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Multiscale Modelling of Fibres Dynamics and Cell Adhesion within Moving Boundary Cancer Invasion

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Recognised as one of the hallmarks of cancer, local cancer cell invasion is a complex multiscale process that combines the secretion of matrix-degrading enzymes with a series of altered key cell processes (such as abnormal cell proliferation and changes in cell–cell and cell–matrix adhesion leading to enhanced migration) to degrade important components of the surrounding extracellular matrix (ECM) and this way spread further in the human tissue. In order to gain a deeper understanding of the invasion process, we pay special attention to the interacting dynamics between the cancer cell population and various constituents of the surrounding tumour microenvironment. To that end, we consider the key role that ECM plays within the human body tissue, and in particular we focus on the special contribution of its fibrous proteins components, such as collagen and fibronectin, which play an important part in cell proliferation and migration. In this work, we consider the two-scale dynamic cross-talk between cancer cells and a two-component ECM (consisting of both a fibre and a non-fibre phase). To that end, we incorporate the interlinked two-scale dynamics of cell–ECM interactions within the tumour support that contributes simultaneously both to cell adhesion and to the dynamic rearrangement and restructuring of the ECM fibres. Furthermore, this is embedded within a multiscale moving boundary approach for the invading cancer cell population, in the presence of cell adhesion at the tissue scale and cell-scale fibre redistribution activity and leading edge matrix-degrading enzyme molecular proteolytic processes. The overall modelling framework will be accompanied by computational results that will explore the impact on cancer invasion patterns of different levels of cell adhesion in conjunction with the continuous ECM fibres rearrangement.
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Bulletin of Mathematical Biology
https://doi.org/10.1007/s11538-019-00598-w
Multiscale Modelling of Fibres Dynamics and Cell Adhesion
within Moving Boundary Cancer Invasion
Robyn Shuttleworth1·Dumitru Trucu1
Received: 27 September 2018 / Accepted: 11 March 2019
© The Author(s) 2019
Abstract
Recognised as one of the hallmarks of cancer, local cancer cell invasion is a complex
multiscale process that combines the secretion of matrix-degrading enzymes with a
series of altered key cell processes (such as abnormal cell proliferation and changes in
cell–cell and cell–matrix adhesion leading to enhanced migration) to degrade impor-
tant components of the surrounding extracellular matrix (ECM) and this way spread
further in the human tissue. In order to gain a deeper understanding of the invasion
process, we pay special attention to the interacting dynamics between the cancer cell
population and various constituents of the surrounding tumour microenvironment. To
that end, we consider the key role that ECM plays within the human body tissue, and
in particular we focus on the special contribution of its fibrous proteins components,
such as collagen and fibronectin, which play an important part in cell proliferation and
migration. In this work, we consider the two-scale dynamic cross-talk between cancer
cells and a two-component ECM (consisting of both a fibre and a non-fibre phase). To
that end, we incorporate the interlinked two-scale dynamics of cell–ECM interactions
within the tumour support that contributes simultaneously both to cell adhesion and to
the dynamic rearrangement and restructuring of the ECM fibres. Furthermore, this is
embedded within a multiscale moving boundary approach for the invading cancer cell
population, in the presence of cell adhesion at the tissue scale and cell-scale fibre redis-
tribution activity and leading edge matrix-degrading enzyme molecular proteolytic
processes. The overall modelling framework will be accompanied by computational
results that will explore the impact on cancer invasion patterns of different levels of
cell adhesion in conjunction with the continuous ECM fibres rearrangement.
Keywords Cancer invasion ·Cell adhesion ·Multiscale modelling ·Computational
modelling
The two authors contributed equally to the paper, the order being purely alphabetical
BDumitru Trucu
trucu@maths.dundee.ac.uk
Extended author information available on the last page of the article
123
R. Shuttleworth, D. Trucu
Mathematics Subject Classification 22E46 ·53C35 ·57S20
1 Introduction
Cancer invasion of the human body is a complex, multiscale phenomenon that incor-
porates both molecular and cellular interactions as well as interconnections within
tissues. Recognised as one of the hallmarks of cancer (Hanahan and Weinberg 2000),
cancer invasion is a process that takes advantage of important changes in the behaviour
of many molecular activities typical for healthy cell, such as the abnormal secretion of
proteolytic enzymes that lead to the degradation of its surrounding environment that
ultimately translate in further tumour progression. Changes in cell adhesion properties
also contribute to the success of tumour invasion.
Led by the proteolytic processes induced by the cancer cells from its outer proliferat-
ing rim, the tumour locally invades neighbouring sites via an up-regulated cell–matrix
adhesion (Berrier and Yamada 2007) concomitant with a loss in cell–cell adhesion.
This local invasion marks the first of a cascade of stages, that ultimately result in the
process of cells escaping the primary tumour and creating metastases at distant sites
in the body. Without treatment, metastasised tumours can lead to organ failure and
eventual death in around 90% of patients (Chaffer and Weinberg 2011).
Acknowledge for the essential role that ECM plays in many vital processes, such
as in embryogenesis (Rozario and DeSimone 2010) and wound healing (Xue and
Jackson 2015), this plays a crucial part also in cancer invasion. The success of tumour
invasion is greatly influenced by the extracellular matrix (ECM), a key biological
structure formed from an interlocking network of proteins including collagen and
elastin, which provide necessary structure and elasticity, as well as proteoglycans that
aid the secretion of growth factors. However, during cancer invasion, the over-secretion
of proteolytic enzymes, such as the urokinase-type plasminogen activator (uPA) and
matrix metallo-proteinases (MMPs) (Parsons et al. 1997) by the cancer cells, followed
by interactions of these enzymes with the ECM components results in the degradation
and remodelling of the ECM (Lu et al. 2011; Pickup et al. 2014), largely contributing
to further tumour progression.
The invasive capabilities of a tumour gather their strength from a cascade of
processes enabled by the cancer cells, which, besides abnormal secretion of matrix-
degrading enzymes, includes also enhanced proliferation and altered cellular adhesion
abilities. Both cell–cell and cell–matrix adhesion play important roles in tumour pro-
gression, and changes to either of these contribute directly to the overall pattern of
invasion. Certain proteins found in the ECM, for example collagen and fibronectin, aid
in the binding of cells to the matrix through the cell–matrix adhesion, process which
is regulated by a family of specific molecules on the cell surface known as calcium
independent cell adhesion molecules (CAMs), or integrins (Humphries et al. 2006).
While collagen is a main component of the ECM, being one of the most common
protein found in the human body, fibronectin plays a crucial role during cell adhesion
having the ability to anchor cells to collagen and other components of the ECM. Thus,
while collagen provides structure and rigidity to the ECM, fibronectin contributes to
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Multiscale Modelling of Fibres Dynamics and Cell Adhesion…
cell migration, growth and proliferation, both ensuring the normal functionality of
healthy cells and being of crucial importance in cancer progression.
On the other hand, calcium-dependent CAMs on the cell surface naturally mediate
cell–cell adhesion. Adhesion is dependent on the cell signalling pathways that are
formed due to interactions between Ca2+ions and the distribution of calcium sens-
ing receptors in the ECM. Specifically, the molecular subfamily of E-cadherins is
responsible for binding with the intra-cellular proteins known as catenins, typically
β-catenin, forming the E-cadherin/catenin complex. The recruitment of cadherins and
β-catenin to the cell cytoskeleton is effectuated by intracellular calcium signalling (Ko
et al. 2001). Evidence suggests that activation of calcium sensing receptors results in
an increase in E-cadherins which in turn increases the binding of β-catenin (Hills et al.
2012). However, any alteration to the function of β-catenin will result in the loss of
the ability of E-cadherin to initiate cell–cell adhesion (Wijnhoven et al. 2000).
As tumour malignancy increases, normal fibroblasts are subverted to promote
tumour growth, known as cancer-associated fibroblasts (CAFs) (Kalluri 2016; Shiga
et al. 2015). CAFs proliferate at a much higher rate than normal fibroblasts in healthy
tissue (Erdogan et al. 2017). Biological evidence shows that CAFs induce tumour
growth, metastasis, angiogenesis and resistance to chemotherapeutic treatments (Tao
et al. 2017). Unlike normal fibroblasts, CAFs are specific to tumour cells and their
microenvironment and possess the ability to change the structure and influence func-
tions within the ECM (Jolly et al. 2016). Many in vitro experiments have shown that
CAFs rearrange both collagen fibres and fibronectin, enabling a smooth invasion of
the cancer cells (Erdogan et al. 2017; Fang et al. 2014; Gopal et al. 2017; Ioachim et al.
2002). The ability to reorganise fibrous proteins in the microenvironment is aided by
the high secretion of collagen types I and II and fibronectin by the fibroblasts (Cirri
and Chiarugi 2011). For that reason, we choose to give here special consideration to
a two-component ECM in the context of cancer invasion, and, to that end, to regard
the ECM as consisting of both a fibre and a non-fibre phase.
Despite increasingly abundant in vivo and in vitro investigations and modelling
for cancer invasion from a variety of standpoints, only a snippet of the interactions
between the cancer cells and components of the extracellular matrix and surrounding
tissues could be so far depicted and understood. However, alongside all these biological
research efforts, the past 25 years have witnessed increasing focus on the mathematical
modelling of cancer invasion (Andasari et al. 2011; Anderson 2005; Anderson et al.
2000; Chaplain et al. 2011,2006; Gerisch and Chaplain 2008; Peng et al. 2017;Ramis-
Conde et al. 2008; Szyma´nska et al. 2009; Trucu et al. 2013), addressing various
processes of cancer cells and their interactions with the surrounding environment
through a variety of approaches ranging from discrete, local and non-local continuous
models to hybrid and multiscale models. Among these models, we note here the ones
concerning the secretion and transport of proteolytic enzymes such as uPAs and MMPs,
with direct impact upon the degradation of ECM (Andasari et al. 2011; Chaplain and
Lolas 2005; Peng et al. 2017; Trucu et al. 2013) as well as those exploring the direct
effects of chemotaxis, proliferation and adhesion on tumour invasion (Bitsouni et al.
2017; Chauviere et al. 2007; Domschke et al. 2014; Gerisch and Chaplain 2008; Painter
2008; Ramis-Conde et al. 2008), all these aspects being of direct interest for us in the
current investigation.
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R. Shuttleworth, D. Trucu
There are several models which have previously focused on the components of
the surrounding microenvironment of tumours and how these contribute to inva-
sion (Perumpanani et al. 1998; Scianna and Preziosi 2012). A model describing the
mesenchymal motion of cells in a fibre network and suggesting that the cells will pref-
erentially follow the direction of the fibres was proposed in Hillen (2006). Chemotactic
and haptotactic effects between cells and the fibrous environment of the ECM where
considered in Chauviere et al. (2007) and explored two scenarios, namely that either
cancer cells will try to gather in to high-density regions of fibres, or they will try to
avoid these regions altogether.
Finally, as the invasion process is genuinely multiscale, with its dynamics ranging
from molecular sub-cellular and cellular-scale to intercellular- and tissue-scale, the
multiscale modelling of cancer invasion has witnessed major advances over the past
15 years (Anderson et al. 2007; Peng et al. 2017; Ramis-Conde et al. 2008; Trucu
et al. 2013). However, while recognised by most previous works that a combination
of information from different scales would pave the way for a better understanding
of cancer invasion, the naturally interlinked multiscale dynamics of this process was
for the first time addressed in a genuinely spatially multiscale fashion in Trucu et al.
(2013), where a novel multiscale moving boundary model was developed by explor-
ing the double feedback link between tissue-scale tumour dynamics and the tumour
invasive edge cell-scale matrix-degrading enzymes (MDEs) activity. In that multiscale
model, while the tissue-scale macro-dynamics of cancer cells induces the source for
the leading edge cell-scale molecular micro-dynamics of MDEs, in turn, through its
proteolytic activity, this molecular micro-dynamics causes significant changes in the
structure of the ECM in the peritumoural region that ultimately translate in a tissue-
scale relocation of the tumour boundary. Later on, that model was adapted in Peng et al.
(2017) to capture the influence within the tumour invasion process of the proteolytic
dynamics of urokinase-plasminogen activator (uPA) system, exploring various sce-
narios for ECM degradation and proliferation of cancer cells. More recently, a further
extension of that modelling was developed in Shuttleworth and Trucu (2018), where
the dynamics of cell adhesion within a two-cell population heterogeneous context was
explored by adopting the non-local modelling proposed in Domschke et al. (2014)
and Gerisch and Chaplain (2008) as the macro-scale part of the multiscale platform
introduced in Trucu et al. (2013).
Building on the modelling platform introduced in Trucu et al. (2013) and extended
in Shuttleworth and Trucu (2018), in this paper we will pay a special attention to the
complicated structure of the ECM, and to that end we will propose a novel multiscale-
moving boundary model to account upon the multiscale dynamics of a two-component
ECM, considered here to consist of both a fibre and a non-fibre phase. This way we will
highlight the significance of the fibrous structure of the invading tumour and explore
not only the influence of these fibres within the macroscopic cancer cell dynamics,
but also capture their microscopic rearrangement. This new two-scale fibres dynamics
will be considered in the context of the multiscale moving boundary invasion process
as formulated in Trucu et al. (2013), leading this way to a novel multiscale moving
boundary framework, with two simultaneous but different in nature micro-dynamic
processes that are each connected through two double feedback loops to a shared
tissue-scale cancer macro-dynamics.
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Multiscale Modelling of Fibres Dynamics and Cell Adhesion…
2 The Novel Multiscale Modelling Framework
Building on the multiscale moving boundary framework initially introduced in Trucu
et al. (2013), in the following we describe the novel modelling platform for cancer
invasion. Besides the underlying tumour invasive edge two-scale proteolytic activity
of the matrix-degrading enzymes considered in Trucu et al. (2013), the new modelling
framework will now incorporate and explore the multiscale ECM fibre dynamics within
the bulk of the invading tumour, accounting in a double feedback loop for their micro-
scopic rearrangement as well as their macro-scale effect on cancer cell movement.
Let us denote the support of the locally invading tumour by Ω(t)and assume that this
evolves within a maximal environmental tissue cube YRN, with N=2,3, which
is centred at the origin of the space. In this context, at any tissue-scale spatio-temporal
node (x,t)Ω(t)×[0,T], we consider the tumour as being a dynamic mixture
consisting of a cancer cell distribution c(x,t)combined with a cumulative extracellular
matrix density v(x,t):= F(x,t)+l(x,t)whose multiphase configuration (F,l)will
be detailed in Sects. 2.22.3.
2.1 The Multiscale Moving Boundary Perspective
While postponing for the moment the precise details of the macro-dynamics (leaving
this to be fully introduced and explored in Sect. 2.4), since the novel modelling plat-
form that we develop here builds on the initial multiscale moving boundary framework
proposed in Trucu et al. (2013), let us start by devoting this entire section to briefly
revisiting and summarising the main features of the general two-scale moving bound-
ary approach that was introduced in Trucu et al. (2013). In this context, let us express
for the moment the tissue-scale tumour macro-dynamics on the evolving Ω(t)in the
form of the following pseudo-differential operator equation
T(c,F,l)=0(1)
where T(·,·,·)denotes an appropriately derived reaction-diffusion-taxis operator
whose precise form will be completely defined in Sect. 2.4. Furthermore, as detailed
in Trucu et al. (2013), the key multiscale role played by the tumour invasive prote-
olytic enzymes processes in cancer invasion is captured here in a multiscale moving
boundary approach where the link between the tumour macro-dynamics (1) and the
cell-scale leading edge proteolytic molecular micro-dynamics is captured via a double
feedback loop. This double feedback loop is realised via a top-down and a bottom-up
link, as illustrated schematically in Fig. 1and detailed below.
The top-down link. As discussed previously, cancer invasion is a multiscale process
in which the matrix-degrading enzymes (MDEs), such as matrix metallo-proteinases
(MMP) which are secreted by the cancer cells from within the outer proliferation rim of
the tumour, are responsible for the degradation of the peritumoural ECM, enabling fur-
ther tumour expansion. Thus, adopting the terminology and framework developed in
Trucu et al. (2013), this tumour invasive edge molecular micro-dynamics, which occurs
within a cell-scale neighbourhood of the tumour interface ∂Ω(t), can be explored on
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R. Shuttleworth, D. Trucu
Fig. 1 Schematic showing the interactions between the macro- and the proteolytic MDEs micro-scale
dynamics and the role this plays in boundary reallocation
an appropriately constructed bundle of -size half-way overlapping micro-domains
{Y}YP(t)satisfying some naturally arising topological requirements. These require-
ments ensure that each Y “sits on the interface” and captures relevant parts of both
inside and outside regions of the tumour where the proteolytic activity takes place (as
briefly detailed in Appendix Eand illustrated in schematic Fig. 19, while for complete
details we refer the reader to Trucu et al. (2013)). This allows us to decouple this
leading edge proteolytic activity in a bundle of corresponding MDE micro-processes
occurring on each Y. In this context, a source of MDEs arises at each zYΩ(t0)
as a collective contribution of all the cells that (subject to macro-dynamics (1)) arrive
within the outer proliferating rim at a spatial distance from zsmaller than a certain
radius γ>0 (representing the maximal thickness of the outer proliferating rim). Thus,
the source of MDEs that is this way induced by the macro-dynamics at the micro-scale
on each Yrealises a significant top-down link that can be mathematically expressed
as
1.gY(z,τ)=
B(z,γ )Ω(t0)
αc(x,t0+τ)dx
λ(B(z,γ)Ω(t0)) ,zYΩ(t0),
2.gY(z,τ)=0,zY\Ω(t0)+{zY|||z||2}),
(2)
where B(z,γ) := {ξY|zξζ}and αis an MDE secreting rate for the
cancer cell population. In the presence of this source, a cross-interface MDEs transport
takes place. As in this paper we only consider the micro-dynamics of a single class of
MDEs, such as MMPs, this simply results in a diffusion-type transport over the entire
Ymicro-domain, and so denoting the MDE molecular density by m(z,τ), this can
be mathematically formulated as
m
∂τ =DmΔm+gY(z,τ), zY∈[0t],(3)
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Multiscale Modelling of Fibres Dynamics and Cell Adhesion…
The bottom-up link. During the micro-dynamics (3), the MDEs transported across the
interface in the peritumoural region interact with ECM distribution that they meet in the
immediate tumour proximity outside the cancer region within each boundary micro-
domain Y. On each microdomain Y, provided that a sufficient amount of MDEs have
been transported across the cancer invading edge enclosed in this microdomain, it is the
pattern of the front of the advancing spatial distribution of MDEs that characterises the
way in which the ECM is locally degraded. As introduced and described in Trucu et al.
(2013), within each Y, the pattern of degradation of ECM caused by the significant
levels of the advancing front of MDEs give rise to a direction ηYand displacement
magnitude ξY(detailed in Appendix E), which determine the cancer boundary move-
ment characteristics represented back at macro-scale through the movement of the
appropriately defined boundary mid-points x
Yto their new spatial positions
x
Y,see
Fig. 1. Thus, over a given time perspective [t0,t0+Δt],thebottom-up link of the
interaction between the proteolytic tumour invasive edge micro-dynamics and macro-
scale is realised through the macro-scale boundary movement characteristics that are
provided by the micro-scale MDEs activity, leading to the expansion of the tumour
boundary Ω(t0)to an enhanced domain Ω(t0+Δt)where the multiscale dynamics
is continued.
2.2 The Multiscale and Multi-component Structure of the ECM
To gain a deeper understanding of the invasion process, in this work we pay special
attention to the ECM structure within the overall multiscale dynamics. While in pre-
vious multiscale approaches (such as those proposed in Peng et al. (2017) and Trucu
et al. (2013)), the ECM has been considered as a “well mixed” matrix distribution,
with no individual components taken in to consideration, in the following we account
for the structure of the ECM by regarding this as a two-component media. The first
ECM component that we distinguish accounts for all significant ECM fibres such as
collagen fibres or fibronectin fibrils. This will be denoted by F(x,t)and will simply
be referred to as the fibres component. Finally, the second ECM component that we
distinguish consists of the rest of ECM constituents bundled together. This will be
referred to as the non-fibres component and will be denoted by l(x,t).
While from the tissue-scale (macro-scale) stand point the fibres are regarded as a
continuous distribution at any xY, from the cell-scale (micro-scale) point of view, a
specific micro-structure can be in fact distinguished as a mass-distribution of the ECM
micro-fibres f (z,t)that are spatially distributed on a small micro-domain of micro-
scale size δ>0 centred at any macroscopic point xY, namely on δY(x):= δY+x.
In this context, as we will detail below, the microscopic mass-distribution of ECM
micro-fibres will be able to supply important macro-scale fibre characteristics, both in
terms of their associated macroscopic fibre orientation θf(x,t)and magnitude F(x,t),
which will be introduced in Sect. 2.3. Figure 2illustrates such micro-fibres distribution
in micro-domains δY(x),xΩ(t). A concrete example of such micro-scale fibres
pattern is then proposed in Fig. 3, this being given as
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R. Shuttleworth, D. Trucu
Fig. 2 Schematic showing copies of δYcube on the grid with micro-fibres distribution in green and their
induced macroscopic direction θf(x,t)in pale blue (Color figure online)
Fig. 3 Micro-fibre distribution on δY(x)
f(z,t):=
jJ
ψhj(z)(χ2γ)Y(x)ψγ)(z)(4)
where {ψhj}jJare smooth compact support functions of the form
ψhj:δY(x)R
which, at every z:= (z1,z2)δY(x), are given by:
ψhj(z1,z2):= Chje
1
r2(hj(z2)z1)2,if z1∈[hj(z2)r,hj(z2)+r],
0,if z1/∈[hj(z2)r,hj(z2)+r].
(5)
with r>0 being the width of the micro-fibres and Chjbeing constants that determine
the maximum height of ψhjalong the smooth paths {hj}jJin δY(x)that are given in
Appendix B. Finally, χγ)Y(x)(·)represents the characteristic function of the cubic
micro-domain γ)Y(x)centred at xand of size 2γ), with γ>0asmall
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Multiscale Modelling of Fibres Dynamics and Cell Adhesion…
enough radius, while ψγis the usual mollifier defined in Appendix Cthat is smoothing
out this characteristic function to a smooth compact support function on δY.
Furthermore, as we will discuss in the following, while the fibre micro-structure
will be dynamically rearranged at micro-scale by the incoming flux of cancer cell
population, their “on the fly” updated revolving orientation θf(x,t)and magnitude
F(x,t)will be involved in the dynamics at macro-scale.
2.3 Macro-scale Fibres Orientation and Magnitude Induced by the ECM
Micro-fibres Spatial Distribution at Micro-scale: Derivation and
Well-Posedness
On every micro-domain δY(x)centred at a macro-point xΩ(t), at a given time
instance t∈[0,T], the spatial distribution of the micro-fibres f(z,t)on δY(x)natu-
rally provides a cumulative revolving orientation of these with respect to the barycentre
x, and to derive this we proceed as follows.
Considering an arbitrary dyadic decomposition {Dj}jJnof size δ2nfor the micro-
domain δY(x), let us denote by zjthe barycentre of each dyadic cube Dj. Then, for any
jJn, the mass of micro-fibres distributed on Djwill influence the overall revolving
fibre orientation on δY(x)through its contribution in direction of the position vector
−→
xzj:= zjxin accordance with its weight relative standing with respect to the
micro-fibre mass distributed on all other Djcovering δY(x), see Fig. 4. Therefore,
the overall revolving micro-fibres orientation on δY(x)associated with the dyadic
decomposition {Dj}jJnis given by:
θn
fY(x)(x,t):=
jJnDjf(ζ,t)dζ
jJn
Dj
f(ζ,t)dζ
−→
xzj
=
jJn
Dj
f(ζ,t)dζ
δY(x)
f(ζ,t)dζ
−→
xzj
=
jJn
1
λ(Dj)
Dj
f(ζ,t)dζ
λ(Dj)
xzj
δY(x)
f(ζ,t)dζ
=
δY(x)
jJn
1
λ(Dj)
Dj
f(ζ,t)dζ
χDj(z)
xzj
dz
δY(x)
f(ζ,t)dζ
=
δY(x)
jJn
1
λ(Dj)
Dj
f(ζ,t)dζ
χDj(z)(zjx)
dz
δY(x)
f(ζ,t)dζ
(6)
where λ(·)is the usual Lebesgue measure and χDj(·)is the characteristic function of
the dyadic cube Dj. Thus, for any nNdenoting the numerator function
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R. Shuttleworth, D. Trucu
Fig. 4 Schematic of the fibre
micro-domain δY(x)
decomposed into dyadic cubes
shown in green, Dj, with
associated barycentre zjin cyan.
The position vectors are shown
in dark blue (Color figure online)
gn(z):=
jJn1
λ(Dj)
Dj
f(ζ, t)dζχDj(z)(zjx), (7)
let’s observe immediately that {gn}nNis actually a sequence of vector-valued simple
functions that is convergent to f(z,t)(zx)and that its associated sequence of
integrals converges to the Bochner Integral of f(z,t)(zx)on δY(x)with respect to
λ(·)(Yosida 1980), namely
δY(x)
f(z,t)(zx)dz:= lim
n→∞
δY(x)
gn(z)dz.(8)
Hence, from (6)–(8), we obtain that the sequence of revolving {θn
f(x,t)}nNfibres
orientations associated to the entire family of dyadic decompositions {{Dj}jJn}nN
is convergent to the unique revolving barycentral micro-fibres orientation on δY(x)
denoted by θfY(x)(x,t)and given by
θfY(x)(x,t):= lim
n→∞ θn
fY(x)(x,t)
=lim
n→∞
δY(x)
jJn
1
λ(Dj)
Dj
f(ζ,t)dζ
(zjxDj(z)
dz
δY(x)
f(ζ,t)dζ
=
lim
n→∞
δY(x)
gn(z)dz
δY(x)
f(ζ,t)dζ
=
δY(x)
f(z,t)(zx)dz
δY(x)
f(ζ,t)dζ
=
δY(x)
f(z,t)(zx)dz
δY(x)
f(z,t)dz,
(9)
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Multiscale Modelling of Fibres Dynamics and Cell Adhesion…
which is actually precisely the Bochner-mean-value of the position vectors function
δY(x)z→ zxRNwith respect to the measure f(x,t)λ(·)that is induced by
the micro-fibres distribution. Therefore, denoting by θf(x,t)the macroscopic fibres
orientation at (x,t)induced by the revolving barycentral micro-fibres orientation on
δY(x), we have that this is given by
θf(x,t):= 1
λ(δY(x))
δY(x)
f(z,t)dz·θfY(x)(x,t)
θfY(x)(x,t)2
(10)
Finally, the macroscopic representation of the ECM fibres distributed at (x,t)is
denoted by F(x,t)and is given by the Euclidean magnitude of θf(x,t), namely:
F(x,t):= θf(x,t)2,(11)
and so using now (10), from (11) we obtain that
F(x,t):=θf(x,t)2
= 1
λ(δY(x))
δY(x)
f(z,t)dz·θfY(x)(x,t)
θfY(x)(x,t)2
2
=1
λ(δY(x))
δY(x)
f(z,t)dz·θfY(x)(x,t)2
θfY(x)(x,t)2
=1
λ(δY(x))
δY(x)
f(z,t)dz,
(12)
which is precisely the mean value of the micro-fibres distributed on δY(x). There-
fore, the macroscopic fibres orientation at (x,t)induced by the revolving barycentral
micro-fibres orientation on δY(x)has its magnitude given by the mean value of the
micro-fibres on δY(x), and since in (6)–(9) we have ensured the well-posedness of
θfY(x)(x,t), from (10)–(12), we obtain that θf(x,t)and F(x,t)are also well-posed.
With all these preparations, we are now in the position to describe the tumour
macro-dynamics, which will be detailed in full in the next section.
2.4 Tumour Macro-dynamics
To explore mathematically the macro-scale coupled dynamics exercised by the cancer
cells mixed with the ECM, for notation convenience, let’s first gather the macro-
scopic distributions of cancer and the two ECM phases considered here in the
three-dimensional vector
u(x,t):= (c(x,t), F(x,t), l(x,t))T,
and let’s denote tumour’s volume fraction of occupied space by
ρ(u)ρ(u(x,t)) := ϑv(F(x,t)+l(x,t)) +ϑcc(x,t), (13)
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R. Shuttleworth, D. Trucu
with ϑvrepresenting physical space occupied by the fibre and non-fibre phases of
the ECM taken together and ϑcbeing the fraction of physical space occupied by the
cancer cell population c.
Therefore, focusing first upon the cancer cell population, per unit time, under the
presence of a proliferation law, its spatial dynamics is not only due to random motility
(approximated mathematically by diffusion), but this is also crucially influenced by a
combination of cell adhesion processes that include cell–cell adhesion and cell–matrix
adhesion, with cell–matrix adhesion exhibiting distinctive characteristics in relation to
the two ECM phases (namely: the fibres and non-fibres components). Hence, assuming
here a logistic proliferation law, the dynamics of the cancer cell population can be
mathematically represented as
c
t=∇·[D1ccA(t,x,u(·,t), θ f(·,t))]+μ1c(1ρ(u)), (14)
where D1and μ1are nonnegative diffusion and proliferation rates, respectively, while
A(t,x,u(·,t), θ f(·,t)) represents a non-local constitutive flux term accounting for the
critically important cell adhesion processes that influence directly the spatial tumour
movement, whose precise form will be explored as follows.
While generally adopting a similar perspective to the one in Armstrong et al. (2006),
Domschke et al. (2014) and Gerisch and Chaplain (2008) concerning cell–cell adhesion
and cell–ECM–non-fibres substrate, here we move beyond the context considered in
those works by accounting for the crucial role played by the cell–fibres adhesive
interaction. Thus, within a sensing radius R,atagiventimetand spatial location x,
the adhesive flux associated to the cancer cells distributed at (x,t)will account for
not only the adhesive interactions with the other cancer cells and ECM non-fibres
phase distributed on B(x,R), but this will also appropriately consider and cumulate
the adhesive interaction arising between cancer cells and the oriented ECM fibres,
resulting in the following novel non-local adhesion flux term:
A(t,x,u(·,t), θ f(·,t)) =1
RB(0,R)
K(y2)n(y)(Sccc(x+y,t)+Scl l(x+y,t))
n(y)ScF F(x+y,t)(1ρ(u))+(15)
While the influence on adhesive interactions of the distance from the spatial location x
is accounted for through the radial kernel K(·)detailed in Appendix D,n(·)represents
the usual unit radial vector given by
n(y):= y/||y||2if yB(0,R)\{0},
(0,0)if y=(0,0), (16)
along which we consider the cell–cell and cell–ECM–non-fibres adhesion bonds estab-
lished between the cancer cells distributed at xand the cells and non-fibre ECM phase
distributed at x+ywith strengths Scc and Scl , respectively. Specifically, here Scl
is considered to be constant. However, as biological evidence discussed in Gu et al.
(2014) and Hofer et al. (2000) suggests, in direct correlation to collagen levels, it is the
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Multiscale Modelling of Fibres Dynamics and Cell Adhesion…
Fig. 5 Schematic to describe the process of the new cell–fibre adhesion term which includes the distribution
of fibres. It shows the ball B(x,R)centred at xand of radius R, the point x+ywith the direction vector
n(y)in orange, and the fibre orientation θf(x+y,t)in purple (Color figure online)
high level of extracellular Ca2+ions rather than the sole production and presence of
intracellular Ca2+that enables strong and stable adhesive bonds between cells, having
this way a direct impact over the strength of cell–cell adhesion. Therefore, we assume
here that Scc is dependent on the collagen density, smoothly ranging between 0 and a
Ca2+-saturation level Smax, this being taken here of the form
Scc(x,t):= Smax e11
1(1l(x,t))2.
Finally, the last term in (15) considers the crucially important adhesive interactions
between the cancer cells distributed at xand the oriented fibres distributed on B(x,R).
In this context, while the strength of this interaction is proportional to the macro-scale
amount of fibres F(·,t)distributed at x+y, and, as illustrated in Fig. 5, the orientation
θf(·,t)of these fibres biases the direction of these adhesive interactions in the direction
of the vector ˆn(·)defined by
ˆn(y):=
y+θf(x+y)
||y+θf(x+y)||2
if (y+θf(x+y)) = (0,0)
(0,0)R2otherwise.
(17)
Further, per unit time, the fibres distribution are simply degraded by the cancer
cells at macro-scale, and so the dynamics of their macroscopic dynamics is simply
governed by
dF
dt=−γ1cF (18)
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R. Shuttleworth, D. Trucu
where γ1describes the rate of degradation. This macroscopic degradation of fibres is
feed back to the micro-fibres f(·,t)on the micro-domains δY(x)as a factor which
will lower their microscopic height accordingly. To complete the description of the
macroscopic system, the non-fibre ECM is degraded and remodelled by the cancer
cells, and so its dynamics can be mathematically formulated as
dl
dt=−γ2cl +ω(1ρ(u))+(19)
where γ2describes the rate of degradation and ωdenotes the rate of remodelling, while
the volume filling term (1ρ(u))+:= max(0,(1ρ(u)) prevents the overcrowding
of physical space.
2.5 Microscopic Fibre Rearrangement Induced by the Macro-dynamics
As the cancer cells invade, they push the fibres in the direction they are travelling,
thereby influencing the ECM fibres by their own directive movement. Thus, in addition
to the macro-scale fibre degradation (explored in (18)), during the tumour dynamics,
at any instance in time tand spatial location xΩ(t), the cancer cell population is
also pushing and realigning the fibres, causing a micro-scale spatial rearrangement of
the micro-fibres distributed on δY(x). Specifically, this micro-fibres rearrangement is
triggered by the macro-scale spatial flux of migratory cancer cells, namely by
F(x,t)=D1c(x,t)c(x,t)A(t,x,u(·,t), θ f(·,t)). (20)
Naturally magnified in accordance with the amount of cancer cells distributed at (x,t)
relative to the overall macro-scale amount of cancer cells and fibres that they meet at
(x,t), expressed here through the weight
ω(x,t)=c(x,t)
c(x,t)+F(x,t),
the spatial flux of migratory cells F(x,t)gets balanced in a weighted manner by
the macroscopic orientation θf(t,x), resulting in a rearrangement flux vector-valued
function given by
rY(x), t):= ω(x,t)F(x,t)+(1ω(x,t))θ f(x,t), (21)
which acts uniformly upon the micro-fibres distributed on δY(x), leading to an on-the-
fly change in the spatial distribution of micro-fibres on δY(x). In this context, denoting
the barycentric position vector of any micro-scale position zδY(x)by
xdir(z)=
xz,
let’s observe that this micro-scale fibres rearrangement will be exercised provided that
the micro-fibres f(z,t)would not have already reached a certain maximum level fmax
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Multiscale Modelling of Fibres Dynamics and Cell Adhesion…
(when the micro-fibre distribution would be very “stiff” and the cancer cells would
struggle to move through those micro-locations) and that their movement magnitude
will be affected by the micro-fibre saturation fraction
f=f(z,t)
fmax
combined with size of the micro-scale position defect with respect to rY(x), t)that
is given simply by
||rY(x)) xdir(z)||2.
Therefore, under the action of the rearrangement flux rY(x), t), the micro-fibres
distributed at zwill attempt to exercise their movement in the direction of the resulting
vector xdir(z)+rY(x), t), and so their relocation to the corresponding position within
neighbouring micro-domain will be given by the vector-valued function:
νδY(x)(z,t)=(xdir(z)+rY(x), t))·f(z,t)( fmax f(z,t))
f+||rY(x)) xdir (z)||2
·χ{f(·,t)>0}(z)
(22)
where χ{f(·,t)>0}represents the usual characteristic function of the micro-fibres sup-
port set {f(·,t)>0}:={zδY(x)|f(z,t)>0}. Finally, the movement of the
micro-fibres distributed at zto the newly attempted location zgiven by
z:= z+νδY(x)(z,t)
is exercised in accordance with the space available at the new position z. Thus, this
is explored here through the movement probability
pmove := max 0,fmax f(z,t)
fmax
which enables only an amount of pmove f(z,t)of micro-fibres to be transported to
position z(as illustrated in Fig. 6), while the rest of (1pmove)f(z,t)remains at z.
2.6 Schematic Summary of Global Multi-scale Model
In this model, there are two interconnected multi-scale systems, each with their own
distinct cell-scale micro-dynamics, but both of them sharing the same macro-scale
cancer dynamics at the tissue scale, being linked to this through two double feedback
loops, as illustrated in Fig. 7. The macro-scale dynamics governs the spatial distribution
of both the invading cancer cells and the fibrous and non-fibrous density components
of the ECM, and is given by the following non-local coupled system:
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R. Shuttleworth, D. Trucu
Fig. 6 Schematics to describe the process of reallocation of fibre distribution in each δYcube
Fig. 7 Schematic summary of global multiscale model
c
t=∇·[D1ccA(t,x,u(·,t), θ f(·,t))]+μ1c(1ρ(u)), (23a)
dF
dt=−γ1cF,(23b)
dl
dt=−γ2cl +ω(1ρ(u)). (23c)
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Multiscale Modelling of Fibres Dynamics and Cell Adhesion…
Occurring on the cell-scale, the micro-scale part of the first multiscale system controls
the dynamic redistribution of microscopic fibres within the entire cancer region. On
each micro-domain δY(x), the realignment of the existing micro-fibres is triggered by
the spatial flux of cancer cells from the macro-scale and this is realised by weighted
action of this over the oriented macroscopic fibres distribution that they meet at x.
Once all fibre micro-domains within the cancer region have undergone redistribution,
a new macroscopic fibre orientation and mean value per each δY(x)is obtained and
that in its turn will have its effect in the important cell adhesion behaviour that the
cancer exhibits at macro-scale. Finally, in the second multi-scale system, the spatial
distribution of cancer cells induces a source of MDEs on the boundary at the micro-
scale level. In return, the leading edge proteolytic micro-dynamics of MDEs instigates
a change in the position of the tissue-scale tumour boundary that corresponds to the
pattern of the peritumoural ECM degradation, enabling this way the invasion process
to continue on the expanding domain.
3 Numerical Approach: Key Points of the Implementation
Building on the numerical multiscale platform initially introduced in Trucu et al.
(2013), the implementation of the novel multiscale moving boundary model that we
proposed in this work required a number of new major computational steps, which will
be detailed in the next three subsections. These include a special treatment for several
computational aspects, such as those concerned with: macro-scale computations on
the expanding tumour; the macro-scale adhesion term; and a new predictor–corrector
scheme for the cancer dynamics equation (23a).
Finally, the approach for the cross-interface proteolytic micro-dynamics on each
tumour boundary micro-domain Yfollows precisely the steps described in Trucu
et al. (2013), involving a finite element scheme using bilinear shape functions and
square elements, reason for which we do not include that here.
3.1 Macro-scale Computations on the Expanding TumourDomain
While considering a uniform spatial mesh of size Δx=Δy=hfor the maximal
cube Y, recoded on a square grid {(xi,xj)}i,j=1...M, with M:= length(Y)/h+1,
the actual macroscopic computation will be performed exclusively on the expanding
tumour Ω(t0)over every macro–micro-time interval [t0,t0+Δt]as will be detailed in
the following. Specifically, to explore this, let us first denote by I(·,·):{1,...,M
{1,..., M}→{0,1}the on-grid cancer indicator function given as usual by
I(i,j):= 1if(xi,xj)Ω(t0),
0if(xi,xj)/Ω(t0), (24)
Further, let’s observe that the on-grid closest neighbour indicator functions Ix,+1(·,·),
Ix,1(·,·), Iy,+1(·,·),Iy,1(·,·):{2,...,M1}×{2,...,M1}→{0,1}, defined
by
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R. Shuttleworth, D. Trucu
Ix,±1(i,j):= |I(i,j)I(i,j±1)I(i,j),
Iy,±1(i,j):= |I(i±1,j)I(i,j)I(i,j), (25)
enable us to detect on-the-fly the grid positions immediately outside the cancer bound-
ary as the points of nonzero value along each spatial direction, given by the union of
preimages I1
x,1({1})I1
x,+1({1})and I1
y,1({1})I1
y,+1({1})for x- and y-direction,
respectively.
Over each macro-scale time perspective [t0,t0+Δt], the overall macroscopic
scheme for (23) involves the method of lines coupled with a novel predictor–corrector
method for time marching (whose main steps will be detailed in the next subsection),
the discretisation of the spatial operators appearing in the right-hand side of (23a)
is based on central differences and midpoint approximations. For this, considering a
uniform discretisation {tp}p=0...kof [t0,t0+Δt], of time step δt>0, let’s denote by
cp
i,j,Ap
i,j,Fp
i,j,lp
i,jthe discretised values of c,A,F,lat ((xi,xj), tp), respectively.
Thus, at any spatial node (xi,xj)Ω(t0), the no-flux across the moving boundary
dynamics is accounted for via the indicators (24)–(25) on the expanding spatial mesh,
and results into the midpoint approximations
cp
i,j±1
2
:= cp
i,j+Ix,±1(i,j)cp
i,j+I(i,j±1)cp
i,j±1
2,
cp
i±1
2,j:= cp
i,j+Iy,±1(i,j)cp
i,j+I(i±1,j)cp
i±1,j
2,
(26)
and
Ap
i,j±1
2
:= Ap
i,j+Ix,±1(i,j)Ap
i,j+I(i,j±1)Ap
i,j±1
2,
Ap
i±1
2,j:= Ap
i,j+Iy,±1(i,j)Ap
i,j+I(i±1,j)Ap
i±1,j
2,
(27)
while the central differences for cat the virtual nodes (i,j±1
2)and (i±1
2,j)are
given by
[cx]p
i,j+1
2
:= [Ix,+1(i,j)ci,j+I(i,j+1)ci,j+1]ci,j
dx ,
[cx]p
i,j1
2
:= ci,j−[Ix,1(i,j)ci,j+I(i,j1)ci,j1]
dx ,
[cy]p
i+1
2,j:= [Iy,+1(i,j)ci,j+I(i+1,j)ci+1,j]ci,j
dy ,
[cy]p
i1
2,j:= ci,j−[Iy,1(i,j)ci,j+I(i1,j)ci1,j]
dy .
(28)
Therefore, the approximation for the term ∇·[D1ccA(t,x,u(·,t), θ f(·,t))]in
(23a) is obtained as
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Multiscale Modelling of Fibres Dynamics and Cell Adhesion…
(∇·[D1ccA(t,x,u(·,t), θ f(·,t))])p
i,j
D1([cx]p
i,j+1
2
−[cx]p
i,j1
2
)cp
i,j+1
2
·Ap
i,j+1
2
+cp
i,j1
2
·Ap
i,j1
2
Δx
+
D1([cy]p
i+1
2,j−[cy]p
i1
2,j)cp
i+1
2,j·Ap
i+1
2,j+cp
i1
2,j·Ap
i1
2,j
Δy,
(29)
and so, denoting by Fp
i,jthe discretised value of the flux F(·,·)at the spatio-temporal
node ((xi,xj), tp), the spatio-temporal discretisation of ∇·F:= · [ D1c
cA(t,x,u(·,t), θ f(·,t))]given in (29) can therefore be equivalently expressed in
a compact form as
(∇·F)p
i,j
Fp
i,j+1
2
Fp
i,j1
2
Δx+
Fp
i+1
2,jFp
i1
2,j
Δy(30)
where Fp
i,j±1
2
:= D1[cx]p
i,j±1
2
cp
i,j±1
2
·Ap
i,j±1
2
and Fp
i±1
2,j:= D1[cy]p
i±1
2,jcp
i±1
2,j·
Ap
i±1
2,j.
3.2 Adhesive Flux Computation
As already mentioned above, an important aspect within the macroscopic part of our
solver is the numerical approach addressing the adhesive flux A(t,x,u(·,t), θ f(·,t)),
which explores the effects of cell–cell, cell–ECM–non-fibre and cell–fibre adhesion
of cancer cell population. Although to a certain extent we adopt a similar approach to
the one that we previously proposed in Shuttleworth and Trucu (2018) (for a similar
macro-scale invasion context but in the absence of fibre dynamics), the numerical
approximation for the non-local term A(t,x,u(t,·), θ f(·,t)) involves a series of off-
grid computations on a new decomposition of the sensing region, developing further
the approach introduced (Shuttleworth and Trucu 2018) and adapting that to the new
context of the current macro-model. For completeness, we detail this here as follows.
Thus, at a given spatio-temporal node ((xi,xj), tp), we decompose the sensing region
B((xi,xj), R)in
q:=
s
i=1
2m+(i1)annulus radial sectors S1,...,Sq,
which are obtained by intersecting each annulus i∈{1,...,s}annuli with a corre-
sponding number of 2m+(i1)uniformly distributed radial sectors of B((xi,xj), R),
as shown in Fig. 8, while considering the remaining central circle to be of a computa-
tionally negligible radius. Then, using a standard barycentric interpolation approach
for approximating the off-grid values, ν∈{1,...,q}, on each annulus sector Sν,we
calculate the mean values of all the macro-scale densities of cancer cells c(·,tp), ECM
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R. Shuttleworth, D. Trucu
Fig. 8 Sensing region B(x,R)approximated by the annulus radial sectors with the barycentre bSνassoci-
ated to each sector Sνhighlighted with a blue star (Color figure online)
non-fibres component l(·,tp), macroscopic ECM fibres F(·,tp)and their associated
directions θf(·,tp), namely:
Wp
Sν,c:= 1
λ(Sν)
Sν
c, tp)dξ, Wp
Sν,l:= 1
λ(Sν)
Sν
l, tp)dξ,
Wp
Sν,F:= 1
λ(Sν)
Sk
F, tp)dξ, and Wp
Sνf:= 1
λ(Sν)
Sν
θf, tp)dξ,
respectively. Further, ν∈{1,...,q}, denoting by bSνthe barycenter of Sν,this
enable us to evaluate the unit vector denoted by nνthat points from the centre of the
sensing region to bSν, i.e.
nν:= bSν(xi,xj)
bSν(xi,xj)2
,
as well as the corresponding macroscopic vector accounting for the influence of the
cumulative mean value direction of the fibres on Sν, namely
np
ν:=
nν+Wp
Sνf
nν+Wp
Sνf2
Thus, finally, the approximation of the adhesion flux A(t,x,u(·,t), θ f(·,t)) at the
spatio-temporal node ((xi,xj), tp)is denoted by Ap
i,jand is given by
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Multiscale Modelling of Fibres Dynamics and Cell Adhesion…
Ap
i,j=1
R
q
ν=1
bSνΩ(t0)
K(bSν)[nν·(SccWp
Sν,c
+Scl Wp
Sν,l)+np
ν·ScFWp
Sν,F](1ρ(up
bSν))+λ(Sν), (31)
where ν∈{1,...,q}, denoting up
bSν:= [Wp
Sν,c,Wp
Sν,l,Wp
Sν,F]T, we have that
ρ(up
bSν)is volume fraction defined in (13) that corresponds to the discrete vector
up
bSν.
3.3 The Predictor–Corrector Step
For the time discretisation of Eq. (23a), we develop a novel predictor–corrector scheme
involving a non-local trapezoidal corrector. For this, let us denote by H(·,·,·)the right-
hand side spatial operator of (23a), which is defined as follows. At any instance in
time and any corresponding triplet (F,c,u)of given spatially discretised values for
the flux F, the cell population c, and the tumour vector u, by ignoring for simplicity
the time notation we have that His given by
H(Fi,j,ci,j,ui,j):= (∇·F)i,j+μ1ci,j(1ρ(ui,j)), (32)
where the spatial discretisation (∇·F)i,jis given here still by (30) but applied to the
spatial flux F, and ρ(ui,j)is simply the volume fraction defined in (13) evaluated for
the discrete vector ui,j:= [ci,j,Fi,j,li,j],i,j=1...M. In this context, on the
time interval [tp,tp+1], we first predict on-the-fly values for cat tp+1
2, namely
˜cp+1
2
i,j=cp
i,j+δt
2HFp
i,j,cp
i,j,up
i,j.(33)
where up
i,j:= [cp
i,j,Fp
i,j,lp
i,j],i,j=1...M. Further, using these predicted values
˜cp+1
2, we calculate the corresponding predicted flux at tp+1
2, namely ˜
Fp+1
2, and then
we construct a non-local corrector that involves the average of the flux at the active
neighbouring spatial locations
{(xi,xj±1), (xi±1,xj), (xi±1,xj1), (xi±1,xj+1)}∩Ω(t0). (34)
Thus, denoting by Nthe set of indices corresponding to these active locations, we
have that the corrector flux is calculated as
Fp+1
2
i,j=1
card(N)
(σ,ζ )N
˜
Fp+1
2
σ,ζ ,(35)
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R. Shuttleworth, D. Trucu
ultimately enabling us to use the trapezoidal approximation to obtain the corrected
value for cat tp+1
2as
cp+1
2
i,j=cp
i,j+δt
4H(Fp
i,j,cp
i,j,up
i,j)+H(Fp+1
2
i,j,˜cp+1
2
i,j,˜
up+1
2
i,j)(36)
where ˜
up+1
2
i,j:= [˜cp+1
2
i,j,Fp
i,j,lp
i,j],i,j=1...M. Finally, we use the average
¯cp+1
2
i,j:= cp
i,j+cp+1
2
i,j
2(37)
to re-evaluate the flux at tp+1
2, namely Fp+1
2(corresponding to the average values
¯cp+1
2) and then to initiate the predictor–corrector steps described above on this new
time interval [tp+1
2,tp+1]. Thus, following the predictor step, we first obtain the pre-
dicted values at tp+1, namely
˜cp+1
i,jcp+1
2
i,j+δt
2HFp+1
2
i,j,¯cp+1
2
i,j,¯
up+1
2
i,j(38)
where ¯
up+1
2
i,j:= [¯cp+1
2
i,j,Fp
i,j,lp
i,j],i,j=1...M. Finally, we correct these values
at tp+1with the same non-local trapezoidal-type corrector as described in (36), here
involving the corrector flux calculated as average of the predicted flux values ˜
Fp+1
(corresponding to the predicted values ˜cp+1) at the active neighbouring locations given
in (34), namely
Fp+1
i,j=1
card(N)
(σ,ζ )N
˜
Fp+1
σ,ζ .(39)
Thus, this last corrector step gives us ultimately the values that we accept at tp+1,
namely
cp+1
i,jcp+1
i,j+δt
4HFp+1
2
i,j,¯cp+1
2
i,j,¯
up+1
2
i,j+HFp+1
i,j,˜cp+1
i,j,˜
up+1
i,j (40)
where ˜
up+1
i,j:= [˜cp+1
i,j,Fp
i,j,lp
i,j],i,j=1...M.
Lastly, for the discretisation of (23b) and (23c), we follow the same predictor–
corrector method as the one used in Trucu et al. (2013), where we the corrector part
uses simply a second-order trapezoidal scheme on [tp,tp+1].
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Multiscale Modelling of Fibres Dynamics and Cell Adhesion…
4 Computational Simulations and Results
To illustrate our model, we consider the region Y:= [0,4]×[0,4]and we start our
dynamics by adopting here the same initial condition for cas in Trucu et al. (2013),
namely
c(x,0)=0.5exp ||x(2,2)||2
2
0.03 exp(28.125) χB((2,2),0.5γ) ψγ,
(41)
where ψγis the standard mollifier detailed in Appendix Cthat acts within a radius
γ<<Δx
3from B((2,2), 0.5γ) to smooth out the characteristic function
χB((2,2),0.5γ). Thus, initially, the cancer cell population occupies the region Ω(0)=
B((2,2), 0.5)positioned in the centre of Y.
Initial condition for the ECM fibre component For the initial distribution of the ECM
fibre phase, we consider first a generic micro-domain centred a 0 of cell-scale size
δ=h, namely δY, and using the microscopic patterns of fibres defined in (5) and
illustrated in Fig. 3, we replicate and centre this micro-fibre distribution in the cell-
scale neighbourhood of any spatial location (xi,xj)in the discretisation of Yon the
corresponding micro-domain δY(xi,xj):= δY+(xi,xj). The maximal height of the
micro-fibres considered here is appropriately calibrated uniformly across all micro-
domains is so that the resulting macroscopic distribution of fibres F(x,·)represents
a percentage p, of the mean density of the non-fibrous ECM phase. In this context, to
determine the percentage of initial fibres, a sensitivity analysis using varying levels of
pwas performed, detailed in Appendix A. It was concluded that under both a homoge-
neous and heterogeneous non-fibre ECM phase, expansion of the boundary was most
notable when p=0.2, hence it is this value we use for all of the following simula-
tions. Therefore at the initial time t0=0, all fibre micro-domains δY(xi,xj)support
identical distributions of micro-fibres, i,j=1...Mand as a consequence, every
fibre orientation θf((xi,xj), 0)exhibits the same initial orientation and magnitude, as
shown in Fig. 9c.
Finally, for the non-fibre ECM component, we consider both a homogeneous and
a heterogeneous scenario, which will be detailed below.
4.1 Homogeneous Non-fibre ECM Component
The initial distribution of the non-fibre ECM component, l(x,0), will be in the first
instance taken as the homogenous distribution, namely as l(x,0)=min{0.5,1
c(x,0)}. The initial conditions of the cell population c(x,0)given in (41), the full
ECM density v(x,0)=l(x,0)+F(x,0), and the resulting initial fibre orientations
θf(x,0)can be seen in Fig. 9. The adhesive strength coefficients for cell–cell adhesion,
cell–fibre adhesion and cell–non-fibre ECM adhesion, are taken here to be
Smax =0.5,ScF =0.1 and Scl =0.01,(42)
respectively.
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Fig. 9 Initial conditions showing the distribution of cancer cells (a), the homogeneous density of ECM (b)
with the invasive boundary of the tumour represented by the white contour, and the initial macroscopic fibre
orientations per each micro-domain represented by a vector field (c). These vectors have been magnified
from the usual size of the domain for better representation
Using the parameter set Σfrom Appendix Fand adhesion coefficients (42), in
Fig. 10 we show the computational results at macro–micro stage 20Δtfor the evo-
lution of: the cancer cell population in Fig. 10a; the full ECM density in Fig. 10b;
the macro-scale fibre magnitude in Fig. 10c; the vector field of oriented fibres at two
different resolutions, namely coarsened twice and coarsened fourfold in Fig.10d and
f, respectively; and a 3D plot of the macroscopic oriented fibres in Fig. 10e.
Comparing with the initial distributions of cancer cells and ECM displayed in Fig. 9,
the main body of the tumour is increasing in size, while decreasing in overall density,
spreading the initial distribution outwards and creating a plateau of cancer cells, as
shown in Fig. 10a. While in the absence of fibres the boundary of the tumour was
expanding isotropically in the case of homogenous ECM, as showed in Trucu et al.
(2013) and Shuttleworth and Trucu (2018), a different situation we witness here in the
case of homogeneous non-fibre ECM as the presence of the oriented fibres phase of
ECM is now taken into consideration. Specifically, the cancer cell invasion becomes
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Multiscale Modelling of Fibres Dynamics and Cell Adhesion…
Fig. 10 Simulations at stage 20Δtwith a homogenous distribution of the non-fibre ECM component
anisotropic, leading to lobular patterns and having the fibres reaching outwards in the
boundary regions of faster tumour progression. This behaviour is clarified by the fibre
vector plot Fig. 10d where the orientations of the redistributed fibres can be seen to
point in the direction of this lobule on the invasive edge. The orientation of the fibres
is strongly affected during their rearrangement, with their behaviour dependent on the
initial macroscopic density of fibres and the spatial flux of the cancer cells. This flux
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carries a higher weight than the distribution of fibres and thus the cells ultimately have
governance over the direction of realignment. Finally, Alongside the fibre realignment,
the cancer cells also degrade the fibres, this leading to a low-density central region of
fibres Fig. 10c.
As the simulation continues to stage 40Δt, the initial main body of the tumour
(consisting of a high-density region of cells in the centre of Y) is spreading out,
following the initial orientation of the fibres, giving rise to lobular progression pattern
for the cell population in this direction, as shown in Fig. 11a. The boundary of the
tumour has undergone minor changes with respect to stage 20Δtshown in Fig. 10 ,
the main tumour dynamics occurring mainly on the central cluster of cells. The non-
fibrous part of the ECM is further degraded under the presence of cancer cells Fig. 11b,
and the fibres are being pushed to the boundary of the tumour Fig. 11c, creating a larger
region of low-density ECM.
4.2 Heterogeneous Non-fibrous ECM Component
We now introduce an initially heterogeneous non-fibre ECM component. While main-
taining the same initial conditions for c(x,0)specified in (41) as well as for the initial
distributions of ECM micro-fibres (illustrated in Fig. 3), the heterogeneity of the non-
fibre ECM phase will be structured in a similar manner to Domschke et al. (2014) and
Shuttleworth and Trucu (2018) using the initial condition
l(x,0)=min {h(x1,x2), 1c(x,0)},(43)
where
h(x1,x2)=1
2+1
4sinx1x2)3·sinx2
x1
),
(x1,x2)=1
3(x+1.5)∈[0,1]2for xD=7π.
These initial conditions can be seen in Fig. 12.
Computational results at stage 20Δtare shown in Fig. 13, using the initial conditions
shown in Fig. 12 and the parameter set Σwith the adhesive terms (42). Due to the
initial distribution of the non-fibrous component of the matrix, there are patches of
high and low-density areas, and regions of high tumour density correspond to the
areas of high degradation of fibres and the surrounding non-fibre ECM Fig. 13b. The
proliferating edge of the tumour is expanding in a lobular fashion, reaching out to the
high-density patches and encasing the low-density regions in the process, as the higher
ECM density equates to increased opportunity for cell adhesion. This is the natural
direction in which the tumour cells try to invade, pushing out from its centre and into
the surrounding matrix, and causing the tumour to encircle itself with a region of higher
magnitude fibres, as shown in Fig. 13c–f. The macroscopic orientation of the fibres is
refashioned in Fig. 13d as the cancer cells have rearranged and degraded the fibres,
leading to significant changes in the fibre orientations and magnitude patterns near
the boundary of the tumour with respect to their initial state, and causing them both
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Multiscale Modelling of Fibres Dynamics and Cell Adhesion…
Fig. 11 Simulations at stage 40Δtwith a homogenous distribution of the non-fibrous part of the matrix
to increase their magnitude and to point generally towards the fast invading regions
of the cancer boundary. While the fibre is being pushed and rearranged by the cancer
cells outwards, away from the main body of the tumour, in regions of high cancer
density, these are degraded, as evidenced by the low distribution of fibres in the centre
of the tumour, presented in Fig. 13c–f.
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Fig. 12 Initial conditions showing the distribution of cancer cells (a), the heterogeneous density of ECM (b)
with the invasive boundary of the tumour represented by the white contour, and the initial macroscopic fibre
orientations per each micro-domain represented by a vector field (c). These vectors have been magnified
from the usual size of the domain for better representation
Figure 14 illustrates simulations plotted at stage 40Δt. The main body of the tumour
is beginning to form new high distribution regions within the highly degraded patch
of ECM, as shown in Fig. 14a, b. This build up of cells is due to increasingly higher
magnitudes for rearranged fibres with invasion favourable orientations, which result
into a significantly higher effect of cell–fibre adhesion leading to increased transport
of cells towards those areas. Islands are forming within the boundary of the tumour
away from the primary tumour mass due to low ECM density in those regions, which
result in weak levels of both cell–non-fibre ECM and cell–fibre adhesion, and as a
consequence the cells take longer time to advance upon these regions. As shown in
Fig. 14c–f, the fibres persevere in surrounding the tumour, with their oriented fibres
on the central part of the tumour (corresponding to regions of very high cancer cell
density) continuing to be strongly degraded and dominated in their direction by the
flow F. Again, high-density regions of ECM fibres equate to more opportunities for
cell–fibre adhesion, thus creating a preferential direction of invasion. The cancer cells
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Multiscale Modelling of Fibres Dynamics and Cell Adhesion…
Fig. 13 Simulations at stage 20Δtwith a heterogeneous distribution of the non-fibrous part of the matrix
are rearranging the fibres to follow this direction, allowing them an easier route of
invasion. As shown in Fig. 14c, by the gathering of fibre distributions away from the
tumour centre, it is evident that the cancer cells are pushing the fibres outwards to the
boundary of the tumour and in the direction of the invasion front, as found also in
Pinner and Sahai (2008).
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Fig. 14 Simulations at stage 40Δtwith a heterogeneous distribution of the non-fibrous part of the matrix
4.3 Increased Cell–Fibre Adhesion Within the Heterogeneous Non-fibre ECM
Phase Scenario
As we explore the effect of the heterogeneous two phase ECM, it is important to con-
sider the relation between the tumour progression and increased cell–fibre adhesion.
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Multiscale Modelling of Fibres Dynamics and Cell Adhesion…
By increasing cell–fibre adhesion, we expect the cancer cells to advance further into
their surrounding environment. For that, we double the cell–fibre adhesion, taking
ScF =0.2, while maintaining the same initial conditions (i.e. for cancer cell popu-
lation, those given in (41); for fibre ECM phase, those defined in (5) and illustrated
in Fig. 3; and for heterogeneous non-fibrous ECM, those given in (43)) along with
the parameters defined in the parameter set Σand the cell–cell and cell–ECM–non-
fibres adhesion as stated in (42). The computational results shown for this situation in
Figs. 15 and 16 show that the fibres now have more influence over the route of invasion
and its pattern, presenting a leading edge with a highly lobular structure. There are
small islands present in high-density regions of the ECM, suggesting the density is
simply too high and first requires more degradation before the cells can completely
engulf this area. The overall degradation of the matrix has remained centralised to the
central part of the tumour as before. Similar to the previous cases considered here,
the cancer cells have rearranged the fibres also in this situation, strongly degrading
the fibres in the regions with very high cancer density Fig. 15c, while the magnitude
and orientation of the fibres situated closer to the tumour periphery have been altered
and are positioned similar to the case where cell–fibre adhesion ScF =0.1, pointing
clearly outwards in boundary regions of increased invasive behaviour.
Figure 16 displays computations at stage 40Δt. An important difference between
Figs. 15a and 16a is observed within the main body of the tumour. When the cell–fibre
adhesion coefficient is increased, the central part of the tumour has an overall higher
distribution, and is being pulled in different directions, as illustrated by the three areas
of increased cell distribution. Shown in Fig. 16d, the fibre orientations have been
realigned, and in boundary regions of faster invasion the cumulative fibres direction
tends to become almost perpendicular to the fibres from the peritumoural region.
The cancer cells attempt to align the fibres with their own directional preference, i.e.
outwards from the centre and towards the higher density regions of ECM where they
have increased opportunity for adherence. The cells continue to pursue this goal, as
evidenced in Fig. 16b where we see the leading edge advancing on the higher density
areas of ECM. From our simulations, we noted that an increase in cell–fibre adhesion
causes a larger overall invasion of the tumour, suggesting that the fibres presence plays
an important role in the invasion of cancer.
5 Conclusion
We have presented a novel multi-scale moving boundary model which builds on pre-
vious framework first developed in Trucu et al. (2013). This multiscale model is
developed to explore the adhesive dynamics of a cancer cell population within a two-
phase heterogeneous ECM and its impact over the overall invasion pattern during
cancer growth and spread within the surrounding human body tissue. The ECM is
considered here as being a mixture of two constitutive phases, namely a fibre and non-
fibre phase. We pay a special attention to the fibre phase, whose multiscale dynamics
is explored and modelled in an integrated two-scale spatio-temporal fashion, with
the cell-scale micro-dynamics being connected to the tissue-scale tumour dynamic
through an emerging double feedback loop. To that end, we developed a novel multi-
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Fig. 15 Simulations at stage 20Δtwith a heterogeneous distribution of the non-fibrous part of the matrix
and increased cell–fibre adhesion
scale model that explores on the one hand the way the fibre micro-dynamics translates
into the macro-scale level fibre dynamics (by providing on-the-fly at tissue-scale a
spatially-distributed vector field of oriented macroscopic fibre that have direct influ-
ence within the tumour progression) and, on the other hand, the way in which the
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Multiscale Modelling of Fibres Dynamics and Cell Adhesion…
Fig. 16 Simulations at stage 40Δtwith a heterogeneous distribution of the non-fibrous part of the matrix
and increased cell–fibre adhesion
tissue-scale cancer cell population dynamics causes not only fibres degradation at
macro-scale but also fibres rearrangement at micro-scale. Finally, the new multiscale
model is embedded within the multiscale moving boundary framework exploring the
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leading edge proteolytic activity of matrix-degrading enzymes introduced in Trucu
et al. (2013). Thus, we ultimately obtain a novel multi-scale modelling framework that
combines two multiscale sub-systems that contribute to and share the same macro-
dynamics, but have separate micro-scale processes that are simultaneously connected
to the macro-dynamics through two independent feedback loops, with one of them
addressing the cell-scale activity involved in the rearrangement of micro-fibres within
the bulk of the tumour, and the second one exploring the proteolytic activity within a
cell-scale neighbourhood of the tumour boundary.
At the tissue scale, in order to explore the influence of the ECM fibre phase within
the tissue-scale dynamics, besides the usual adhesion terms considered in Domschke
et al. (2014) and Gerisch and Chaplain (2008) concerning cell–cell and cell–ECM–
non-fibre adhesion, we derived and introduced a new non-local term in the macroscopic
equation (23a) for tumour cell population that accounts for the cell–fibres adhesion.
This new term explores the critical influence that the macroscopic fibre vector field has
over the direction of cellular adhesion in the macro-dynamics. Moreover, as this vector
field of oriented ECM fibres is induced from the micro-scale distribution of micro-
fibres, a novel bottom-up feedback link between cell- and tissue-scale dynamics has
this way been identified and explored mathematically.
Further, while the cancer cells degrade both the non-fibre ECM and the fibre ECM
components at macro-scale, it was important to observe that the of flux cancer cell F
given in (20) causes the rearrangement of the micro-fibres at micro-scale. To under-
stand this, at any given macro-scale position xΩ(t)we explored this macro–micro
interacting link on appropriately small cubic micro-domains centred at x, namely
δY(x), where the distribution of the micro-fibres f(z,t)(with zδY(x)) induces
naturally the fibre magnitude F(x,t)and orientation θf(x,t), and whose rigorous
derivation and well-posedness are ensured in Sect. 2.3. Furthermore, while getting
balanced by the initial macro-scale orientation of the existing fibres (induced from the
distribution of the microfibers on δY(x), the macro-scale spatial flux Facts uniformly
on the existing distribution of micro-fibres on any micro-domain δY(x), causing the
micro-fibres initially distributed on δY(x)to be redistributed and rearranged in the
resulting fibres relocation direction given in (22). This fibres relocation direction was
obtained as the contribution of the flux F(x,t)and the fibre vector field θf(x,t)that
are weighted in accordance with the amount of cancer cells transported at (x,t)and
the magnitude of fibre that they meet at (x,t), respectively. Finally, this relocation is
accomplished to the extent in which the local microscopic conditions permit, these
being explored here through an appropriately defined movement probability. This way,
a top down link was established between the macro-dynamics and the dynamics fibres
rearrangement at micro-scale.
To address this new multiscale modelling platform computationally, we extended
significantly the computational framework introduced in Trucu et al. (2013) by bring-
ing in the implementation of the interlinked two-scale fibre dynamics. To that end,
besides the computational approach based on barycentric interpolation that the micro-
scale fibres relocation process has required, the macro-solver needed several extension
to accommodate the new modelling. To that end, alongside the formulation of a new
approach to computing on the moving tumour domain, we proposed a new off-grid
barycentric interpolation approach to calculate the new adhesion term, and finally
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Multiscale Modelling of Fibres Dynamics and Cell Adhesion…
we developed a novel non-local predictor–corrector numerical scheme to address the
challenging macro-scale computational conditions created through the presence of a
multiphase ECM that crucially includes the multiscale dynamics of the oriented fibres.
Using this multiscale computational approach for the proposed model, we were able
to simulate the multiscale nature of cancer invasion by exploring the link between the
macroscopic spatial distribution and orientation of cancer cells and the matrix, and
the microscopic rearrangement of fibres and micro-dynamics of MDEs that occur on
the proliferating edge of the tumour. Overall, we considered the invasion of a cancer
cell population within both homogenous and heterogeneous non-fibrous ECM phase,
investigating the macro-scale dynamics of the cancer population and macroscopic
densities of the ECM components, while considering their influence on both the micro-
scale MDEs molecular dynamics occurring at the cell-scale along the invasive edge and
also the microscopic fibre movement occurring within the boundary of the tumour.
Finally, it is worth remarking at this stage that even in the homogeneous non-fibre
ECM, the ECM as a whole will not be homogeneous, due to the presence of the
oriented fibres that already lead to a constitutive heterogeneous ECM.
The simulations presented in this paper have some similarities with previous work.
We note a general lobular, fingering pattern for the progression of tumour boundary,
aspect that was observed also in Peng et al. (2017) and Trucu et al. (2013)inthe
case of heterogeneous ECM. There is a noticeable increase in this behaviour when the
coefficient of cell–fibre adhesion is increased, suggesting the microscopic fibres play
a key role in the invasion process, aiding in the local progression of the tumour. It is
shown throughout all simulations that, while being degraded by the cancer cells, the
fibres are being pushed outwards from the centre of the domain towards the boundary
(Pinner and Sahai 2008). This behaviour is known for amoeboid cell types, and par-
ticularly occurs in a loose/soft matrix (Krakhmal et al. 2015), which is reminiscent of
our model as the cancer cells flux rearranges the fibres continuously at micro-scale.
We can conclude from our simulations that a heterogeneous ECM non-fibrous phase
permits for an increase in tumour progression compared to an initial homogeneous
distribution. It is clear that the ECM fibres play an important role during invasion, with
an increase in cell–fibre adhesion displaying a larger overall region of invasion. This is
in line with recent biological experiments that suggest the organisation of fibronectin
fibrils promotes directional cancer migration (Erdogan et al. 2017).
Looking forward, this modelling framework enables the opportunity for addressing
questions in a range of directions, such as: accounting for cellular reactions within the
proteolytic micro-scale dynamics involving the fibres, i.e. the chopping/degradation of
the fibres by matrix metallo-proteinases (MMPs); exploring the process of anchoring
of collagen to cells and other components in the ECM; as well as exploring the presence
of a second cancer cell subpopulation. Further work will focus on both fibronectin and
collagen, and their roles and functions, ultimately aiming to gain a better understanding
of the full impact that these have within tumour invasion.
Acknowledgements RS and DT would like to acknowledge the support received through the EPSRC DTA
Grant EP/M508019/1 on the project: Multiscale modelling of cancer invasion: the role of matrix-degrading
enzymes and cell adhesion in tumour progression.
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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna-
tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons license, and indicate if changes were made.
Appendix
A Sensitivity Analysis
To determine the optimal percentage of initial fibres and address sensitivity to the initial
conditions of the model, we present and discuss the results of three different initial
fibre distributions under both a homogeneous and heterogeneous non-fibre ECM.
Considering first a homogeneous non-fibre-ECM phase, l(0,t)=0.5, we construct
the initial distributions of fibres by varying the percentage of the mean density of the
non-fibrous ECM phase, namely p=0.05,0.15,0.2. As we increase the percentage
of the non-fibre ECM phase, the fibrous part of the matrix has a progressively larger
influence on cell movement. This behaviour is expected due to their involvement within
the cellular adhesion process, with increased fibre distribution causing a higher affinity
for cell–fibre adhesion and thus fibre-directed invasion. We observe that main body of
the tumour remains almost symmetrical in the presence of a low initial fibre distribution
p=0.05, Fig. 17a. When the percentage is increased to p=0.15, the boundary of
the tumour has increased in size with the body of the tumour drawing away from the
boundary, Fig. 17b. The main body of the tumour is no longer symmetrical and has
a region of increased density in the direction of initial fibres, Fig. 9c. Continuing on,
Fig. 17c displays results in which the initial fibre distribution is increased to p=0.2.
Here, the main body of the tumour has changed in shape, elongated in the direction
of fibres and showing several regions of high cell distribution. The boundary of the
tumour has also progressed and has become irregular in shape with small protrusions
forming. In the presence of a homogeneous non-fibre ECM phase, the differences in
the invading boundaries are clearly visible, ranging from a symmetric leading edge in
low initial fibre distribution to a fingering boundary in high initial fibre distribution.
Finally, to complement these simulations, we investigate the same cases of initial
fibre distributions as above coupled with a heterogeneous non-fibre ECM phase defined
in (43). Figure 18 displays computations at stage 40Δtof both the cancer cell popu-
lation and macroscopic fibre distribution. Similar to the presence of a homogeneous
non-fibre ECM phase, as the initial fibre distribution is increased, we see the forma-
tion of an overall larger tumour, particularly in the case when p=0.2, Fig. 18c. The
tumour boundary is covering a significantly larger area and exhibiting a highly lobular
pattern on the leading edge. This behaviour is due to both the heterogeneous pattern
of non-fibre ECM and increased initial fibre distributions. Overall, these simulations
conclude that a higher initial distribution of fibres coupled with either a homogeneous
or heterogeneous non-fibre ECM phase results in a faster spreading tumour.
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Fig. 17 Simulation results using different percentages of the mean density value of the homogeneous
non-fibres ECM phase, ap=0.05, bp=0.15 and cp=0.2, showing cancer cell distributions and
macroscopic fibre distributions at stage 40Δt
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Fig. 18 Simulation results using different percentages of the mean density value of the heterogeneous
non-fibres ECM phase, ap=0.05, bp=0.15 and cp=0.2, showing cancer cell distributions and
macroscopic fibre distributions at stage 40Δt
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Multiscale Modelling of Fibres Dynamics and Cell Adhesion…
B Paths for Fibres
As described in Sect. 2, on the microscopic domains δY(x), we consider a fibre distri-
bution given by a combination of five distinctive micro-fibres patterns that are defined
along the following smooth paths {hj}jJ:
h1:z1=z2;h2:z1=1
2;h3:z1=1
5;h4:z2=2
5;and h5:z2=4
5.
C The Mollifier Ã
The standard mollifier ψγ:RNR+(which was used also in Trucu et al. (2013))
is defined as usual, namely
ψγ(x):= 1
γNψx
γ,
where ψis the smooth compact support function given by
ψ(x):=
exp 1
x2
21
B(0,1)
exp 1
z2
21dz,if xB(0,1),
0,if x/B(0,1)
D The Radial Kernel K(·)
To explore the influence on adhesion-driven migration decreases as the distance from
x+yto xwithin the sensing region B(x,r)increases, the expression of the radial
dependent spatial kernel K(·)appearing in (15) is taken here to be:
K(r):= 2πR2
31r
2R.(44)
E Key Aspects Within the Multiscale Moving-Boundary Framework
For completeness, in the following we will briefly describe two key ingredients of the
multiscale moving boundary framework introduced in Trucu et al. (2013), namely: (1)
the computationally feasible boundary tracking; and (2) the direction and magnitude
induced by the micro-scale for the tumour boundary relocation at macro-scale.
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Fig. 19 Schematic diagrams showing in athe spatial cubic region Ycentred at the origin in R3. The solid
red lines represent the family of macroscopic Ycubes placed on the boundary of the tumour ∂Ω(t0),and
the pale green region represents the mass of cancer cells Ω(t0). The appropriately chosen bundle {Y}YP
of micro-domains introduced in Trucu et al. (2013) that covers the boundary ∂Ω(t0)are shown in red and
the corresponding half-way shifted cubes are illustrated by blue dashed lines (Color figure online)
E.1 The Computationally Feasible Boundary Tracking
Following a series of topological constraints as detailed in Trucu et al. (2013), at any
given time t0, for the maximal tissue cube YRN(which includes the growing
tumour Ω(t0)) we appropriately select the coarsest uniform decomposition into a
union of dyadic cubes Ywith the property that from this dyadic decomposition of Y
taken together with all associated families of half-way shifted dyadic cubes (as defined
in Trucu et al. (2013)), a subfamily of overlapping cubes {Y}YPcan be extracted
with the following two key characteristics:
(1) each cube YPconvey a neighbourhood for Y∂Ω(t0)with both the part
inside the tumour YΩ(t0)and the