1.24)
where Tcis called excess stress and mcexcess interaction force. Similar conditions
hold for the other constituents.
Neglecting for sake of simplicity the presence of vessels, one can then write
−φc∇P+∇ · (φcTc) + mc+ρφcbc=0,(1.25)
−φm∇P+∇ · (φmTm) + mm=0,(1.26)
−φℓ∇P+mℓ=0.(1.27)
In the force balance equation for the liquid, as it is usually done when dealing with
porous materials, the excess stress tensor for the extracellular liquid is assumed to be
negligible. This allows to obtain a Darcy’s-like law (Bowen, 1976),(Rajagopal and Tao,
1995).
In fact, the interaction forces of all the constituents with the liquid can be taken to
be proportional to the velocity difference between the liquid and the other constituents
through invertible matrices Mcand Mm, so that
Liquid-Solid Interactions in a Saturated Mixture
mℓc =−Mc(vℓ−vc),(1.28)
mℓm =−Mm(vℓ−vm),(1.29)
where Mαwith α=c, m are invertible matrices. Therefore, (1.27) can be written as
Mm(vℓ−vm) + Mc(vℓ−vc) = −φℓ∇P , (1.30)
or
vℓ=M−1 X
α=c,m
Mαvα−φℓ∇P!,(1.31)
where M=Mc+Mm, that explicitly gives the liquid velocity in terms of the other
velocities and of the pressure gradient.
In particular, in the case of a rigid porous material vc=vm=0, defining M−1=
K
µφ2
ℓ
where Kis the permeability tensor and µis the viscosity of the extracellular
liquid, one has the classical Darcy’s law
vℓ=−K
µφℓ∇P . (1.32)
Using (1.6) one can also determine the following equation relating the pressure to
the other variables
∇ · K
µ∇P=∇ · "X
α=c,m φαI+K
φℓµMαvα#−X
α=c,m,ℓ
Γα.(1.33)
By the action-reaction principle assumed at the beginning of this section, the same
terms as (1.29) are found with the opposite sign in the equations for the cells, the
ECM constituents and possibly the vessels. These equations also contain the pressure
gradient terms, that because of (1.27) are of the same order as the interaction force
with the extracellular liquid. It can be argued that these terms are much smaller than
the interaction forces acting between cells and between cells and ECM, so that the
momentum equations can be simplified into
∇ · (φcTc) + mcm +ρφcbc=0,(1.34)
∇ · (φmTm)−mcm =0.(1.35)
Under these hypotheses the equations above do not depend either on the interstitial
pressure or on the liquid velocity. Therefore, they can be solved in principle without
solving (1.31) and (1.33), which is only required if we want to describe the evolution
of either the interstitial pressure, or of the liquid velocity. They are solvable in cascade
after integrating eqs (1.34) and (1.35).
Mechanical Aspects of Tumour Growth: Multiphase Modelling, Adhesion, and Evolving Natural Configurations
We observe that summing them gives the force balance equation for the tissue
∇ · (φcTc+φmTm) + ρφcbc=0,(1.36)
with the pressure gradient term and the interaction forces with the liquid neglected,
compatibly with the assumptions done before writing Eqs.(1.34) and (1.35). This might
be a more convenient equation to use because the mixture stress present in the diver-
gence is the stress that can be actually measured experimentally.
We also observe that if one considers an unsaturated mixture, for instance neglect-
ing the interstitial liquid and its mechanical interactions with the other constituents,
one could get again (1.34) and (1.35). Of course, the unilateral geometrical condition
on the volume ratio (1.1) need still to be assured by the model.
1.5 Modelling Adhesion Forces
In order to properly model the adhesion forces exertable by cells, it is useful to report
what is known form the experimental point of view. In fact, several researchers tried
to measure the strength of single or clustered adhesion bonds formed by a cell (see,
for instance, (Baumgartner et al., 2000), (Canetta et al., 2005), (Sun et al., 2005)).
The typical experiment is done using an atomic force microscopy cantilever with a tip
that can be possibly functionalised with proper adhesion molecules to check the specific
interaction of the cell adhesion molecules with those placed on the tip of the cantilever.
After putting the tip in contact with the cell for some time, either the cantilever or
the plate with the cell are pulled away at a constant speed, typically in the range
0.2–5 µm/sec. If the tip of the cantilever does not attach to the cell, when they are
taken apart there is no deflection of the cantilever. This is experimentally obtained,
for instance, by the addition of an antibody attaching to the external domain of the
adhesion molecule (Baumgartner et al., 2000), or by interfering with the links between
the adhesion molecules and the cell cytoskeleton (Canetta et al., 2005), or by disrupting
the actin cytoskeleton (Sun et al., 2005). On the other hand, adhesion gives rise to the
deflection of the cantilever that can be related to the stretching force exerted by the
cell. Of course, with time the distance between the cell and the cantilever increases,
increasing the deflection angle and the stretching force. It is then observed that after
some time the adhesive bond breaks causing a characteristic jump in the deflection of
the cantilever that returns to its undeformed configuration. In this way it is possible
to evaluate the maximum force of an adhesion bond.
(Baumgartner et al., 2000) measured the strength of the adhesion bonds to be in
the range 35–55 pN giving a distribution function of the critical unbinding force like
the one shown in Fig.1.2a.
Similar results were obtained by (Canetta et al., 2005), and Sun et al. (Sun et al.,
2005). In particular, Sun et al. (Sun et al., 2005) did not functionalize the microsphere
and allowed a longer resting period on the cell surface, ranging from 2 to 30 seconds.
Again, pulling away the cantilever at a constant speed in the range 3–5 µm/sec caused
the rupture of one or more adhesive bonds. They used different cell types (Chinese
hamster ovary cells, endothelial cells and human brain tumour cells), all showing an
adhesive strength of a single bond slightly below 30 pN (see Fig.1.2b).
Modelling Adhesion Forces
(a)
(b)
Fig. 1.2 Distribution function of the force of unbinding events (a) when a single bond is
acting (red) and when more adhesion bonds are clustering (blue) (Data from (Baumgartner
et al., 2000)) and (b) for different types of cells (Data from (Sun et al., 2005)).
Mechanical Aspects of Tumour Growth: Multiphase Modelling, Adhesion, and Evolving Natural Configurations
Unfortunately, all these measurements are done at the cellular scale. In order to
use this information in the continuum model introduced in the previous section, one
needs to upscale the results of the above experiments at the macroscopic scale.
A way to do that is suggested by (Olz and Schmeiser, 2010), (Olz et al., 2008) who
solved a similar problem when dealing with the actin cytoskeleton. In order to do that,
as a starting point let us focus on a single bond that is formed at a certain time between
a cell and the extracellular fibre the cell is moving on. While moving, this bond, initially
unstretched, will stretch and exert a force on the cell that depends on the amount of
space travelled from the instance of formation of the bond. This length of time is the
age of the bond aand the force is Fmic =Fmic[x(t)−x(t−a)]. For instance, if we
can assume Hooke’s law for the bond Fmic =−kmic[x(t)−x(t−a)] and if the velocity
of the cell is constant Fmic =−kmic ˙
xa. However, from Fig.4 in (Baumgartner et al.,
2000) it is evident that there is a hardening effect, i.e., the unbinding force increases
with the relative velocity of the cantilever with respect to the cell.
Of course, at time tthe cell does not have a single bond active but an ensemble of
bonds and, during motion, bonds continuously form and are broken. Following (Olz
and Schmeiser, 2010), (Olz et al., 2008) and focusing on a one-dimensional situation,
we then introduce a probability distribution f(t, a) over the age of the bonds, so that
the number of bonds formed at time tis
N(t) = Z+∞
0
f(t, a)da .
If, as time goes, the bonds break at a rate ζ, that in general is not constant but is a
function of Fmic, then df
dt =−ζf ,
where d/dt is the material derivative following the bond during its evolution. Taking
into account the obvious fact that after a time interval ∆tthe bond ages of the same
amount, i.e., da/dt = 1, one has that the evolution equation for the bond distribution
is
∂f
∂t +∂f
∂a =−ζ f , (1.37)
as in the classical structured population models (Iannelli et al., 2005) or in transport
theory.
The above PDE needs a boundary condition for a= 0 related to the law of bond
formation. We could take it to be constant, but as will be shown in the following,
this will give rise in some cases to unreasonable results. A better boundary condition
should take into account the fact that the cell can form a maximum number of bonds
Nmax, so that one can assume that the formation of new bonds is proportional to the
bonds that can still be formed, i.e.,
f(t, a = 0) = βNmax −Z+∞
0
f(t, a)da.(1.38)
The final output of the computation it the total force that is given by
Modelling Adhesion Forces
F=−kmic Z+∞
0
[x(t)−x(t−a)]f(t, a)da . (1.39)
Before proceeding it is worthwhile to write the equations in dimensionless form in
order to justify the quasistationary version of the problem used later on. We then scale
lengths with the typical cell size Land time with T=L/ ˙x, i.e., the time needed by
a cell to move across a cell length. It is useful to introduce the ratio ǫ=A/T where
Ais the typical age of a bond. We assume that the characteristic time related to cell
motion is much larger than the typical age of a bond, so that ǫ≪1. We also scale f
by Nmax/A.
The dimensionless form of Equation (1.37) is then
ǫ∂˜
f
∂˜
t+∂˜
f
∂˜a=−˜
ζ(˜
Fmic)˜
f , (1.40)
where ˜
ζ=Aζ, that need to be joined to the boundary condition
˜
f(˜
t, ˜a= 0) = ˜
β1−Z+∞
0
˜
f(˜
t, ˜a)d˜a,(1.41)
where ˜
β=Aβ. Finally, set L˜x=x
F=−kmicNmax LZ+∞
0
[˜x(T˜
t)−˜x(T(˜
t−ǫ˜a))] ˜
f(˜
t, ˜a)d˜a . (1.42)
In the limit ǫ→0,
Fmic ≈ −kmic ˙xa (1.43)
and the problem reduces to its quasistationary version
∂˜
f
∂˜a=−˜
ζ˜
f ,
˜
f(˜a= 0) = ˜
β1−Z+∞
0
˜
f(˜a)d˜a,
(1.44)
with the macroscopic force that can be approximated by
F=−ǫkmicNmax ˙
˜xZ+∞
0
˜af(˜a)d˜a , (1.45)
where ˙
˜x=T˙x/L.
We prefer, however, to work with dimensional variables and go back to the dimen-
sional quasi-stationary problem
∂f
∂a =−ζ(kmic ˙xa)f ,
f(a= 0) = β1−Z+∞
0
f(a)da,
F=−kmic ˙xZ+∞
0
af(a)da .
(1.46)
Mechanical Aspects of Tumour Growth: Multiphase Modelling, Adhesion, and Evolving Natural Configurations
Before proceeding we observe that in the case of non-Hookean microscopic force of
the bond, the last equation in (1.46) is replaced by
F=Z+∞
0
Fmic( ˙xa)f(a)da . (1.47)
For any ζ, the partial differential equation in (1.46) can be solved giving
f(a) = Cexp −Za
0
ζ(kmic ˙xα)dα,(1.48)
where Ccan be determined through the boundary condition obtaining
C=β
1 + βR+∞
0exp −Ra
0ζ(kmic ˙xα)dαda .(1.49)
Hence,
f(a) =
βexp −Za
0
ζ(kmic ˙xα)dα
1 + βZ+∞
0
exp −Za
0
ζ(kmic ˙xα)dαda
,(1.50)
and
F=−kmicβ˙xZ+∞
0
aexp −Za
0
ζ(kmic ˙xα)dαda
1 + βZ+∞
0
exp −Za
0
ζ(kmic ˙xα)dαda
.(1.51)
For instance, if ζ=ζ0constant, then
f(a) = βζ0
β+ζ0
e−ζ0a,(1.52)
and
F=−kmic
β
ζ0(β+ζ0)˙x . (1.53)
One then finds the classical drag law asserting that the interaction force acting on the
cell moving with a velocity ˙xis proportional to its velocity.
However, more general forces can be obtained. For instance, if ζis taken to be
proportional to the microscopic force (i.e., ζ=ckmic ˙xa), then
f(a) = βexp[−ckmica2/2]
1 + β/√2ckmic ˙xπ (1.54)
and
F=−β
c
1
1 + β/√2ckmic ˙xπ (1.55)
More interesting are the cases in which the bonds break only if Fmic overcomes
a threshold Fm, because from the data by (Baumgartner et al., 2000), (Sun et al.,
Modelling Adhesion Forces
Fig. 1.3 Macroscopic adhesion laws for different cell-ECM microscopic interaction laws.
2005) it seems that there is a non-vanishing Fmranging between 5pN and 10pN. If,
for instance,
ζ(Fmic) = ζ0H(Fmic −Fm) (1.56)
where His the Heavyside function, then
F=−βF2
0+FmF0+1
2F2
m
(β+ζ0)F0+βFm
,(1.57)
where
F0=kmic ˙x
ζ0
.(1.58)
For small velocities |F|tends to Fm/2 while for large velocities it goes to βkmic ˙x/[(β+
ζ0)ζ0]. This behaviour, shown in Fig.1.3, is compatible with the one proposed in
(Preziosi and Tosin, 2009) where it is argued that if cells are not pulled strong enough
to detach from the ECM, they remain attached to it. If they detach, the force in ex-
cess can be assumed initially to be proportional to the relative velocity of the cell with
respect to the ECM (see also Fig.1.4).
It should be noticed here that if the rate of formation of bonds were simply constant,
i.e., if the integral in the boundary condition were absent, than the above procedure
would yield a force blowing up for small velocity. This biologically corresponds to the
fact that if the cell barely moves, bonds always form but never break. So, in the limit
an infinite number of bonds form, corresponding to an infinite force. This is of course
unphysical and justifies the presence of a saturation term in the boundary condition
(1.38).
Mechanical Aspects of Tumour Growth: Multiphase Modelling, Adhesion, and Evolving Natural Configurations
It is also known that adhesion bonds have a limited strength and that all bonds
break for large enough microscopic forces (e.g., 200 pN in Baumgardner experiments).
In the model above this means that ζgoes to infinite above a critical value FM, or
better that the breaking time (1/ζ) goes to zero. This assumption leads to
F=−βF2
0+FmF0+1
2F2
m−(FM+F0)F0exp [−(FM−Fm)/F0]
ζ0F0+β[Fm+F0−F0exp [−(FM−Fm)/F0]] ,(1.59)
that is characterized by a behaviour similar to (1.57) for small velocities but it goes
back to zero for large velocities due to the breaking of all the bonds.
Different breaking distributions b(Fmic) would give rise to different macroscopic
forces, taking into account that the breaking distribution (e.g., those in Fig.1.2) are
related to ζby
ζ=b(Fmic)
B(Fmic)where B(Fmic ) = ZFM
Fmic
b(φ)dφ (1.60)
is the survivial function and FMis the sup of the support of Fmic.
1.6 Modelling Cell-ECM Interaction Force
Coming back to the original problem of modelling the interaction between cells and
ECM and keeping in mind the discussion in the previous section, the simplest as-
sumption of a constant rate of detachment of focal adhesion sites implies (1.53) and
therefore a linear relationship between the interaction term and the relative velocity
of the cells with respect to the ECM, that can be rephrased in vector form as
mcm =−Mcm(vc−vm),(1.61)
similarly to (1.29).
The introduction of a threshold before breaking the bonds as in (1.57) is instead re-
flected, for instance, in the following simplified constitutive model proposed in (Preziosi
and Tosin, 2009)
vc=vm,if |mcm| ≤ σcm ,
(|mcm| − σcm )mcm
|mcm|=−Mcm(vc−vm),if |mcm|> σcm .
(1.62)
The coefficient σcm is expected to depend on the adhesion mechanisms and on the
volume ratio of the actors, the cells and the ECM.
Neglecting body forces, from (1.34)
mcm =−∇ · (φcTc),
and therefore Eq. (1.62) can be rewritten as
vc−vm=Kcm 1−σcm
|∇ · (φcTc)|+∇ · (φcTc) (1.63)
where (·)+stands for the positive part of the parenthesis and Kcm =M−1
cm is called
in this chapter motility coefficient of the cell population. Equation (1.63) replaces
Eq. (1.34) and has to be solved jointly with Eq. (1.35) or (1.36).
Modelling Cell-ECM Interaction Force
Fig. 1.4 Viscoplastic cell–ECM interaction.
To better understand the meaning of the constitutive equations above we can make
some calculations for |mcm|> σcm . Taking the modulus of Equation (1.62), one has
Mcm|vm−vc|=|mcm| − σcm .(1.64)
Replacing |mcm|in the same equation, it can be rewritten as
Mcm|vc−vm|mcm
σcm +Mcm|vc−vm|=Mcm(vm−vc),(1.65)
or
mcm =−σcm
vc−vm
|vc−vm|−Mcm(vc−vm),(1.66)
that allows to distinguish in mcm a static contribution in the direction of the relative
motion (the first term) from a drag contribution proportional to the velocity difference.
Equation (1.66) can also be compared with (1.57). Of course, a viscous drag force is
recovered in the limit σcm = 0.
As already mentioned the friction force σcm strongly depends on the concentration
of ECM. For instance, increasing the concentration of ECM leads to an increase in
activated adhesion sites and therefore in a stronger friction threshold. In addition, it
is known that there is an optimal concentration of ECM favouring motility (Palecek
et al., 1997), because the content of ECM can not become too small, otherwise the
lack of substratum would lead to a decrease in cell motility. Then the observation that
cells hardly move when there is little or too much ECM can be translated as σcm
increasing for small and “large” φm, thus, effectively, prohibiting cellular motion.
We can take the interaction between cell and ECM to be proportional to the volume
ratio of cells and ECM and assume that there is a threshold above which cells can not
pass through the ECM. A function with the above properties is
Mechanical Aspects of Tumour Growth: Multiphase Modelling, Adhesion, and Evolving Natural Configurations
Fig. 1.5 Growth of a tumour mass in presence of thick basal membranes at t= 30,60,90,120.
σcm =ˆσcmφcφm
φ∗
m−φm
,(1.67)
with φ∗
m<1. This is important to consider the case of cell compartimentalization by
ECM barriers or basal membranes as shown in Fig.1.5.
1.7 Tumour Cell Constituent as a Liquid
A reasoning similar to the one above can be reproposed for the deduction of the
constitutive equation for the stress tensor of the cellular constituent. Qualitatively
speaking, the adhesive interaction force will be replaced by the stress within the mul-
ticell spheroid and relative velocities between the interacting constituents by a rate of
strain tensor. For the moment we will not do that and assume, as most of the papers
in the literature, the easiest constitutive equation for the ensemble of cells, i.e., that
it behaves like an elastic fluid
Tc=−ΣcI,
where Σcis taken positive in compression. The use of this constitutive equation would
result in a multicellular spheroid that in absence of ECM is not able to sustain shear.
In this case one can substitute (1.63) in the mass balance equation (1.4) to obtain
the following model
Tumour Cell Constituent as a Liquid
Fig. 1.6 Examples of contact inhibition of growth reported in the experiments by Tsukatani
et al. (Tzukatani et al., 1997) using human breast epithelial cells and by Orford et al. (Orford
et al., 1999) using canine kidney-derived nontransformed epithelial cells.
∂φc
∂t +∇ · (φcvm) = ∇ · "φcKcm 1−σcm
|∇(φcΣc)|+∇(φcΣc)#+ Γc,
∂φm
∂t +∇ · (φmvm) = Γm,
∇ · (φmTm)− ∇(φcΣc) = 0.
(1.68)
A possible extension is to consider a viscous behaviour as done in (Byrne and
Preziosi, 2004), (Franks et al., 2003a), (Franks et al., 2003b), (Franks and King, 2003)
ˆ
Tc= (−Σc+λ∇ · vc)I+ 2µDc,(1.69)
where D= (∇vc+∇vT
c)/2 is the rate of strain tensor. This constitutive equation has
the advantage to confer more stability to the growing mass.
1.7.1 Contact Inhibition of Growth
As an example and still having in mind the focus of this chapter on mechanical aspects
of tumour growth, we describe here the phenomenon of contact inhibition of growth
(Dietrich et al., 1997), (Kato et al., 1997), (Nelson and Chen, 2003), (Polyak et al.,
1994), (StCroix et al., 1998), that consists in the properties of normal cells to decrease
their proliferation rate when they realize that they are coming in contact with other
cells. A quantification of this phenomenon is represented in Fig. 1.7.1 that reports
some experimental results by Tsukatani et al. (Tzukatani et al., 1997) on human
Mechanical Aspects of Tumour Growth: Multiphase Modelling, Adhesion, and Evolving Natural Configurations
breast epithelial cells grown in vitro over a suitable substratum and by Orford et al.
(Orford et al., 1999) on canine kidney-derived nontransformed epithelial cells.
It can be seen that after an initial exponential growth cell density saturates forming
a monolayer of cells. This phenomenon is called growth to confluence. On the other
hand, tumor cells continue proliferating forming a multilayer leading to the conjecture
that they need to feel more contacts or a larger pressure to stop their proliferation
program.
A key role in transferring this information is played by the mechano-transduction
pathways involving cadherin-cadherin junctions. Cadherins are transmembrane recep-
tors involved in homophilic cell-cell interactions. By several experiments (Warchol,
2002), (Caveda et al., 1996), (Castilla et al., 1999) it is proved that if a cell is not
so sensitive to the control mechanisms above it is subject to deregulated growth, a
phenomenon that is considered such an important milestone in the development of
tumors to deserve to be named cadherin switch.
In fact, it is known that loss of contact responsiveness is commonly associated with
the formation of hyperplasia and malignant transformation such as gastric carcinoma
(Becker et al., 1994), (Oda et al., 1994), adenocarcinoma (Tzukita et al., 1993), epithe-
lial tumors (Cavallaro et al., 2002), (Christofori and Semb, 1999), colon polyps and
carcinoma (Gottardi et al., 2001), gynecological cancers (Risinger et al., 1994), intimal
thickening (Uglow et al., 2000) (see also the review by (Harja and Fearon, 2002)).
However, cadherins only represent the tip of the iceberg. They are more visible
than other hidden players for their transmembrane location, but there are many other
candidates that can be responsible of a possible incorrect mechano-transduction. The
second family of suspects are the catenins, the proteins cadherins link to for a func-
tional cell-to-cell adhesion. In fact, (Stockinger et al., 2001) showed that epithelial
cells exhibited a strong β-catenin activity at low densities (≤40% confluency), which
was five- to seven-fold reduced when cells reached a confluency >80%. In fact, it is
tought that in physiological conditions upon reaching confluency the expressed cad-
herins sequester catenins downregulating their activity. Since it is known that the
upregulation of catenins is necessary for cell duplication, the final result is that cell
adhesion negatively affects cell proliferation.
More in detail, (Dietrich et al., 1997) explain the mechanism of contact inhibition
of growth as follows:
•tissue compression and overexpression of cadherins cause the underexpression of
catenins, that are sequestered by cadherin at the cell membrane;
•the underexpression of catenins determines the accumulation of the cyclin-dependent
kinase (cdk) inhibitors p16, p21, and p27;
•their overexpression inhibits the entry in the S phase causing cell cycle arrest in
the G1 phase. More in detail, referring to Fig. 1.7.1
- p16 blocks the activity of cdk4 by dissociating cyclin D from cdk4 and binding to
cdk4;
- p27 inhibits cdk2-cyclin E activity directly by binding to the complex.
The above biochemical description of the mechano-transduction pathway involved
in contact inhibition can be introduced in the model stating that mitosis stops when
Tumour Cell Constituent as a Liquid
Fig. 1.7 Sketch of the mechanotransduction pathway. Arrows and blockades respectively
indicate stimulatory and inhibitory activities. cdk stands for cyclin-dependent kinase, pRB
for hypophosphorilated retinoblastoma, and the added p’s indicate its phosphorilation. On
the left, the inhibitory response related to contact inhibition. On the right, the activation of
the duplication program due to lack of cell-cell contact.
the volume ratio (or the compression) overcomes a given threshold.
The behaviour of the cells in terms of growth and motion then crucially depends
on how they feel the presence of other cells and how they translate the mechanical
cues. In (Chaplain et al., 2005), (Graziano and Preziosi, 2007), (Galle et al., 2009) it is
shown that a fault in the mechanotrasduction pathway might lead to a misperception
of the compression state of the local tissue, and can then determine a clonal advantage
on the surrounding cells leading to the replacement and the invasion of the healthy
Mechanical Aspects of Tumour Growth: Multiphase Modelling, Adhesion, and Evolving Natural Configurations
tissue with the formation of hyperplasia and therefore tumour lesions.
In the description to follow we denote by Ωt(t= 0) the region initially occupied
by tumour cells with volume ratio φtand by Ωn(t= 0) = Ω \Ωt(t= 0) that occupied
by normal host cells with volume ratio φn. The interface ∂Ωt(t) between tumour and
environment is a material surface moving with the common velocity of the cells
n·dxt
dt =n·vct=n·vcn,on ∂Ωt(t).(1.70)
It can be shown that the two cell populations, initially segregated, stay segregated at
all times.
From the mathematical point of view the phenomenological description of contact
inhibition of growth can be formalized saying that the threshold value for tumour cells
to overcome the restriction point and commit themselves to divide is slightly larger
than the physiological one. Actually, it may even tend to infinity, meaning that the
cells are completely insensitive to compression and continue replicating independently
of the compression level.
We shall then consider the following growth terms
Γci= [γiHσ(ψi−ˆ
ψi)−δi(ψi)]φci, i =n, t , (1.71)
where ψi=φci+φmand Hσ(ψi−ˆ
ψi) is a mollifier of the step function, which is at
least continuous, is constantly equal to 1 for ψsmaller than the threshold value ˆ
ψi,
and vanishes for ψ > ˆ
ψi+σ.
Of course, cellular mechano-trasduction is not the only cause of formation of hy-
perplasia and tumours. In fact, chemical factors operate to regulate the reproduction
rates so that the growth terms crucially depend on the presence of growth promoting
factors, growth inhibitory factors and, of course, nutrients.
However, the aim of this chapter is on the mechanical aspects of tumour growth and
therefore we assume that all the constituents required to sustain growth and mitosis
can be abundantly found in the extracellular liquid.
According to the discussion above the threshold values ˆ
ψnand ˆ
ψtare such that
ˆ
ψn<ˆ
ψt. For the following discussion it is useful to observe that a balance between
cell growth and death occurs when γiHσ(ψi−ˆ
ψi) = δi(ψi), or, in the case in which δi
is considered constant as in the following simulations
ψi=ˆ
ψi+H−1
σδi
γi.(1.72)
We assume that what makes the difference between a normal and a tumour cell
stays in the growth term and in its dependence from the stress level. Taking into
account that the interface conditions are enforced through continuity of stress and
velocity, and treating for sake of simplicity the ensemble of cells as elastic fluids, we
have the following free boundary problem
Tumour Cell Constituent as a Liquid
Fig. 1.8 Growth of two colonies which differ in their cell motility. The clone on the left is
more motile and therefore able to relax the stress much faster, triggering growth. Colours
represent the growth rate that is positive near the boundary. The center of the clones is
contact inhibited as well as the region between them when they come into contact. Plots are
after 6 and 8 days after seeding.
∂φci
∂t =∇ · φciKcim1−σcim
|∇ · (φciΣci)|+∇(φciΣci)!+ Γci,in Ωi(t),
vct·n=vcn·n,on ∂Ωt(t),
φctΣct(φct) = φcnΣcn(φcn),on ∂Ωt(t),
(1.73)
where i=t, n and we have assumed that the ECM is rigid (vm=0) and non-
remodelling (φmconstant in time). This makes it unnecessary to detail its stress
tensor because the internal stress is indeterminate due to the rigidity constraint.
In Figure 1.8, following (Graziano and Preziosi, 2007),(Galle et al., 2009), we ap-
plied the model above to the growth of Widr colonies (a colorectal adenocarcinoma
cell line). Two clones of tumor cells are virtually seeded at a dimensional distance of
320µm. They are characterized by different motilities, which is the only difference be-
tween the two clones. Specifically, the left one has a ten times larger motility coefficient
than the right one. We mention that comparable results would be achieved if cells of
Mechanical Aspects of Tumour Growth: Multiphase Modelling, Adhesion, and Evolving Natural Configurations
the colony on the left were assumed to be stiffer. At the very beginning there is almost
no difference in the growth of the two colonies. However, while all cells duplicate in the
left colony, in the right one the cells in the center duplicate less. This is due to the fact
that the less motile colony on the right does not release the inner stress so fast. The
radius of the colony on the right is a bit smaller than that on the left. Going on with
time cells in the center become too compressed to activate their duplication program.
The difference between the two clones becomes more and more evident, with cell pro-
liferation confined to the outer proliferating rim (see Fig. 1.8, left). This is similar to
what is observed in many papers on tumor growth as an effect of nutrient availability
inside the tumor. However, we remind that nutrient distribution is not important in
these experiments because the colonies are growing on a flat Petri dish with nutrients
coming from above. The proliferating rim is simply due to contact inhibition and for
this reason duplication is mainly restricted to cells close to the boundary.
After nearly 6 days the two colonies touch and growth is inhibited on the contact
line and in the center also of the clone on the left (see the figure on the right of
Fig. 1.8). However, cells on the outer border keep duplicating, and those on the left
are always more active, so at the end the colony on the right is almost engulfed by the
one on the left.
1.7.2 Integrin Switch and Fibrosis
As a second example we consider ECM remodelling and the formation of fibrosis. In
fact, a variety of stromal cells, mainly fibroblasts are involved in constantly renewing
the ECM through the production of new ECM components and of matrix metallopro-
teinases (MMP). This is a physiologically functional process because it allows to keep
the stroma young and reactive. In fact, as everybody knows, prolonged rest is detri-
mental for bones and muscles, while exercise and physical training have an opposite
effect.
In stationary conditions the remodelling of ECM is a slow process. For instance,
in the human lung the physiological turnover of ECM is 10-15% per day (Johnson,
2001), which leads to an estimated complete turnover in a period of nearly a week.
However, when a new tissue has to be formed, e.g. to repair a wound, then the rate of
production is one or two orders of magnitude faster (Caveda et al., 1996). In addition,
this process is strongly affected by stress as it is well known that for bones, teeth, and
muscles.
The percentage of ECM content changes considerably from tissue to tissue (see
Table 1.1) from normal to tumor tissues, and also within the same tumor with tumor
progression (see (Zhang et al., 2003)). For instance, (Takeuchi et al., 1976) found that
breast tumors presented a denser and more fibrous stroma with several differences in
the chemical composition. In fact, it is well known to everybody that the first hints
on the possible presence of a breast nodules are obtained by palping the breast and
feeling stiffer regions. A strong variability in the collagen content was also found in
prostate cancer (Zhang et al., 2003) where it can range from 7% to 26% according to
the grade of the tumor.
Increased presence of ECM characterizes also other pathologies such as cardiac
hyperthrophy, intima hyperplasia, cardiac fibrosis, liver fibrosis, pulmonary fibrosis,
Tumour Cell Constituent as a Liquid
Tissue Elastic modulus (Pa)
Normal mammary gland 167 ±31
Average breast tumor 4049 ±938
Stroma attached to tumor 916 ±269
Reconstituted basement membrane 175 ±37
Collagen (2.0 mg/ml) 328 ±87
Collagen (4.0 mg/ml) 1589 ±380
Table 1.1 Examples of elastic moduli of normal and abnormal breast tissue and stroma
(data from (Paszek et al., 2005)).
asthma, glomerulonephritis, colon cancer (Johnson, 2001).
The alteration in the ECM composition can be due to several probably concurring
reasons:
•increased synthesis of ECM proteins;
•decreased activity of matrix degrading enzymes (MDEs);
•upregulation of tissue-specific inhibitors of metalloproteinases (TIMPs).
On the other hand, excessive degradation of ECM due to excessive production of
MMP-13 characterizes chronic inflammatory diseases such as osteoarthritic cartilage,
rheumatoid synovium, chronic ulcer, intestinal ulcerations, periodontitis, and many
malignant tumors.
The interaction between cells and ECM is very important because cells need to
properly adhere in order to survive. They only duplicate if they are anchored to the
ECM. The mechano-transduction cascade is mainly activated by integrins.
On the other hand, in the process of invasion and formation of metastases tumor
cells detach from the original site, invade the sorrounding tissue, intravasate entering
the blood or lymphatic system, and extravasate to reach a secondary site. It is then
clear that the formation and diffusion of mestatases require that cells acquire the
ability of surviving without interacting with the ECM. In fact, like for cadherins,
it is found that tumors have altered integrins, which, in turn, alter the downstream
signalling pathway, so that one could argue that there is an integrin switch in addition
to the mentioned cadherin and angiogenic switches.
Actually, (Paszek et al., 2005) prove that through the integrin signalling pathway
the stiffness of the ECM promote malignant behavior consisting in growth enhance-
ment and loss of tissue polarity which for instance leads to the absence of lumen
formation in ductal carcinoma and the formation of hyperplasia, the first step toward
tumorigenesis.
Remodeling and degradation of ECM is due to the motion of the cells within the
scaffold and to the action of MMPs, whose concentration per unit volume is denoted
by e. One can then argue that
Γm=µt(φct, φm)Hǫ(ψm−ψ)φct+µn(φcn, φm)Hǫ(ψm−ψ)φcn−νeφm,(1.74)
where ψ=φct+φcn+φm. Here, µα,α=t, n, is a nonnegative, nonincreasing function
representing the net matrix production rate by the cell population cαtempered by the
Mechanical Aspects of Tumour Growth: Multiphase Modelling, Adhesion, and Evolving Natural Configurations
Tissue C1(Pa) C2
Fat 4460 ±2345.6 7.4±4
Normal mammary gland 15174.5±6750.7 12.3±7.4
Phyllodes tumour 50312.8 11.9
Papilloma 17765.2±4201.6 21.4±2.8
Lobular carcinoma 28269.6 20.9
Fibroadenoma 37572.4±6047.4 20.0±1.4
Infiltrating ductal carcinoma 37958.7±6146.7 19.9±5.5
Ductal carcinoma in situ 55776 24.4
Table 1.2 Average fit parameters for the exponential fit Tc=C1
C2(eC2ǫ−1) determined by
(Wellman et al., 1999) for several normal and cancer breast tissues. No standard deviation is
reported for those cases with only one specimen.
free space function Hǫ, and ν > 0 is the degradation rate by the MMPs. As usual, the
evolution of the concentration of MMPs is governed by a reaction-diffusion equation
∂e
∂t =D∇2e+πnφcn+πtφct−e
τ,(1.75)
for net production rates πα>0, α=t, n, and enzyme half-life τ > 0. Actually,
enzyme dynamics is much faster than that involving cell growth and death, hence it is
possible to work under a quasi-stationary approximation. Furthermore, enzyme action
is usually very local, so that also diffusion can be neglected and finally
e=τ(πnφcn+πtφct).(1.76)
Inserting this expression into Eq. (1.74) and defining να=ν τ πα,α=t, n, allows
to write the mass balance equation for the ECM (1.12)
∂φm
∂t =X
α=t,n
[µα(φcα, φm)Hǫ(ψm−ψ)−ναφm]φcα,(1.77)
The pathological cases possibly leading to fibrosis are either µt(·)> µn(·) or νt<
νn. In (1.77) it is important that the production coefficients of ECM by normal and
tumour cells be different in order to describe the formation of fibrosis characterizing
many tumours. In particular, µt(·)> µn(·). Alternatively, νt< νnimplies that tumor
cells produce less MMPs than healthy cells.
One of the by-product of this model is the description of the formation of fibrotic
tissues and of tissues stiffer than normal so that they may be sometimes felt with a
self-test. This is the aim of the simulation shown in Figure 1.9. The ECM is initially
distributed homogeneously with m= 0.2. On the other hand, while proliferating tu-
mour cells will produce matrix degrading enzyme as the host cells but will produce
more extracellular matrix than normal. This leads to the formation of a tumour char-
acterized by an amount of ECM with a volume ratio close to m= 0.3. From the
mechanical point of view this increase in the percentage of ECM would lead to an
increase of almost one order of magnitude in tissue stiffness.
The Tumour Mass as a Solid: Evolving Natural Configurations
(a) (b)
Fig. 1.9 Growth of a fibrotic tumour in a homogeneous tissue surrounding a bone. (a) Cell
volume ratio (b) ECM volume ratio. The line delimits the tumour from the host tissue.
1.8 The Tumour Mass as a Solid: Evolving Natural Configurations
Of course, tumours are not liquids. However, treating cells like a liquid brings many
useful simplifications, starting from the fact that it is possible to use an Eulerian
approach and to deal with velocities rather than deformation with respect to a not so
well specified natural configuration. Though the models are relatively easy, the fact
that the cellular liquid is contained in a solid structure made by the ECM, implies
that the tissue as a whole behaves like a viscoelastic solid, or even a rigid solid if the
newtork of ECM fibers is assumed to be rigid. However, this is still far from reality
because of the presence of adhesion bonds between cells.
A big theoretical difficulty is instead encountered in describing tumors as solid
masses because the cells forming them duplicate and die, the stroma and in particu-
lar the extracellular matrix continuously remodel and even in absence of growth and
death the ensemble of cells undergoes an internal re-organization in response to defor-
mation. There is then a difficulty in defining a reference configuration and in using a
Lagrangean coordinate system. In particular, also the meaning of deformation looses
the immediate meaning it had in classical continuum mechanics when dealing with
inert matter. In fact, when dealing with a living tissue like a tumour, it is not clear
with respect to what we should measure deformations, because the material is always
changing. That is why the concept of evolving natural configuration introduced by Ra-
jagopal and coworkers can be of help. Actually, the basic idea of this formalism, also
present in plasticity theory, started being applied in a biomechanical context to de-
scribe growth in (Klisch and Hoger, 2003), (Rodriguez et al., 1994). In the recent past,
Humphrey and Ra jagopal applied this concept to describe the growth and remodelling
of several tissues (Humphrey and Rajagopal, 2002), (Humphrey and Rajagopal, 2003),
(Malik et al., 2008), (Rao et al., 2003). Ambrosi and Mollica (Ambrosi and Mollica,
2002),(Ambrosi and Mollica, 2004) used a purely elastic one–component model to eval-
uate residual stress formation in a growing multicellular spheroid. This approach was
developed in (Ambrosi and Preziosi, 2009) working in a multiphase framework and
taking also internal re-organisation and ECM deformation into account. This gave rise
to an elasto-viscoplastic description for the cell population and a compressive elastic
description for the ECM.
Mechanical Aspects of Tumour Growth: Multiphase Modelling, Adhesion, and Evolving Natural Configurations
Fig. 1.10 Multiple natural configuration.
In this section we follow (Ambrosi and Preziosi, 2009) distinguishing in the defor-
mation gradient of the tumour component Fcthe contributions due to pure growth,
plastic deformation and elastic deformation by a multiplicative decomposition, as
shown in Fig. 1.10.
This splitting is suggested also by the biological observation that growth occurs on
a much longer time scale (hours up to days) than deformation. Therefore, it is possible
to separate the contibution due to growth from the one due to deformation (without
growth) and to model each of them individually not only from the theoretical point of
view, but also from the experimental point of view.
The deformation gradient Fcis a mapping from a tangent space onto another
tangent space, and it indicates how the body is deforming locally going from the
initial (reference) configuration K0to the current configuration Kc. An imaginary
intermediate configuration can be introduced assuming that a point of the body can
relieve its state of stress while relaxing the continuity, i.e. the integrity of the body. It
then relaxes to a stress-free configuration. The atlas of these pointwise configurations
forms what we define natural configuration with respect to Kcand denote by Kn.
Referring to Fig. 1.10, we identify this deformation without growth with the tensor
Fn, which then describes how the body is deforming locally while going from the
natural configuration Knto Kc.
The particle in the configuration Knhas possibly undergone growth and plastic
The Tumour Mass as a Solid: Evolving Natural Configurations
deformation. One can then again consider the map from K0to Knas composed of
two parts: the first one related to growth/death processes (therefore to mass variations
in the volume element), the second one due to internal reorganisation, which implies
re-arranging of the adhesion links among the cells, without change of mass in the
volume element. Denoting by Kpthe “grown configuration”, i.e., the intermediate
configuration of the body between K0and Kn, we will assume that for any given
point the volume ratio in Kpis the same as in the natural configuration Knand in
the original reference configuration K0, i.e., φp=φc(t= 0) = φn.
According to the three-steps process outlined above, the deformation gradient is
split as
Fc=FnFpGc.(1.78)
For sake of simplicity we also take growth to be isotropic Gc=gI, so that Jg=
det Gc=g3.
In (Ambrosi and Preziosi, 2009) it was shown that
˙
Jg
Jg
= 3 ˙g
g=Γc
φc
,(1.79)
where the dots indicate the time derivative following the cell population. This relation
links the evolution of gto the death/growth term Γcappearing on the right hand side
of the mass balance equation for the cellular constituent, i.e. Eq.(1.4).
Introducing
Dp= sym(˙
FpF−1
p),(1.80)
from standard tensor calculus one has that
˙
Jp=Jptr Dp,(1.81)
but since it is assumed that the volume ratio does not change during plastic re-
organisation, Jp= det Fp= 1, and therefore
tr Dp= 0 .(1.82)
Deriving
Bn=FnFT
n,(1.83)
with respect to time, one has
˙
Bn=LnBn+BnLT
n,(1.84)
while, deriving Fin time, one has
˙
F= ˙gg−1F+g˙
FnFp+gFn˙
Fp= ( ˙gg−1I+Ln+FnLpF−1
n)F.(1.85)
that, through the definition of L, can be re-written as
Ln=L−˙gg−1I−FnLpF−1
n.(1.86)
Substituing it back in (1.84) gives
Mechanical Aspects of Tumour Growth: Multiphase Modelling, Adhesion, and Evolving Natural Configurations
˙
Bn=LBn+BnLT−2 ˙gg−1Bn−2FnDpFT
n,(1.87)
or, if the definition of upper convected Maxwell derivative,
DM
Dt =˙
M−LM −MLT,(1.88)
is introduced DBn
Dt =−2 ˙gg−1Bn−2FnDpFT
n.(1.89)
We want now to include elasto-visco-plastic effects in the mechanics of cell aggre-
gates. We do that on the basis of the following observations
1. when and where the cell populations is sub ject to a moderate amount of stress,
then the body behaves elastically;
2. when and where the stress overcomes a threshold yield stress then the body un-
dergoes visco-plastic deformations.
Given the resistance of a single bond, the threshold of the onset of plastic defor-
mations is proportional to the area of the cell membranes in contact, that depends on
the number of cells per unit volume. We call yield stress this threshold value τ(φ) and
compare it with a frame invariant measure fof the stress of the cellular constituent
φcTc. It has to be noticed that a very small volume ratio can correspond to much
dispersed single cells or very clusterized ensembles. In the former case, the yield stress
is expected to be very low. In the second case, borrowing ideas from the dynamics
of colloidal particles and flocculated suspensions, the yield stress should increase with
the second or the third power of the volume ratio (Buscall et al., 1988), (Snabre and
Mills, 1996). Iordan et al. measured τ(φc) to be proportional to φ8.4
c.
On this basis, the following elastic-type constitutive equation can be suggested in
the elastic regime
Tc=ˆ
Tc(Bn),if f(φTc)≤τ(φ).(1.90)
As a frame invariant measure of the stress, (Basov and Shelukhin, 1999) suggest to
use
t(n) = φ[Tcn−(n·Tcn)n],(1.91)
that represents the tangential stress vector relative to the surface identified by the
normal n. In particular, we will use
f(φcTc) = max
|n|=1 |t(n)|,(1.92)
that represents the maximum shear stress magnitude occurring in the plane identified
by the eigenvector corresponding to the maximum of |t(n)|. It can be proved that f
is given by half of the difference between the maximum and the minimum eigenvalue
of φcTc.
Following (Ambrosi and Preziosi, 2009), above the yield stress the tension in excess
originates from cell unbinding at the microscopic scale and then cell rearrangement
The Tumour Mass as a Solid: Evolving Natural Configurations
at the macroscopic scale. Such a pictorial description is put into formal terms by the
following constitutive equation
1−τ(φc)
f(φT′
c)α(φcT′
c) = 2η(φc)FnDpF−1
n,if f(φcT′
c)> τ(φc),(1.93)
where T′
c=Tc−1
3(trTc)Ioperates on the current configuration and for compatibility
we mapped Dpinto the same configuration using Fn. We can merge the above equation
to the condition that there is no evolution for shear stresses smaller than the yield
stress by writing
FnDpF−1
n=1
2˜η1−1
˜
f(T′
c)+
T′
c,(1.94)
where [·]+stands for the positive part of the argument, ˜η=η(φc)/φc, and ˜
f(T′
c) =
[f(φcT′
c)/τ(φc)]α.
As it should be,
tr(FnDpF−1
n) = tr Dp= 0 ,(1.95)
hence the l.h.s. of (1.94) is traceless as the r.h.s.
It is worth to state, according to (DiCarlo and Quiligotti, 2002), how the quantities
introduced above transform under a change of frame. Denoting with a star (*) the value
of a field after an euclidean change of frame, one has
F∗
n=QFn,
G∗
c=Gc,
F∗
p=Fp,
where Qis an orthogonal tensor.
Thanks to the previous relations
FnDpF−1
n∗=QFnDpF−1
nQT,
being Tcobjective, one has the objectivity of (1.94).
If we take the following elastic-type constitutive equation
T′
c=µBn−1
3trBnI,(1.96)
then, Fpevolves according to
˙
Fp=g−1
λ"1−1
µ˜
fBn−1
3(trBn)I#+
F−1
nBn−1
3(trBn)IF,(1.97)
where λ=˜η
µis called the cell re-organisation time.
Eq. (1.97) can be explained phenomenologically in the following way: Assuming for
a moment that there is no growth, if the body undergoes a deformation corresponding
to a stress below the yield stress, then Fpdoes not change, i.e., the intermediate
Mechanical Aspects of Tumour Growth: Multiphase Modelling, Adhesion, and Evolving Natural Configurations
configuration does not evolve and all the energy is elastically stored. If the measure of
tension ftakes a value larger than the yield stress, then the reference configuration
changes to release the stress in excess, until the yield surface defined by fis reached
again. The ratio λ= ˜η/µ gives an indication of the characteristic time needed to relax
the internal stress through the re-organisation of cells and to reach the yield surface.
Deriving (1.96) one can write
˙
T′
c=µ[LBn+BnLT−2 ˙gg−1Bn−2FnDpFT
n
−1
3tr(LBn+BnLT−2 ˙gg−1Bn−2FnDpFT
n)I].
Using (1.94), one then has
˙
T′
c+µ
˜η1−1
˜
f(T′
c)+T′
cBn−1
3tr(T′
cBn)I=
µ[LBn+BnLT−2 ˙gg−1Bn−1
3tr(LBn+BnLT−2 ˙gg−1Bn)I],
which can be written in terms of upper convective derivative as
DT′
c
Dt +1
λ1−1
˜
f(T′
c)+T′
cBn−1
3tr(T′
cBn)I=
2
3µ[tr(Bn)D−tr(BnD)I−˙gg−1(3Bn−(trBn)I)] .
(1.98)
Note that the Maxwell derivative of an objective tensor is itself ob jective.
We now assume that the deformations from the natural configuration are small
throughout the evolution of the system. The small deformation assumption applies
depending on the value of the yield stress: it has to be small enough so that the
condition |Bn·I−3| ≪ 1 is always satisfied during the motion. The experiments by
(Iordan et al., 2008) give an indication of the order of magnitude of the yield stress,
that depends on the volume ratio and is below 1 Pa (for φc= 0.6, the maximum
volume ratio tested). The advantage of this hypothesis is that for small elastic strain
one can use linear elasticity. For larger stresses, the natural configuration evolves.
In the limit of small deformations,
LBn+BnLT≈2D,and T′
cBn≈T′
c,(1.99)
and therefore (1.98) simplifies to
˙
T′
c+1
λ1−1
˜
f(T′
c)+
T′
c= 2µD−1
3trD I,(1.100)
where the trace of Dis related to the growth term through the mass balance equation.
We observe that in (1.100), the term containing the yield stress plays the role
of a stress relaxation term that switches on as soon as the stress is above the yield
value. Otherwise, for ˜
f(T′
c)<1, (1.100) can be integrated to give back an elastic-like
constitutive equation.
Response to Shear Tests
Fig. 1.11 Yield stress measurement as a function of the cell volume ratio as measured by
(Iordan et al., 2008).
The limit λmuch larger then the characteristic time of the process of interest, in
principle would lead to the models used in (Araujo and McElwain, 2005b),(Araujo and
McElwain, 2005c),(Araujo and McElwain, 2004). However, in this case the procedure
is incompatible with the small deformation assumption because the stress relaxes very
slowly and so large stresses and deformation can build up.
On the other hand, rewriting (1.100) as
λ˙
T′
c+1−1
˜
f(T′
c)+
T′
c= 2ηD−1
3trD I,(1.101)
it is easy to realise that for processes with characteristic times larger than λand
stresses much larger than τ(i.e., ˜
f≫1) the model behaves like the viscous models
used in (Franks et al., 2003a), (Franks et al., 2003b),.
Following the same argument proposed in (Preziosi and Joseph, 1987), one can
state that in transient phenomena for times much larger than cell reorganisation time
the natural configuration has evolved relaxing the stress, leaving the material in a
state of stress living at most on the yield surface.
1.9 Response to Shear Tests
In these last two sections we will propose some responses of the materials satisfying
(11) to several tests. More comparisons can be found in (Preziosi et al., 2010).
For pure shear tests (1.100) rewrites
˙
Tc+1
λ1−ˆτ
|Tc|α+
Tc=µ˙γ , (1.102)
Mechanical Aspects of Tumour Growth: Multiphase Modelling, Adhesion, and Evolving Natural Configurations
Fig. 1.12 Unrecovered deformation due to stress history above the yield stress.
where ˙γis the shear rate, f(Tc) = |Txy |=|Tc|and ˆτ=τ(φ)/φ. The case α= 1
corresponds to a Bingham fluid while the limit case α= 0 corresponds to an elastic
fluid. In the following, sometimes the particular case α= 1 will be considered in more
detail, because it allows to write analytical solutions in an easy way and to easily
understand the main feature of the constitutive model.
When a cell aggregate undergoes responding to the constitutive model above is
sheared with a smoothly increasing loading, initially the body deforms elastically and
the stress grows until the yield value ˆτis reached. In this initial regime the cell bonds
are simply stretched, no internal re-organization occurs, the natural configuration does
not evolve, and the body would be able to return to the initial stress-free configuration,
if allowed to do so, for instance releasing the stress (see upper plots in Fig.1.12). When
locally the stress overcomes the yield value, some adhesion bonds break, new ones
form and the natural configuration evolves. The body is then not able to return to the
original configuration any longer, as shown in the lower plots in Fig.1.12.
1.9.1 Steady shear
As an example, consider the case of shear increasing linearly in time, i.e., γ= ˙γ0tand
α= 1. Integration in time of (1.102) gives
Tc(t) =
µ˙γ0t , for t≤tτ=ˆτ
µ˙γ0;
ˆτ+µ˙γ0λ1−e(−t+tτ)/λ ,for t > tτ.
(1.103)
If t→ ∞ then Tc(t)→ˆτ+ ˜η˙γ0. This means that plotting the apparent viscosity
η( ˙γ0) = Tc/˙γ0as a function of the shear rate ˙γ0, the plot should depend on ˙γ−1
0for
Response to Shear Tests
Fig. 1.13 Comparison of the model vs. the experiments by (Iordan et al., 2008). Viscosity
versus shear rate at different volume ratios.
small ˙γ0. This is the case of experiments shown in (Iordan et al., 2008) and reported
in Fig.1.13.
In general, for 0 ≤α≤1, the relation between ˙γ0and the apparent viscosity η( ˙γ0)
is given by
˙γ0=ˆτ
η( ˙γ0)1−˜η
η( ˙γ0)−1/α
.(1.104)
If η→ ∞ then ˙γ0goes to zero as ˆτ /η( ˙γ0) which implies that for small ˙γ0, the
apparent viscosity η( ˙γ0) behaves like ˙γ−1
0. On the other hand, if η→˜ηthen ˙γ0
goes to infinity, that means that at high shear rates, the apparent viscosity η( ˙γ0) =
T / ˙γ0goes towards its limiting value ˜η. This limit is concentration dependent and has
been obtained experimentally (Chien et al., 1967), (Iordan et al., 2008). A plot of the
viscosity dependence against shear rate is presented in Fig.1.14. One can note the
limiting behaviors at small shear rates (slope −1 on this log–log figure). At high ˙γ0,
the viscosity reaches its limit ˜η.
In order to validate the model with a true biological aggregate, in (Preziosi et al.,
2010) the result of the model are compared with the experiment performed by (Iordan
et al., 2008) and reported here in Fig.1.13. The only adjustable parameters are found
to be ˆτ / ˜ηand α, by simple scaling arguments. As ˜ηis fixed to be the culture medium
viscosity ˜η= 0.0013 P a ·s, only ˆτand αare allowed to vary. For instance, for the 42%
concentration ˆτ= 0.05P a and α= 0.01. It is found that the values of the yield stress
are close to those obtained by (Iordan et al., 2008) and show a typical dependence of
the type ˆτ∼962.2φ11, at large concentrations 0.4≤φ≤0.6.
Mechanical Aspects of Tumour Growth: Multiphase Modelling, Adhesion, and Evolving Natural Configurations
Fig. 1.14 Apparent viscosity η( ˙γ0) = T / ˙γ0as a function of ˙γ0for different values of the
parameter α.
1.9.2 Stress relaxation tests
A standard relaxation test is obtained applying a sudden constant deformation γ0. If
γ0≤ˆτ /µ, then the stress in the body is Tc=µγ0(<ˆτ) at any time corresponding to
the straight line in Fig.1.15a, an elastic response. If, on the other hand, γ0>ˆτ /µ the
solution of Eq.(1.102) is
T(t) = ˆτn1 + hµγ0
ˆτα−1ie−αt/λ o1/α
.(1.105)
In particular, for α= 1
T(t) = ˆτ+ (µγ0−ˆτ)e−t/λ ,(1.106)
that is plotted in Fig.1.15 for increasing values of γ0.
Hence for small strains the body behaves elastically, while for large strains part
of the stress relaxes to the yield value ˆτ, regardless of the magnitude of the applied
strain. The decrease toward the asymptotic state is faster for larger exponents.
(Forgacs et al., 1998) obtained similar plots when compressing a multicellular
spheroid in a uniaxial compression test that will be analysed more deeply in the next
section. They observe that when a fixed deformation is applied to the cell aggregate,
the internal stress relaxes until an asymptotic value. They interpret this long time
behavior as the effect of surface tension. This idea is supported by the observation
that doubling the deformation would give rise to the measurement of a similar surface
Uniaxial Compression Tests
0
0.5
1
1.5
2
2.5
3
0 1 2 3 4 5 6
T/τ
t/λ
3
2.5
2
1.5
1
0.7
0.4
Fig. 1.15 Stress relaxation response T /ˆτfor different values of µγ0/ˆτand α= 1. For
γ0≤ˆτ/µ the response is elastic. Time is normalized with respect to λ.
tension coefficient, while a constitutive model for a linear elastic solid would give rise
to a doubling of the stress. They also observe that the coefficient linearly depends
on the density of cadherins on the cell surface, that is closely related to the adhe-
sion properties of the cell. This information and the above results suggest that the
same result can be interpreted using the constitutive model presented here and relat-
ing the asymptotic behavior to a measure of the yield stress. In fact, similarly to the
experiments in (Forgacs et al., 1998), in the present virtual experiment doubling the
deformation will give rise to the same asymptotic value measuring the yield stress. In
addition the present model would also be in agreement with the results by (Iordan
et al., 2008), that cannot be explained by the concept of surface tension. Hence, using
the values in (Forgacs et al., 1998) the yield stress for the different tissues tested there
ranges between 1 to 100 Pa. The results obtained by (Iordan et al., 2008) for a cell
suspension with a volume ratio of 0.6 (the maximum one they used) give a yield stress
around 1P a, and higher concentrations will probably lead to data in the same range,
especially since the yield stress is concentrated dependent like ˆτ∼φ11 , as observed
previously.
1.10 Uniaxial Compression Tests
1.10.1 Uniaxial compression
To compare the results that can be obtained by the model with the experimental
results described in (Forgacs et al., 1998),(Foty et al., 1996),(Winters et al., 2005) we
assume that the deformation generated by a compressive stress Pappl along the z-axis
is homogeneous taking the following form
Mechanical Aspects of Tumour Growth: Multiphase Modelling, Adhesion, and Evolving Natural Configurations
x=X
pψ(t), y =Y
pψ(t), z =ψ(t)Z , (1.107)
with Fpgiven by
Fp= diag (1
pΨ(t),1
pΨ(t),Ψ(t)).(1.108)
How Ψ(t) differs from 1 then is a measure of how much the aggregate has re-organized
and the natural configuration has evolved. Hence Fcand Fnare respectively given by
Fc= diag (1
pψ(t),1
pψ(t), ψ(t)),(1.109)
and
Fn= diag (sΨ(t)
ψ(t),sΨ(t)
ψ(t),ψ(t)
Ψ(t)).(1.110)
Therefore, dropping tfor sake of simplicity,
Bn= diag Ψ
ψ,Ψ
ψ,ψ2
Ψ2,(1.111)
and
Tc=µdiag −Σ + Ψ
ψ,−Σ + Ψ
ψ,−Σ + ψ2
Ψ2= diag{0,0, Pappl},(1.112)
where Pappl is the time–dependent applied stress in the z-direction.
In conclusion,
Σ = Ψ
ψ,and Pappl =µψ3−Ψ3
ψΨ2.(1.113)
On the other hand,
Dp=˙
FpF−1
p= diag −1
2,−1
2,1˙
Ψ
Ψ,(1.114)
and therefore from (1.94)
˙
Ψ
Ψ=1
3˜η1−1
˜
f(T′)+
Pappl =1
3˜η1−2ˆτ
|Pappl|α+
Pappl .(1.115)
Substituting (1.113) in the equation above, one has the evolution equation for Ψ, i.e.,
for the natural configuration
Uniaxial Compression Tests
˙
Ψ
Ψ=µ
3˜η1−2ˆτ
µ
ψΨ2
|ψ3−Ψ3|α+
ψ3−Ψ3
ψΨ2.(1.116)
In order to mimick Forgacs’ experiment assume that a given strain ψ0<1 is
imposed on the cell aggregate. Then, if the compression is sufficiently strong so that
the square parenthesis above is positive, i.e.,
1
ψ0−ψ2
0>2ˆτ
µ,(1.117)
Ψ will decrease from Ψ = 1 according to
˙
Ψ = −µ
3˜η1−2ˆτ
µ
ψ0Ψ2
Ψ3−ψ3
0αΨ3−ψ3
0
ψ0Ψ,(1.118)
that is integrated in Fig. 1.16.
If the compression applies for a long enough time, then Ψ will tend towards that
value Ψ∞> ψ0corresponding to the vanishing of the square parenthesis in (1.118),
i.e.,
Ψ3
∞−ψ3
0
ψ0Ψ2
∞
=2ˆτ
µ,(1.119)
that is independent of α. In particular, Pappl will tend to
Pappl,∞=µψ3
0−Ψ3
∞
ψ0Ψ2
∞
=−2ˆτ , (1.120)
(see bottom Fig. 1.16) that in addition of being independent of αis also independent
of ψ0as in the experiments in (Forgacs et al., 1998).
In Fig. 1.16 different compressions are applied. If ψ0>0.8351 the compression
is not strong enough to trigger cell re-organization and therefore Ψ(t) = 1 does not
evolve and |Pappl|is constantly below the yielding value that in this case is 2ˆτ= 0.5.
For larger ψ0the natural configuration evolves and Pappl again tends to the constant
value given in (1.120).
The behaviour is then similar to the one described for stress relaxation tests in
Section 9.2, though (1.118) cannot be easily solved. Finally, if at an instant t2(when
Ψ=Ψ2>Ψ∞), the compression is suddenly released, then ψwill readily adjust to
the value ψ= Ψ2.
As already discussed in the previous section this result can be compared with
the stress relaxation tests performed by (Forgacs et al., 1998). They observe that the
measured stress decreases asymptotically to a value that depends on the tissue but
is independent of the deformation imposed. They interpret this long time behavior
as the effect of surface tension, but as just shown the same result can be interpreted
using the constitutive model above and relating the asymptotic behavior to a measure
of the yield stress.
Furthermore, (Foty et al., 1996) correlate the surface tension coefficient obtained
through the asymptotic value with the density of surface cadherin per cell. The same
Mechanical Aspects of Tumour Growth: Multiphase Modelling, Adhesion, and Evolving Natural Configurations
Fig. 1.16 Uniaxial compression for ˆτ /µ = 0.25. Time is normalized with respect to ˜η/µ and
the applied stress is reported normalized with respect to µ. If ψ0>0.8351 the compression
is not strong enough to trigger cell re-organization and therefore Ψ(t) = 1 does not evolve
and Pappl/µ is constantly equal to ψ2
0−1/ψ0>−0.5. The curves represent the behaviour for
ψ0<0.8351 when a stronger deformation triggers cell re-organisation and stress relaxation.
The asymptotic value of the volume ratio is in all cases 0.8351, that of Ψ is given by (1.119),
and that of Pappl/µ is always -0.5 corresponding to the yield value. αis fixed to 1.
Uniaxial Compression Tests
proportionality obviously holds for the yield stress Ibecause this coefficient is clearly
related to the number of adhesion bonds. In our opinion the relation between yield
stress and surface tension also explains the observation by Winters et al (Winters et al.,
2005) that cell invasiveness is related to the inverse of surface tension: the smaller the
adhesion between cells, the smaller the yield stress, the more invasive the cell clone.
We also mention that Forgacs et al. (Forgacs et al., 1998) point out the existence
of two relaxation times in the cellular matter (one of the order of few seconds the
other of the order of tens of seconds), that in the present approach would be reached
considering two re-organization mechanisms or the detachment of different types of
adhesion proteins. From the modelling point of view this means generalizing the model
above allowing two relaxation times.
1.10.2 Elastic recovery
A second part of the experiment by (Foty et al., 1996) consists in releasing the imposed
stress after some time from the beginning of the experiment. As already stated, if the
imposed deformation is small as compared to the yield condition, then the body will
go back to the original configuration. Otherwise, it induces an internal re-organization
of the cells and the body will not recover its initial configuration, because in the
meantime the natural configuration has changed. In fact, denoting by T2the value of
stress before the sudden stress release (that is much faster than the cell re-organization
time λ) the yield stress is reached for
γ=γ2−T2−ˆτ
µ,(1.121)
independently of α. After that, the body will relax the stress, reaching the stress-free
configuration when
γfin =γ2−T2
µ.(1.122)
To be more specific, if in the stress relaxation experiment described by Eq.(1.105),
at time tcompr the compressing plate is removed so that the specimen is stress-free
γfin =γ0−ˆτ
µn1 + hµγ0
ˆτα−1ie−αtcompr /λ o1/α
.(1.123)
In particular, if tcompr ≪λ, the exponential and the power in (1.123) can be
approximated to give
γfin ≈γ01−ˆτ
µγ0αtcompr
λ.(1.124)
This means that if the compression is kept for such a small time that the body does
not have enough time to re-organize, then the body will recover almost everything and
Mechanical Aspects of Tumour Growth: Multiphase Modelling, Adhesion, and Evolving Natural Configurations
return close to the original configuration mainly showing an elastic like behavior. If,
on the other hand, tcompr ≫λ, then
γfin =γ0−ˆτ
µ,(1.125)
meaning that the body will still recover the amount ˆτ /µ corresponding to the elastic
component. In other words, it will not keep the value γ=γ0imposed, even if this is
done for a very long time.
The above description is consistent with the observations in (Foty et al., 1996)
(their figures 3 and 5) where they see that if the spheroid is compressed for few
seconds then it will bounce back almost to the original configuration (there is only a
small flattening at the poles). If it is compressed for a longer time (few hours in their
experiments) this does not occur though a minor shape recovery is observed. This
can not be explained using the concept of surface tension, but is compatible with the
model presented here.
On the other hand, we have to point out that in this model the process is instan-
taneous, while it seems that in the experiment in (Foty et al., 1996) it takes some
times for the spheroid to return to the natural configuration as in standard indenta-
tion tests. This might be due to the fact that here we completely neglected the fact
that the spheroid is a porous material filled with the liquid in which the experiment
is done. Such an effect can be included for instance taking into account that Tcis
only one of the components of the stress and a further viscous component needs to
be added to obtain the stress for the mixture as a whole, similarly to what done in
(Saramito, 2007), (Saramito, 2008). This would give rise to a constitutive model that
can more properly describe creep tests.
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