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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 Y∈RN, 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. 2.2–2.3.

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}Y∈P(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 z∈Y∩Ω(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)) ,z∈Y∩Ω(t0),

2.gY(z,τ)=0,z∈Y\Ω(t0)+{z∈Y|||z||2<ζ}),

(2)

where B(z,γ) := {ξ∈Y|z−ξ∞≤ζ}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), z∈Y,τ∈[0,Δt],(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 x∈Y, 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 x∈Y, 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),x∈Ω(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):=

j∈J

ψhj(z)(χ(δ−2γ)Y(x)∗ψγ)(z)(4)

where {ψhj}j∈Jare 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}j∈Jin δY(x)that are given in

Appendix B. Finally, χ(δ−γ)Y(x)(·)represents the characteristic function of the cubic

micro-domain (δ −γ)Y(x)centred at xand of size (δ −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 x∈Ω(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}j∈Jnof size δ2−nfor the micro-

domain δY(x), let us denote by zjthe barycentre of each dyadic cube Dj. Then, for any

j∈Jn, 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:= zj−xin 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}j∈Jnis given by:

θn

f,δY(x)(x,t):=

j∈JnDjf(ζ,t)dζ

j∈Jn

Dj

f(ζ,t)dζ

−→

xzj

=

j∈Jn

Dj

f(ζ,t)dζ

δY(x)

f(ζ,t)dζ

−→

xzj

=

j∈Jn⎛

⎝1

λ(Dj)

Dj

f(ζ,t)dζ⎞

⎠λ(Dj)−−→

xzj

δY(x)

f(ζ,t)dζ

=

δY(x)⎡

⎣

j∈Jn⎛

⎝1

λ(Dj)

Dj

f(ζ,t)dζ⎞

⎠χDj(z)−−→

xzj⎤

⎦dz

δY(x)

f(ζ,t)dζ

=

δY(x)⎡

⎣

j∈Jn⎛

⎝1

λ(Dj)

Dj

f(ζ,t)dζ⎞

⎠χDj(z)(zj−x)⎤

⎦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 n∈N∗denoting 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):=

j∈Jn1

λ(Dj)

Dj

f(ζ, t)dζχDj(z)(zj−x), (7)

let’s observe immediately that {gn}n∈N∗is actually a sequence of vector-valued simple

functions that is convergent to f(z,t)(z−x)and that its associated sequence of

integrals converges to the Bochner Integral of f(z,t)(z−x)on δY(x)with respect to

λ(·)(Yosida 1980), namely

δY(x)

f(z,t)(z−x)dz:= lim

n→∞

δY(x)

gn(z)dz.(8)

Hence, from (6)–(8), we obtain that the sequence of revolving {θn

f(x,t)}n∈N∗ﬁbres

orientations associated to the entire family of dyadic decompositions {{Dj}j∈Jn}n∈N∗

is convergent to the unique revolving barycentral micro-ﬁbres orientation on δY(x)

denoted by θf,δY(x)(x,t)and given by

θf,δY(x)(x,t):= lim

n→∞ θn

f,δY(x)(x,t)

=lim

n→∞

δY(x)⎡

⎣

j∈Jn⎛

⎝1

λ(Dj)

Dj

f(ζ,t)dζ⎞

⎠(zj−x)χDj(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)(z−x)dz

δY(x)

f(ζ,t)dζ

=

δY(x)

f(z,t)(z−x)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→ z−x∈RNwith 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·θf,δY(x)(x,t)

θf,δY(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·θf,δY(x)(x,t)

θf,δY(x)(x,t)2

2

=1

λ(δY(x))

δY(x)

f(z,t)dz·θf,δY(x)(x,t)2

θf,δY(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

θf,δY(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=∇·[D1∇c−cA(t,x,u(·,t), θ f(·,t))]+μ1c(1−ρ(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))

+ˆn(y)ScF F(x+y,t)(1−ρ(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 y∈B(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 e1−1

1−(1−l(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(1−ρ(u))+(19)

where γ2describes the rate of degradation and ωdenotes the rate of remodelling, while

the volume ﬁlling term (1−ρ(u))+:= max(0,(1−ρ(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 x∈Ω(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)=D1∇c(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

r(δY(x), t):= ω(x,t)F(x,t)+(1−ω(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 z∈δY(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 r(δY(x), t)that

is given simply by

||r(δY(x)) −xdir(z)||2.

Therefore, under the action of the rearrangement ﬂux r(δY(x), t), the micro-ﬁbres

distributed at zwill attempt to exercise their movement in the direction of the resulting

vector xdir(z)+r(δY(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)+r(δY(x), t))·f(z,t)( fmax −f(z,t))

f∗+||r(δY(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}:={z∈δY(x)|f(z,t)>0}. Finally, the movement of the

micro-ﬁbres distributed at zto the newly attempted location z∗given 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 (1−pmove)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=∇·[D1∇c−cA(t,x,u(·,t), θ f(·,t))]+μ1c(1−ρ(u)), (23a)

dF

dt=−γ1cF,(23b)

dl

dt=−γ2cl +ω(1−ρ(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,...,M−1}×{2,...,M−1}→{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 I−1

x,−1({1})∪I−1

x,+1({1})and I−1

y,−1({1})∪I−1

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,j−1

2

:= ci,j−[Ix,−1(i,j)ci,j+I(i,j−1)ci,j−1]

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

i−1

2,j:= ci,j−[Iy,−1(i,j)ci,j+I(i−1,j)ci−1,j]

dy .

(28)

Therefore, the approximation for the term ∇·[D1∇c−cA(t,x,u(·,t), θ f(·,t))]in

(23a) is obtained as

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Multiscale Modelling of Fibres Dynamics and Cell Adhesion…

(∇·[D1∇c−cA(t,x,u(·,t), θ f(·,t))])p

i,j

D1([cx]p

i,j+1

2

−[cx]p

i,j−1

2

)−cp

i,j+1

2

·Ap

i,j+1

2

+cp

i,j−1

2

·Ap

i,j−1

2

Δx

+

D1([cy]p

i+1

2,j−[cy]p

i−1

2,j)−cp

i+1

2,j·Ap

i+1

2,j+cp

i−1

2,j·Ap

i−1

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:= ∇ · [ D1∇c−

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,j−1

2

Δx+

Fp

i+1

2,j−Fp

i−1

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,j−cp

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+(i−1)annulus radial sectors S1,...,Sq,

which are obtained by intersecting each annulus i∈{1,...,s}annuli with a corre-

sponding number of 2m+(i−1)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(ξ, tp)dξ, Wp

Sν,l:= 1

λ(Sν)

Sν

l(ξ, tp)dξ,

Wp

Sν,F:= 1

λ(Sν)

Sk

F(ξ, tp)dξ, and Wp

Sν,θ f:= 1

λ(Sν)

Sν

θf(ξ, 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

Sν,θ f

nν+Wp

Sν,θ f2

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

bSν∈Ω(t0)

K(bSν)[nν·(SccWp

Sν,c

+Scl Wp

Sν,l)+np

ν·ScFWp

Sν,F](1−ρ(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(1−ρ(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,xj−1), (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

F∗p+1

2

i,j=1

card(N)

(σ,ζ )∈N

˜

Fp+1

2

σ,ζ ,(35)

123

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(F∗p+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,j=¯cp+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

F∗p+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,j=¯cp+1

i,j+δt

4HFp+1

2

i,j,¯cp+1

2

i,j,¯

up+1

2

i,j+HF∗p+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),0.5−γ) ∗ψγ,

(41)

where ψγis the standard molliﬁer detailed in Appendix Cthat acts within a radius

γ<<Δx

3from ∂B((2,2), 0.5−γ) to smooth out the characteristic function

χB((2,2),0.5−γ). 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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R. Shuttleworth, D. Trucu

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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R. Shuttleworth, D. Trucu

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), 1−c(x,0)},(43)

where

h(x1,x2)=1

2+1

4sin(ζ x1x2)3·sin(ζ x2

x1

),

(x1,x2)=1

3(x+1.5)∈[0,1]2for x∈D,ζ=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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R. Shuttleworth, D. Trucu

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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R. Shuttleworth, D. Trucu

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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R. Shuttleworth, D. Trucu

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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R. Shuttleworth, D. Trucu

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 x∈Ω(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 z∈δY(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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Multiscale Modelling of Fibres Dynamics and Cell Adhesion…

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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R. Shuttleworth, D. Trucu

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}j∈J:

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:RN→R+(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

2−1

B(0,1)

exp 1

z2

2−1dz,if x∈B(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

31−r

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}Y∈P

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 Y⊂RN(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}Y∈Pcan be extracted

with the following two key characteristics:

(1) each cube Y∈Pconvey a neighbourhood for Y∩∂Ω(t0)with both the part

inside the tumour Y∩Ω(t0)and the