Szabolcs Suveges
email: assuveges@dundee.ac.uk

Kismet Hossain-Ibrahim

Douglas Steele
email: d.steele@dundee.ac.uk

Raluca Eftimie
email: raluca.eftimie@univ-fcomte.fr

Dumitru Trucu
email: btrucu@maths.dundee.ac.uk

J Douglas Steele

Mathematical Modelling of Glioblastomas Invasion within the Brain: A 3D Multi-Scale Moving-Boundary Approach

Keywords: Cancer invasion, Cell adhesion, Multi-scale modelling, 3D computational modelling, T1 weighted image, DTI, Glioblastoma

published or not. The documents may come

Introduction

Glioblastoma multiforme is a highly invasive and aggressive type of brain tumour, typically with poor patient prognosis [START_REF] Burri | The Evolving Role of Tumor Treating Fields in Managing Glioblastoma[END_REF][START_REF] Davis | Glioblastoma: Overview of Disease and Treatment[END_REF][START_REF] Klopfenstein | Cell lines and immune classification of glioblastoma define patient[END_REF][4][START_REF] Meneceur | Establishment and Characterisation of Heterotopic Patient-Derived Xenografts for Glioblastoma[END_REF][START_REF] Preusser | Current concepts and management of glioblastoma[END_REF][START_REF] Sottoriva | Intratumor heterogeneity in human glioblastoma reflects cancer evolutionary dynamics[END_REF] (median survival rate is less than 1 year [? ]). These tumours arise from abnormal glial cells located in the central nervous system, and shortly after their appearance they invade the surrounding tissues in a heterogeneous fashion. This heterogeneous invasion pattern leads to tumours whose outer edges are difficult or impossible to determine with current imaging technologies, including for instance magnetic resonance imaging (MRI) and diffusion tensor imaging (DTI), both of which measure the diffusion of water molecules and enable the study of brain structures.

Due to the limited experimental approaches that one can use to study the brain, researchers have started using mathematical models to provide certain biological insights that otherwise would be difficult to obtain experimentally. Such models can help predict how tumours grow for specific patients, aiding clinicians in decision-making, or they can help test and provide new hypotheses about potential anti-tumour treatments. Mathematical modelling of tumours has seen significant advances over the last few decades, which broadened our understanding of tumour dynamics and how cells interact with their environment . Although the majority of these models do not restrict themselves to a specific tumour type and rather focus on general tumours, there are some models that focus on the evolution of gliomas within the brain [START_REF] Alfonso | Why one-size-fits-all vasomodulatory interventions fail to control glioma invasion: in silico insights[END_REF][START_REF] Engwer | Glioma follow white matter tracts: a multiscale DTI-based model[END_REF][START_REF] Hunt | A Multiscale Modeling Approach to Glioma Invasion with Therapy[END_REF][START_REF] Painter | Mathematical modelling of glioma growth: The use of Diffusion Tensor Imaging (DTI) data to predict the anisotropic pathways of cancer invasion[END_REF][START_REF] Scribner | Effects of Anti-Angiogenesis on Glioblastoma Growth and Migration: Model to Clinical Predictions[END_REF][START_REF] Swanson | A quantitative model for differential motility of gliomas in grey and white matter[END_REF][START_REF] Swanson | A mathematical modelling tool for predicting survival of individual patients following resection of glioblastoma: a proof of principle[END_REF][START_REF] Swanson | Quantifying the Role of Angiogenesis in Malignant Progression of Gliomas: In[END_REF][START_REF] Syková | Diffusion in Brain Extracellular Space[END_REF]. Recently, models also started to incorporate the structure of the brain, by including MRI and DTI scans [START_REF] Engwer | Glioma follow white matter tracts: a multiscale DTI-based model[END_REF][START_REF] Hunt | A Multiscale Modeling Approach to Glioma Invasion with Therapy[END_REF][START_REF] Painter | Mathematical modelling of glioma growth: The use of Diffusion Tensor Imaging (DTI) data to predict the anisotropic pathways of cancer invasion[END_REF][START_REF] Clatz | Realistic simulation of the 3-D growth of brain tumors in MR images coupling diffusion with biomechanical deformation[END_REF][START_REF] Cobzas | Tumor Invasion Margin on the Riemannian Space of Brain Fibers[END_REF][START_REF] Jbabdi | Simulation of anisotropic growth of low-grade gliomas using diffusion tensor imaging[END_REF][START_REF] Konukoglu | Extrapolating glioma invasion margin in brain magnetic resonance images: Suggesting new irradiation margins[END_REF]. Even though these images are generated in 3D, most of these models are simulating the tumour growth in 2D and only a few of them are 3D models [START_REF] Clatz | Realistic simulation of the 3-D growth of brain tumors in MR images coupling diffusion with biomechanical deformation[END_REF][START_REF] Suarez | Mathematical Modeling of Human Glioma Growth Based on Brain Topological Structures: Study of Two Clinical Cases[END_REF][START_REF] Yan | 3D Mathematical Modeling of Glioblastoma Suggests That Transdifferentiated Vascular Endothelial Cells Mediate Resistance to Current Standardof[END_REF]. Moreover, the majority of published models focus on tumour progression only on one spatio-temporal scale. However, tumour progression is characterised by various biological processes occurring on different scales, and thus their effects on the overall tumour dynamics cannot be neglected. Hence, recent efforts have been made to establish new multi-scale frameworks for tumour progression [START_REF] Shuttleworth | Multiscale Modelling of Fibres Dynamics and Cell Adhesion within Moving Boundary Cancer Invasion[END_REF][START_REF] Shuttleworth | Multiscale dynamics of a heterotypic cancer cell population within a fibrous extracellular matrix[END_REF][START_REF] Shuttleworth | Cell-Scale Degradation of Peritumoural Extracellular Matrix Fibre Network and Its Role Within Tissue-Scale Cancer Invasion[END_REF][START_REF] Suveges | Directionality of Macrophages Movement in Tumour Invasion: A Multiscale Moving-Boundary Approach[END_REF][START_REF] Suveges | Re-polarisation of macrophages within a multi-scale moving boundary tumour invasion model[END_REF][START_REF] Engwer | Glioma follow white matter tracts: a multiscale DTI-based model[END_REF][START_REF] Hunt | A Multiscale Modeling Approach to Glioma Invasion with Therapy[END_REF][START_REF] Painter | Mathematical modelling of glioma growth: The use of Diffusion Tensor Imaging (DTI) data to predict the anisotropic pathways of cancer invasion[END_REF][START_REF] Peng | A multiscale mathematical model of tumour invasive growth[END_REF], which were able to capture some of these multi-scale underlying biological processes usually involving the extracellular matrix (ECM).

In this paper, we extend the general 2D multi-scale moving-boundary modelling framework introduced in [START_REF] Trucu | A Multiscale Moving Boundary Model Arising In Cancer Invasion[END_REF][START_REF] Shuttleworth | Multiscale Modelling of Fibres Dynamics and Cell Adhesion within Moving Boundary Cancer Invasion[END_REF] to capture the invasion of glioblastomas within a 3D fibrous brain environment. To this end, we incorporate the information provided by both the T1 weighted and DTI scans into our multi-scale framework and use the resulting model to simulate numerically the growth of 3D gliomas within the brain. We focus on a few cases showing tumour growth in different regions in the brain, with different distributions of grey/white matter densities, which leads to different tumour invasion patterns.

The model presented in this paper is a first to advance the multi-scale moving boundary modelling of 3D brain gliomas in two key ways compared to previous ones, see for instance [START_REF] Engwer | Glioma follow white matter tracts: a multiscale DTI-based model[END_REF][START_REF] Hunt | A Multiscale Modeling Approach to Glioma Invasion with Therapy[END_REF][START_REF] Painter | Mathematical modelling of glioma growth: The use of Diffusion Tensor Imaging (DTI) data to predict the anisotropic pathways of cancer invasion[END_REF]. On the one hand, using this modelling framework, we explicitly address the micro-fibres and their rearrangement dynamics caused by the macro-scale movement of the cancer cell population (a process of great importance as the micro-scale structural changes of the underlying tissue can significantly influence the cell population motility). On the other hand, we also model another vital proteolytic process that occurs along the invasive edge of the tumour, allowing us to explicitly model tumour morphological changes and how the tumour boundary progresses over a time interval.

The paper is organised as follows. First, we formulate our extended multi-scale moving boundary framework in Section 2. Then, following a brief description of the numerical methods, we present the computational simulation results in Section 3. Finally, in Section 4 we summarise and discuss these results.

Multi-Scale Modelling of the Tumour Dynamics

To model the evolution of glioblastomas within a 3-dimensional brain, we employ a multi-scale moving boundary model that was initially introduced in [START_REF] Trucu | A Multiscale Moving Boundary Model Arising In Cancer Invasion[END_REF] and later expanded in several other works [START_REF] Shuttleworth | Multiscale Modelling of Fibres Dynamics and Cell Adhesion within Moving Boundary Cancer Invasion[END_REF][START_REF] Shuttleworth | Multiscale dynamics of a heterotypic cancer cell population within a fibrous extracellular matrix[END_REF][START_REF] Shuttleworth | Cell-Scale Degradation of Peritumoural Extracellular Matrix Fibre Network and Its Role Within Tissue-Scale Cancer Invasion[END_REF][START_REF] Suveges | Directionality of Macrophages Movement in Tumour Invasion: A Multiscale Moving-Boundary Approach[END_REF][START_REF] Suveges | Re-polarisation of macrophages within a multi-scale moving boundary tumour invasion model[END_REF][START_REF] Peng | A multiscale mathematical model of tumour invasive growth[END_REF]. To account for the brain's structure, we aim to use 3D T1 weighted and DTI scans that ultimately influence the migration of the cancer cells as well as affect both micro-scale dynamics. Hence, here we aim to explore the impact of the brain structure on the interlinked macro-scale and micro-scale tumour dynamics.

Macro-Scale Dynamics

Since in this work we extend the 2-dimensional (2D) modelling framework introduced in [START_REF] Trucu | A Multiscale Moving Boundary Model Arising In Cancer Invasion[END_REF][START_REF] Shuttleworth | Multiscale Modelling of Fibres Dynamics and Cell Adhesion within Moving Boundary Cancer Invasion[END_REF], we begin by describing briefly some of the key features of this framework and by giving a few useful notations. First, we denote by Ω(t) the expanding 3-dimensional (3D) tumour region that progresses over the time interval [0, T] within a maximal tissue cube Y ⊂ R N with N = 3, i.e., Ω(t) ⊂ Y, ∀t ∈ [0, T]; see also Figure 1. Then at any macro-scale spatio-temporal point (x, t) ∈ Y × [0, T] we consider a cancer cell population c(x, t) that is placed within and interacts with a two-phase ECM: the non-fibre l(x, t) and fibre F(x, t) ECM phases [START_REF] Shuttleworth | Multiscale Modelling of Fibres Dynamics and Cell Adhesion within Moving Boundary Cancer Invasion[END_REF][START_REF] Shuttleworth | Multiscale dynamics of a heterotypic cancer cell population within a fibrous extracellular matrix[END_REF][START_REF] Shuttleworth | Cell-Scale Degradation of Peritumoural Extracellular Matrix Fibre Network and Its Role Within Tissue-Scale Cancer Invasion[END_REF][START_REF] Suveges | Directionality of Macrophages Movement in Tumour Invasion: A Multiscale Moving-Boundary Approach[END_REF][START_REF] Suveges | Re-polarisation of macrophages within a multi-scale moving boundary tumour invasion model[END_REF]. On the one hand, the fibre ECM phase accounts for all major fibrous proteins such as collagen and fibronectin, whose micro-scale distribution induces the spatial orientation of the ECM fibres. Hence, the macro-scale spatio-temporal distribution of the ECM fibres is represented by an oriented vector field θ f (x, t) that describes their spatial bias, as well as by F(x, t) := θ f (x, t) which denote the amount of fibres at a macro-scale point (x, t) [START_REF] Shuttleworth | Multiscale Modelling of Fibres Dynamics and Cell Adhesion within Moving Boundary Cancer Invasion[END_REF][START_REF] Shuttleworth | Multiscale dynamics of a heterotypic cancer cell population within a fibrous extracellular matrix[END_REF][START_REF] Shuttleworth | Cell-Scale Degradation of Peritumoural Extracellular Matrix Fibre Network and Its Role Within Tissue-Scale Cancer Invasion[END_REF][START_REF] Suveges | Directionality of Macrophages Movement in Tumour Invasion: A Multiscale Moving-Boundary Approach[END_REF][START_REF] Suveges | Re-polarisation of macrophages within a multi-scale moving boundary tumour invasion model[END_REF]. On the other hand, in the non-fibre ECM phase we bundle together every other ECM constituent such as non-fibrous proteins (for example amyloid fibrils), enzymes, polysaccharides and extracellular Ca 2+ ions [START_REF] Shuttleworth | Multiscale Modelling of Fibres Dynamics and Cell Adhesion within Moving Boundary Cancer Invasion[END_REF][START_REF] Shuttleworth | Multiscale dynamics of a heterotypic cancer cell population within a fibrous extracellular matrix[END_REF][START_REF] Shuttleworth | Cell-Scale Degradation of Peritumoural Extracellular Matrix Fibre Network and Its Role Within Tissue-Scale Cancer Invasion[END_REF][START_REF] Suveges | Directionality of Macrophages Movement in Tumour Invasion: A Multiscale Moving-Boundary Approach[END_REF][START_REF] Suveges | Re-polarisation of macrophages within a multi-scale moving boundary tumour invasion model[END_REF]. Furthermore, in this new modelling study we incorporate the structure of the brain by extracting data from DTI and T1 weighted brain scans, and then using this data to parametrise the model. Specifically, we denote by D Water (x) the water diffusion tensor that is induced by the DTI scan. Also, we denote by w(x) the white matter density and by g(x) the grey matter density, both of which are extracted from the T1 weighted image. Finally, to facilitate the description of the model and to make the mathematical notations more compact, we denote by u the global tumour vector which at each (x, t) is given by

u := (c(x, t), l(x, t), F(x, t)) .
With this notation, the total space occupied by the macroscopic tissue is denoted by ρ(u) and is defined as ρ(u) = ρ(c(x, t), l(x, t), F(x, t)) := c(x, t) + l(x, t) + F(x, t).

Cancer cell population dynamics

To describe the spatio-temporal evolution of the cancer cell population c(x, t), we first assume a logistic-type growth with rate µ [START_REF] Shuttleworth | Multiscale Modelling of Fibres Dynamics and Cell Adhesion within Moving Boundary Cancer Invasion[END_REF][START_REF] Shuttleworth | Multiscale dynamics of a heterotypic cancer cell population within a fibrous extracellular matrix[END_REF][START_REF] Shuttleworth | Cell-Scale Degradation of Peritumoural Extracellular Matrix Fibre Network and Its Role Within Tissue-Scale Cancer Invasion[END_REF][START_REF] Suveges | Directionality of Macrophages Movement in Tumour Invasion: A Multiscale Moving-Boundary Approach[END_REF][START_REF] Suveges | Re-polarisation of macrophages within a multi-scale moving boundary tumour invasion model[END_REF][START_REF] Laird | Dynamics of Tumour Growth[END_REF][START_REF] Laird | Dynamics of Tumour Growth: Comparison of Growth Rates and Extrapolation of Growth Curve to One Cell[END_REF][START_REF] Tjorve | The use of Gompertz models in growth analyses, and new Gompertz-model approach: An addition to the Unified-Richards family[END_REF]. For the movement of this cell population, we use the structure of the brain by taking into account both the T1 weighted and DTI scans (from the IXI Dataset [? ]), as well as the various adhesion mediated processes [START_REF] Chen | Macrophage Binding to Receptor VCAM-1 Transmits Survival Signals in Breast Cancer Cells that Invade the Lungs[END_REF][START_REF] Condeelis | Macrophages: Obligate Partners for Tumor Cell Migration, Invasion, and Metastasis[END_REF][START_REF] Huda | Lévy-like movement patterns of metastatic cancer cells revealed in microfabricated systems and implicated in vivo[END_REF][START_REF] Petrie | Random versus directionally persistent cell migration[END_REF][START_REF] Weiger | Real-Time Motion Analysis Reveals Cell Directionality as an Indicator of Breast Cancer Progression[END_REF][START_REF] Wu | Three-dimensional cell migration does not follow a random walk[END_REF]. Hence, the spatio-temporal dynamics of the cancer cell population is described by

∂c ∂t = ∇∇ : D T (x)c fully anisotropic diffusion -∇ • cA c (x, t, u, θ f ) adhesion processes + µc 1 -ρ(u) + logistic-type proliferation . (1)
Here, the operator ∇∇ : denotes the full second order derivative [START_REF] Engwer | Glioma follow white matter tracts: a multiscale DTI-based model[END_REF], i.e., it is defined as

∇∇ : D T (x)c := N ∑ i,j=1 ∂ ∂x i ∂ ∂x j D i,j c , N = 3,
with D i,j denoting the components of the tumour diffusion tensor D T . Since classical diffusion models with constant coefficient cannot capture any directional cues, as those provided by the DTI data, in Eq. ( 1) we use a tensor model (involving a fully anisotropic diffusion term) that is able to incorporate the anisotropic nature of the cancer cell movement. These tensor models were proposed in [START_REF] Basser | Diagonal and off-diagonal components of the self-diffusion tensor:their relation to and estimation from the NMR spin-echo signal[END_REF][START_REF] Basser | Diffusion tensor echo-planar imaging of human brain[END_REF][START_REF] Basser | Estimation of the Effective Self-Diffusion Tensor from the[END_REF][61] and have been used to mathematically model the gliomas within the brain; see for instance [START_REF] Engwer | Glioma follow white matter tracts: a multiscale DTI-based model[END_REF][START_REF] Hunt | A Multiscale Modeling Approach to Glioma Invasion with Therapy[END_REF][START_REF] Painter | Mathematical modelling of glioma growth: The use of Diffusion Tensor Imaging (DTI) data to predict the anisotropic pathways of cancer invasion[END_REF]. The main idea of this approach is to use the measured water diffusivity in the structured, fibrous brain environment characterised by a symmetric water diffusion tensor [? ]

D Water (x) =   d xx (x) d xy (x) d xz (x) d xy (x) d yy (x) d yz (x) d xz (x) d yz (x) d zz (x)   , (2)
and appropriately construct a macroscopic diffusion tensor for the cancer cell population. Since this water tensor (2) is naturally symmetric (due to the DTI scans) [? ], it can be diagonalised. Denoting its eigenvalues by λ 1 (x) ≥ • • • ≥ λ N (x) and the associated eigenvectors by φ 1 (x), • • • φ N (x), we follow [START_REF] Painter | Mathematical modelling of glioma growth: The use of Diffusion Tensor Imaging (DTI) data to predict the anisotropic pathways of cancer invasion[END_REF][START_REF] Hillen | Moments of von mises and fisher distributions and applications[END_REF][START_REF] Mardia | Directional statistics[END_REF] and construct the 3D tumour diffusion tensor as

D T (x) := D c D WG (x) r + (1 -r) coth k(x) k(x) - 1 k(x) 2 I 3 + (1 -r) 1 - 3 coth k(x) k(x) + 3 k(x) 2 φ 1 (x)φ 1 (x) . ( 3
)
Here D c > 0 is the diffusion coefficient, r ∈ [0, 1] specifies the degree of isotropic diffusion, I 3 denotes the 3 × 3 identity matrix, k(x) is given by k(x) := K FA FA(x), with K FA ≥ 0 being a proportionality constant measuring the sensitivity of the cells to the environments' direction, and FA(x) denotes the fractional anisotropy index [START_REF] Hagmann | Understanding Diffusion MR Imaging Techniques: From Scalar Diffusion-weighted Imaging to Diffusion Tensor Imaging and Beyond[END_REF] given by

FA(x) := (λ 1 -λ 2 ) 2 + (λ 2 -λ 3 ) 2 + (λ 1 -λ 3 ) 2 2(λ 2 1 + λ 2 2 + λ 2 3 )
. Finally, it is well known that the malignant glioma cells positioned in the white matter exercise quicker motility than those situated in the grey matter [START_REF] Swanson | A quantitative model for differential motility of gliomas in grey and white matter[END_REF][START_REF] Chicoine | Assessment of brain tumor cell motility in vivo and in vitro[END_REF][START_REF] Kelly | The limited value of cytoreductive surgery in elderly patients with malignant gliomas[END_REF][START_REF] Silbergeld | Isolation and characterization of human malignant glioma cells from histologically normal brain[END_REF]. To account for this effect, in (3), we use a regulator term D WG (•) that is given by

D WG (x) = a + (1 -a) D G g(x) + w(x) * ψ ρ (x), (4)
where 0 ≤ D G ≤ 1 is the grey matter regulator coefficient, * is the convolution operator [START_REF] Damelin | The Mathematics of Signal Processing[END_REF], ψ ρ := ψ(x)/ρ N denotes the standard mollifier and g(x) and w(x) are the grey and white matter densities provided by the T1 weighted image (following an image segmentation process). Finally, 0 ≤ a ≤ 1 is a parameter that distinguishes between different cases (see Section 3).

In addition, the movement of the cancer cells is further biased by various adhesion mediated process [START_REF] Chen | Macrophage Binding to Receptor VCAM-1 Transmits Survival Signals in Breast Cancer Cells that Invade the Lungs[END_REF][START_REF] Condeelis | Macrophages: Obligate Partners for Tumor Cell Migration, Invasion, and Metastasis[END_REF][START_REF] Huda | Lévy-like movement patterns of metastatic cancer cells revealed in microfabricated systems and implicated in vivo[END_REF][START_REF] Petrie | Random versus directionally persistent cell migration[END_REF][START_REF] Weiger | Real-Time Motion Analysis Reveals Cell Directionality as an Indicator of Breast Cancer Progression[END_REF][START_REF] Wu | Three-dimensional cell migration does not follow a random walk[END_REF]. Due to the increasing evidence that gliomas induce a fibrous environment within the brain [START_REF] Gondi | Adenovirus-Mediated Expression of Antisense Urokinase Plasminogen Activator Receptor and Antisense Cathepsin B Inhibits Tumor Growth[END_REF][START_REF] Gregorio | Collagen VI in healthy and diseased nervous system[END_REF][START_REF] Kalinin | Cell -extracellular matrix interaction in glioma growth[END_REF][START_REF] Mohanam | Biological significance of the expression of urokinase-type plasminogen activator receptors (uPARs) in brain tumors[END_REF][START_REF] Persson | Urokinase-Type Plasminogen Activator Receptor as a Potential PET Biomarker in Glioblastoma[END_REF][START_REF] Pointer | Association of collagen architecture with glioblastoma patient survival[END_REF][START_REF] Pullen | Current insights into matrix metalloproteinases and glioma progression: transcending the degradation boundary[END_REF][START_REF] Ramachandran | Expression and prognostic impact of matrix metalloproteinase-2 (MMP-2) in astrocytomas[END_REF][START_REF] Veeravalli | MMP-9 and uPAR regulated glioma cell migration[END_REF][START_REF] Veeravalli | Integrin α9β1-mediated cell migration in glioblastoma via SSAT and Kir4.2 potassium channel pathway[END_REF][START_REF] Young | Sphingosine-1-Phosphate Regulates Glioblastoma Cell Invasiveness through the Urokinase Plasminogen Activator System and CCN1[END_REF], in (1) we model the overall adhesion process using a non-local flux term that was introduced in [START_REF] Shuttleworth | Multiscale Modelling of Fibres Dynamics and Cell Adhesion within Moving Boundary Cancer Invasion[END_REF] (see also [START_REF] Domschke | Mathematical modelling of cancer invasion: Implications of cell adhesion variability for tumour infiltrative growth patterns[END_REF][START_REF] Shuttleworth | Multiscale dynamics of a heterotypic cancer cell population within a fibrous extracellular matrix[END_REF][START_REF] Shuttleworth | Cell-Scale Degradation of Peritumoural Extracellular Matrix Fibre Network and Its Role Within Tissue-Scale Cancer Invasion[END_REF][START_REF] Suveges | Directionality of Macrophages Movement in Tumour Invasion: A Multiscale Moving-Boundary Approach[END_REF][START_REF] Suveges | Re-polarisation of macrophages within a multi-scale moving boundary tumour invasion model[END_REF][START_REF] Peng | A multiscale mathematical model of tumour invasive growth[END_REF][START_REF] Armstrong | A continuum approach to modelling cell-cell adhesion[END_REF][START_REF] Gerisch | Mathematical modelling of cancer cell invasion of tissue: Local and non-local models and the effect of adhesion[END_REF] for similar terms). Specifically, we explore the adhesive interactions of the cancer cells at x ∈ Ω(t) with other neighbouring cancer cells, with the distribution of the non-fibre ECM phase [START_REF] Ghosh | p53 amyloid formation leading to its loss of function: implications in cancer pathogenesis[END_REF][START_REF] Gras | Chapter 6 -Surface-and Solution-Based Assembly of Amyloid Fibrils for Biomedical and Nanotechnology Applications[END_REF][START_REF] Gras | Functionalised amyloid fibrils for roles in cell adhesion[END_REF][START_REF] Jacob | Cell Adhesion on Amyloid Fibrils Lacking Integrin Recognition Motif[END_REF] as well as with the oriented fibre ECM phase [START_REF] Wolf | Collagen-based cell migration models in vitro and in vivo[END_REF][START_REF] Wolf | Extracellular matrix determinants of proteolytic and non-proteolytic cell migration[END_REF], all located within a sensing region B(x, R) of radius R > 0. For this, we define the non-local flux term as

A c (x, t, u, θ f ) := 1 R B(0,R) K(y) n(y) S cc c(x + y, t) + S cl l(x + y, t) + n(y, θ f (x + y, t))S cF F(x + y, t) 1 -ρ(u) + dy, (5)
where S cc , S cl , S cF > 0 are the strengths of the cell-cell, cell-non-fibre ECM and cell-fibre ECM adhesions, respectively. While we take both S cl and S cF as positive constants, we consider the emergence of strong and stable cell-cell adhesion bonds to be positively correlated with the level of extracellular Ca +2 ions (one of the non-fibre ECM component) [START_REF] Gu | Soft matrix is a natural stimulator for cellular invasiveness[END_REF][START_REF] Hofer | Intercellular communication mediated by the extracellular calcium-sensing receptor[END_REF]. Hence, following the approach in [START_REF] Shuttleworth | Multiscale Modelling of Fibres Dynamics and Cell Adhesion within Moving Boundary Cancer Invasion[END_REF][START_REF] Shuttleworth | Multiscale dynamics of a heterotypic cancer cell population within a fibrous extracellular matrix[END_REF][START_REF] Shuttleworth | Cell-Scale Degradation of Peritumoural Extracellular Matrix Fibre Network and Its Role Within Tissue-Scale Cancer Invasion[END_REF][START_REF] Suveges | Directionality of Macrophages Movement in Tumour Invasion: A Multiscale Moving-Boundary Approach[END_REF][START_REF] Suveges | Re-polarisation of macrophages within a multi-scale moving boundary tumour invasion model[END_REF], we describe the cell-cell adhesion strength by

S cc (x, t) := S min + (S max -S min ) exp 1 - 1 1 -(1 -l(x, t)) 2 ,
where S min > 0 and S max > 0 are the minimum and maximum levels of Ca +2 ions. Furthermore, in [START_REF] Meneceur | Establishment and Characterisation of Heterotopic Patient-Derived Xenografts for Glioblastoma[END_REF] we denote by n(•) and n(•, •) the unit radial vector and the unit radial vector biased by the oriented ECM fibres [START_REF] Shuttleworth | Multiscale Modelling of Fibres Dynamics and Cell Adhesion within Moving Boundary Cancer Invasion[END_REF][START_REF] Shuttleworth | Multiscale dynamics of a heterotypic cancer cell population within a fibrous extracellular matrix[END_REF][START_REF] Shuttleworth | Cell-Scale Degradation of Peritumoural Extracellular Matrix Fibre Network and Its Role Within Tissue-Scale Cancer Invasion[END_REF][START_REF] Suveges | Directionality of Macrophages Movement in Tumour Invasion: A Multiscale Moving-Boundary Approach[END_REF][START_REF] Suveges | Re-polarisation of macrophages within a multi-scale moving boundary tumour invasion model[END_REF] defined by

n(y) :=    y y 2 if y ∈ B(0, R) \ {0}, 0 if y = 0, n(y, θ f (x + y)) :=      y + θ f (x + y, t) y + θ f (x + y, t) 2 if y ∈ B(0, R) \ {0}, 0 if y = 0,
respectively (for details on the fibre orientation θ f see Section 2.2.1). Also, to account for the gradual weakening of all adhesion bonds as we move away from the centre point x within the sensing region B(x, t) in ( 5), we use a radially symmetric kernel K(•) [START_REF] Suveges | Directionality of Macrophages Movement in Tumour Invasion: A Multiscale Moving-Boundary Approach[END_REF][START_REF] Suveges | Re-polarisation of macrophages within a multi-scale moving boundary tumour invasion model[END_REF] given by

K(y) = ψ y R , ∀y ∈ B(0, R),
where ψ(•) is the standard mollifier. Finally, in (5) a limiting term [1ρ(u)] + := max(0, 1ρ(u)) is used to prevent the contribution of overcrowded regions to cell migration [START_REF] Gerisch | Mathematical modelling of cancer cell invasion of tissue: Local and non-local models and the effect of adhesion[END_REF]. For a schematic of this adhesion process, we refer the reader to Figure 1(a).

Two phase ECM macro-scale dynamics

In addition to the cancer cell population, the rest of the macro-scale tumour dynamics are described by the two-phase ECM. Here, both fibres and non-fibres ECM phases are assumed to be simply described by a degradation term due to the cancer cell population. Hence, per unit time, their dynamics is governed by

∂F ∂t = -β F cF, ∂l ∂t = -β l cl, (6)
where β F > 0 and β l > 0 are the degradation rates of the fibre and non-fibre ECM phases, respectively.

The complete macro-dynamics

In summary, equations (1) for cancer cells dynamics and ( 6) for the two-phase ECM dynamics lead to the following non-dimensional PDE system describing the evolution of tumour at macro-scale:

∂c ∂t =∇∇ : D T (x)c -∇ • cA c (x, t, u, θ f ) + µc 1 -ρ(u) , ∂F ∂t = -β F cF, ∂l ∂t = -β l cl. (7)
To complete the macro-scale model description, we consider zero-flux boundary conditions and appropriate initial conditions (for instance the ones given in Section 3).

Micro-Scale Processes and the Double Feedback Loop

Since the cancer invasion process is genuinely a multi-scale phenomenon, several micro-scale processes are closely linked to the macro-scale dynamics [START_REF] Weinberg | The Biology of Cancer[END_REF]. In this work, we consider two of these micro-processes, namely the rearrangement of the ECM fibres micro-constituents [START_REF] Shuttleworth | Multiscale Modelling of Fibres Dynamics and Cell Adhesion within Moving Boundary Cancer Invasion[END_REF] and the cell-scale proteolytic processes that occur at the leading edge of the tumour [START_REF] Trucu | A Multiscale Moving Boundary Model Arising In Cancer Invasion[END_REF]. Here we briefly outline these micro-processes, in addition to the naturally arising double feedback loop that ultimately connects the micro-scale and the macro-scale.

Two-scale representation and dynamics of fibres

To represent the oriented fibres on the macro-scale, we follow [START_REF] Shuttleworth | Multiscale Modelling of Fibres Dynamics and Cell Adhesion within Moving Boundary Cancer Invasion[END_REF]. There, the authors characterised not only the amount of fibres F(x, t), but also their ability to withstand incoming cell fluxes and forces through their spatial bias. By considering a cell-scale microdomain δY(x) := x + δY of appropriate micro-scale size δ > 0, both of these characteristics are induced by the microscopic fibre distribution f (z, t), with z ∈ δY(x). In fact, both of them are captured though a vector field representation θ f (x, t) of the ECM micro-fibres [START_REF] Shuttleworth | Multiscale Modelling of Fibres Dynamics and Cell Adhesion within Moving Boundary Cancer Invasion[END_REF] that is defined as:

θ f (x, t) := 1 λ(δY(x)) δY(x) f (z, t)dz • θ f ,δY(x) (x, t) θ f ,δY(x) (x, t) 2 , ( 8
)
where λ(•) is the Lebesgue measure in R d and θ f ,δY(x) (•, •) is the revolving barycentral orientation given by [25]

θ f ,δY(x) (x, t) := δY(x) f (z, t)(z -x)dz δY(x) f (z, t)dz .
Hence, the fibres' ability to withstand forces is naturally defined by this vector field representation [START_REF] Anderson | Mathematical modelling of tumour invasion and metastasis[END_REF] and their amount distributed at a macro-scale point (x, t) is given by

F(x, t) := θ f (x, t) 2 ,
which is precisely the mean-value of the micro-fibres distributed on δY(x). Since both of these macro-scale oriented ECM fibre characteristics (F(x, t) and θ f (x, t)) that we use in the macro-scale dynamics [START_REF] Sottoriva | Intratumor heterogeneity in human glioblastoma reflects cancer evolutionary dynamics[END_REF], genuinely emerge from the micro-scale distribution of the ECM fibres f (z, t), we refer to this link as the fibres bottom-up link. An illustration of a micro-domain δY(x) and its corresponding macro-scale orientation θ f (x, t) can be seen in Figure 1(b).

On the other hand, there is also a naturally arising link that connects the macro-scale to this micro-scale, namely the fibres top-down link. This connection is initiated by the movement of the cancer cell population that trigger a rearrangement of ECM fibres microconstituents on each micro-domain δY(x) (enabled by the secretion of matrix-degrading enzymes that can break down various ECM proteins). Hence, using the fact that the fully anisotropic diffusion term can be rewritten as

∇∇ : D T (x)c = ∇ • [D T (x)∇c + c∇ • D T (x)],
the fibre rearrangement process is kicked off by the cancer cell spatial flux

F c (x, t) := D T (x)∇c + c∇ • D T (x) -cA c (x, t, u, θ f ), (9)
which is generated by the tumour macro-dynamics [START_REF] Sottoriva | Intratumor heterogeneity in human glioblastoma reflects cancer evolutionary dynamics[END_REF]. Then, at any spatio-temporal point (x, t) ∈ Ω(t) × [0, T] this flux (9) gets naturally balanced in a weighted fashion by the macro-scale ECM fibre orientation θ f (•, •), resulting in a rearrangement flux [START_REF] Shuttleworth | Multiscale Modelling of Fibres Dynamics and Cell Adhesion within Moving Boundary Cancer Invasion[END_REF] r(δY(x), t)

:= ω(x, t)F c (x, t) + (1 -ω(x, t))θ f (x, t), (10)
with ω(x, t) := c(x, t)/(c(x, t) + F(x, t)), that acts uniformly upon the micro-fibre distribution on each micro-domain δY(x). Ultimately, this macro-scale rearrangement vector (10) induces a micro-scale reallocation vector ν δY(x) (z, t) [START_REF] Shuttleworth | Multiscale Modelling of Fibres Dynamics and Cell Adhesion within Moving Boundary Cancer Invasion[END_REF], enabling us to appropriately calculate the new position z * of any micro-node z ∈ δY(x) as

z * := z + ν δY(x) (z, t). (11)
For further details on the micro-fibre rearrangement process, we refer the reader to Appendix B and [START_REF] Shuttleworth | Multiscale Modelling of Fibres Dynamics and Cell Adhesion within Moving Boundary Cancer Invasion[END_REF][START_REF] Shuttleworth | Multiscale dynamics of a heterotypic cancer cell population within a fibrous extracellular matrix[END_REF][START_REF] Shuttleworth | Cell-Scale Degradation of Peritumoural Extracellular Matrix Fibre Network and Its Role Within Tissue-Scale Cancer Invasion[END_REF][START_REF] Suveges | Directionality of Macrophages Movement in Tumour Invasion: A Multiscale Moving-Boundary Approach[END_REF][START_REF] Suveges | Re-polarisation of macrophages within a multi-scale moving boundary tumour invasion model[END_REF].

MDE micro-dynamics and its links

The second micro-scale process that we take into consideration is the proteolytic molecular process that occurs along the invasive edge of the tumour and is driven by the cancer cells' ability to secrete several types of matrix-degrading enzymes (MDEs) (for instance, matrix-metalloproteinases) within the proliferating rim [START_REF] Hanahan | The hallmarks of cancer[END_REF][START_REF] Hanahan | Hallmarks of cancer: The next generation[END_REF][START_REF] Lu | Extracellular matrix degradation and remodeling in development and disease[END_REF][START_REF] Parsons | Matrix metalloproteinases[END_REF][START_REF] Pickup | The extracellular matrix modulates the hallmarks of cancer[END_REF]. Subsequent to the secretion, these MDEs are subject to spatial transport within a cell-scale neighbourhood of the tumour interface and, as a consequence, they degrade the peritumoral ECM, resulting in changes of tumour boundary morphology [START_REF] Weinberg | The Biology of Cancer[END_REF].

To explore such a micro-scale process, we adopt the approach that was first introduced in [START_REF] Trucu | A Multiscale Moving Boundary Model Arising In Cancer Invasion[END_REF] where the emerging spatio-temporal MDEs micro-dynamics is considered on an appropriate cell-scale neighbourhood of the tumour boundary ∂Ω(t). This neighborhood is represented by a time-dependent bundle of overlapping cubic micro-domains { Y} Y∈P (t) , with > 0 being the size of each micro-domain Y, which allows us to decompose the overall MDE micro-process, transpiring on Y∈P (t) Y, into a union of proteolytic micro-dynamics occurring on each Y; see also Figure 1(c). Hence, choosing an arbitrary micro-domain Y and a macroscopic time instance t 0 ∈ [0, T], we follow the evolution of the MDE micro-dynamics during the time period [t 0 , t 0 + ∆t], with appropriately chosen ∆t > 0 and within the associated micro-domain Y. By denoting the spatio-temporal distribution of the MDEs by m(y, τ) at any micro-point (y, τ) ∈ Y × [0, ∆t], we observe that the cancer cell population, located within an appropriately chosen distance γ h > 0 from y ∈ Y, induce a source h(y, τ) of MDEs which can be mathematically described via a non-local term [START_REF] Trucu | A Multiscale Moving Boundary Model Arising In Cancer Invasion[END_REF]

h(y, τ) =        B(y,γ h )∩Ω(t 0 ) c(x, t 0 + τ) dy λ(B(y, γ h ) ∩ Ω(t 0 )) y ∈ Y ∩ Ω(t 0 ), 0 y / ∈ Y \ (Ω(t 0 ) + {z ∈ Y | z 2 < ρ}), ( 12
)
where 0 < ρ < γ h is a small mollification range and B(y, γ h ) denotes the • ∞ ball of radius γ h centred at a micro-node y. Since the calculation of this micro-scale MDE source [START_REF] Basanta | The role of IDH1 mutated tumour cells in secondary glioblastomas: an evolutionary game theoretical view[END_REF] directly involves the macro-scale cancer cell population c(•, •), we observe a naturally arising MDE top-down link that connects the macro-scale to the MDE micro-scale. In fact, such source term [START_REF] Basanta | The role of IDH1 mutated tumour cells in secondary glioblastomas: an evolutionary game theoretical view[END_REF] allows us to describe the spatio-temporal evolution of the MDEs micro-scale distribution m(•, •) by [START_REF] Trucu | A Multiscale Moving Boundary Model Arising In Cancer Invasion[END_REF]

∂m ∂τ = D m ∆m + h(y, τ), m(y, 0) = 0, ∂m ∂n ∂ Y = 0, (13)
where D m > 0 is the constant MDEs diffusion coefficient and n denotes the outward normal vector. As it was shown in [START_REF] Trucu | A Multiscale Moving Boundary Model Arising In Cancer Invasion[END_REF], we can use the solution of the MDEs micro-dynamics (13) to acquire both movement direction and magnitude of a tumour boundary point x * Y within the peritumoral area covered by the associated boundary micro-domain Y. This ultimately causes a boundary movement, and as a consequence we obtain a new evolved tumour macro-domain Ω(t 0 + ∆t), the link of which we refer to as the MDE bottom-up link. For illustration of the boundary movement we refer the reader to Figure 1(c) and for further details of the MDE micro dynamics see Appendix C or [START_REF] Trucu | A Multiscale Moving Boundary Model Arising In Cancer Invasion[END_REF][START_REF] Shuttleworth | Multiscale Modelling of Fibres Dynamics and Cell Adhesion within Moving Boundary Cancer Invasion[END_REF][START_REF] Shuttleworth | Multiscale dynamics of a heterotypic cancer cell population within a fibrous extracellular matrix[END_REF][START_REF] Shuttleworth | Cell-Scale Degradation of Peritumoural Extracellular Matrix Fibre Network and Its Role Within Tissue-Scale Cancer Invasion[END_REF][START_REF] Suveges | Directionality of Macrophages Movement in Tumour Invasion: A Multiscale Moving-Boundary Approach[END_REF][START_REF] Suveges | Re-polarisation of macrophages within a multi-scale moving boundary tumour invasion model[END_REF].

Computational Results: Numerical Simulations in 3D

We start this section by briefly discussing the numerical method that we use to solve the macro-scale dynamics [START_REF] Sottoriva | Intratumor heterogeneity in human glioblastoma reflects cancer evolutionary dynamics[END_REF], and for details on the numerical approach used for the two micro-scale dynamics (fibres and MDE) we refer the reader to [START_REF] Suveges | Directionality of Macrophages Movement in Tumour Invasion: A Multiscale Moving-Boundary Approach[END_REF][START_REF] Suveges | Re-polarisation of macrophages within a multi-scale moving boundary tumour invasion model[END_REF]. Here, we use the method of lines approach to discretise the macro-scale tumour dynamics [START_REF] Sottoriva | Intratumor heterogeneity in human glioblastoma reflects cancer evolutionary dynamics[END_REF] first in space, and then, for the resulting system of ODEs, we employ a non-local predictor-corrector scheme [START_REF] Shuttleworth | Multiscale Modelling of Fibres Dynamics and Cell Adhesion within Moving Boundary Cancer Invasion[END_REF]. In this context, we carry out the spatial discretisation on a uniform grid, where both spatial operators (fully anisotropic diffusion and adhesion) are accurately approximated in a convolution-driven fashion. First, we note that the fully anisotropic diffusion term can be split into two parts

∇∇ : D T (x)c = ∇ • D T (x)∇c diffusive + ∇ • c∇ • D T (x) advective , (14)
which enables us to use a combination of two appropriate distinct schemes for an accurate approximation. While for the diffusive part in [START_REF] Chaplain | Mathematical modelling of cancer cell invasion of tissue: the role of the urokinase plasminogen activation system[END_REF], we use the symmetric finite difference scheme [START_REF] Van Es | Finite-difference schemes for anisotropic diffusion[END_REF][START_REF] Günter | Modelling of heat transport in magnetised plasmas using non-aligned coordinates[END_REF], for the combination of the advective ( 14) and adhesion operators (5

) (i.e., ∇ • c A c (x, t, u, θ f ) + ∇ • D T (x)
) we use the standard first-order upwind finite difference scheme which ensures positivity and helps avoiding spurious oscillations in the solution.

Finally, to approximate the adhesion integral A c (x, t, u, θ f ), we consider an approach similar to [START_REF] Suveges | Directionality of Macrophages Movement in Tumour Invasion: A Multiscale Moving-Boundary Approach[END_REF][START_REF] Suveges | Re-polarisation of macrophages within a multi-scale moving boundary tumour invasion model[END_REF], and use N s random points located within the sensing region B(0, R) and sums of discrete-convolutions.

Initial Conditions

For the numerical simulations presented in this paper, we consider the tissue cube Y = [0, 4] × [0, 4] × [0, 4] with the following initial condition for the cancer cells

c(x, 0) = 1 2 exp -x 2 2 0.02 • χ B((2,2,2),0.25) ,
and for the non-fibre ECM phase, the initial condition l(x, 0) is acquired by appropriately scaling the T1 weighted image via a normalising constant. Current DTI scans do not provide suitable resolution to determine the underlying micro-fibre distributions, and so here, we describe the initial micro-fibre distribution within a micro-domain δY(x) as follows. When the macro-scale point x that corresponds to the micro-domain δY(x) is located in the grey matter, then within δY(x) we randomly draw straight lines until the ratio between the points that belong and the points that do not belong to the collection of lines is about 35% : 65%. On the other hand, when the point x is located within the white matter, we use a set of predefined lines with the same point ratio (35% : 65%), ultimately achieving a random orientation within the grey matter and an aligned orientation within the white matter [? ]. Finally, the grey matter's fibre density is assumed to be 1/D G times smaller than the density in the white matter [? ]. A schematics of this initial condition for the micro-fibres can be seen in Figure 2. Hence, we also incorporate the information about the white and grey matter tracks provided by the T1 weighted image into our micro-scale fibre distribution.

Numerical Simulations in 3D

Here, we present the 3D numerical solutions of the multi-scale model described above, for the parameter values listed in Table 1 in Appendix A (any alteration from these values will be stated accordingly). To display the advanced tumours at time 50∆t, we show four panels for each simulation results. In the first three panels we show the three classical cross-section planes i.e., the coronal plane (the head of the subject is viewed from behind), the axial plane (the head of the subject is viewed from above) and the sagittal plane (the head of the subject is viewed from the left). In the last panel of each simulation we show the 3D image of the brain with the embedded tumour alongside the 3D tumour in isolation.

The three Figures shown below investigate tumour evolution when the initial tumour starts in different regions of the brain. To present the simulations, we divide each result into four panels: coronal, axial, sagittal and 3D view. Within each coronal, axial and sagittal views, we show the tumour embedded within the brain on the left, the cancer cell density on the top-right and the ECM density on the bottom-right. In the 3D view (the most right panel in each results) we show the cross-section of the whole brain with the tumour on the bottom-left corner and on the top-right corner we show the isolated tumour.

In Figure 3 we present three distinct cases obtained by varying different parameters 1 that appear in the tumour diffusion tensor D T (x) defined in (3). In Figure 3 (a) we assume 2 that the tensor D T (x) depends on the white-grey matter and for that purpose we set r = 1 3 in ( 3) and a = 0 in (4); this results in isotropic tumour diffusion. In Figure 3 (b) we use 4 the DTI data (i.e., there is no a-priory assumption about the preferential direction for cell 5 movement in white matter) and thus we set a = 1 in (4) (with r = 0.1, as in Table 1); this 6 results in an anisotropic diffusion that does not depend explicitly on the white-grey matter. In Figure 3 (c) we use both DTI data and the white-grey matter dependency (i.e., r = 0.1

8
and a = 0), with the baseline parameters from Table 1. Here, it is worth mentioning that 9 even though we do not use the T1 weighted image to obtain functions w(x) and g(x) that In all these simulations shown in Figure 3, we place the small initial tumour in the tumour that is growing away from the skull towards the centre of the brain as well as 30 it is mainly following the white matter. This creates a highly heterogeneous elongated 31 tumour with many branching outgrowths. On the other hand, in Figure 4 we only see 32 slight differences between the three cases. This contradicts the results from Figure 3 and

33
suggests that both the DTI data and white-grey matter dependency may not always be 34 decisive factor in tumour morphology.

35

Similarly to Figure 3 and Figure 4, in Figure 5 we keep the same three cases (Figure 5 36 that branched to both the left and right side of the brain with some asymmetry. Also, as in 40 Figure 4 we can see that all three cases are quite similar, and so the additional information 41 provided by both the DTI data and white-grey matter dependency seems to be unnecessary 42 for this initial condition. However, we must note that the initial conditions (fibre and 43 non-fibre ECM) still uses the information provided by the T1 weighted image, and so here,

44
we only investigate the effect of changing the diffusion tensor.

Discussion and Final Remarks 61

In this study, we have further extended the 2D multi-scale moving-boundary frame-62 work previously introduced in [START_REF] Trucu | A Multiscale Moving Boundary Model Arising In Cancer Invasion[END_REF][START_REF] Shuttleworth | Multiscale Modelling of Fibres Dynamics and Cell Adhesion within Moving Boundary Cancer Invasion[END_REF], by developing it to 3D and applying it to the study 63 of glioma invasion within the brain. Since experiments are limited within the brain, we fo-64 cused on incorporating DTI and T1 weighted scans into our framework to provide insights 65 into the structure of the brain, the tumour, and the surrounding tissue.

66

The original framework developed in [START_REF] Trucu | A Multiscale Moving Boundary Model Arising In Cancer Invasion[END_REF][START_REF] Shuttleworth | Multiscale Modelling of Fibres Dynamics and Cell Adhesion within Moving Boundary Cancer Invasion[END_REF] modelled a generic tumour in a 2D Therefore, compared to other previous 3D models such as those in [START_REF] Engwer | Glioma follow white matter tracts: a multiscale DTI-based model[END_REF][START_REF] Hunt | A Multiscale Modeling Approach to Glioma Invasion with Therapy[END_REF][START_REF] Painter | Mathematical modelling of glioma growth: The use of Diffusion Tensor Imaging (DTI) data to predict the anisotropic pathways of cancer invasion[END_REF], here we can To that end, we used this new 3D model to explore the effects of the anisotropic 94 diffusion term for the cancer cell population. Our numerical simulations in Figure 3 95 showed that including an anisotropic diffusion term may lead to significant changes in the 96 overall tumour morphology. However, it seems that these changes depend on the position To conclude this study, we mention that further investigation is needed to determine Although a movement direction and a displacement magnitude can be this way determined

Figure 1 .

 1 Figure 1. Schematics of the multi-scale model. (a) Illustration of the sensing region B(x, R), the two vectors y and y + θ f (y + x, t) and the overall travelling direction A c . (b) An example of a fibre micro-domain δY(x) containing fibres (blue lines) that induces an overall orientation θ f (x, t) for δY(x). (c) An example of a boundary micro-domain Y(x) where the blue volume represents the tumour volume at the current time-step with boundary point x and the red volume represents the evolved tumour at next time-step with shifted boundary point x * Y .

Figure 2 .

 2 Figure 2. Schematics of the initial condition of the micro-fibres (blue lines) within a micro-domain δY(x) of orientation θ f (x, t) located in the (a) white matter and in the (b) grey matter.

Figure 3 .

 3 Figure 3. 3D computer simulation results (a) with only white-grey matter dependency (r = 1), (b) with only DTI data used (D G = 1) (c) with both white-grey matter dependency and DTI data incorporated. To present the simulations, we divide each result into four panels: coronal, axial, sagittal and 3D view. Within each coronal, axial and sagittal views, we show the tumour embedded within the brain on the left, the cancer cell density on the top-right and the ECM density on the bottom-right. In the 3D view (the most right panel in each results) we show the cross-section of the whole brain with the tumour on the bottom-left corner and on the top-right corner we show the isolated tumour.

7

 7

10 appear

 10 in D WG (as D WG = 1 in Figure 3 (b), since a = 1) we still use the T1 weighted image 11 to initialise the micro-scale non-fibre initial density as well as the initial micro-scale fibre 12 distributions as described above.

13

 13

14 middle- 16 Figure 3 17 Figure 4 . 25 In Figure 4

 14163174254 Figure 3 (a) to (b) and Figure 3 (b) to (c), we see that when we include the white-grey matter17

39 Figure 5 .

 395 Figure 5. 3D computer simulation results (a) with only white-grey matter dependency (r = 1), (b) with only DTI data used (D G = 1) (c) with both white-grey matter dependency and DTI data incorporated. To represent the simulation results, we use the same format as in Figure 3.

67 setting,

 67 and so to model gliomas within a 3D brain, we extended this modelling approach 68 by considering the structural information provided by both DTI and T1 weighted scans.69We used both DTI and T1 weighted scans to construct the tumour diffusion tensor D T (x) 70 defined in[START_REF] Klopfenstein | Cell lines and immune classification of glioblastoma define patient[END_REF], which resulted in a fully anisotropic diffusion term. While the T1 weighted 71 image can give different diffusion rates based on whether the cancer cells are located in the 72 white or grey matter, the DTI data is used to incorporate the underlying brain structure 73 and to give higher diffusion rates along specific directions based on how the measured 74 water molecules behaved within the brain. The T1 weighted image, which provided the 75 white-grey matter densities, were also used in our initial conditions for both ECM phases.76Hence, the initial density of the non-fibre ECM phase was taken as a normalised version 77 of the T1 weighted image, and the initial condition of the micro-fibre distribution and 78 magnitude were also considered to be dependent on the white-grey matter structure.79Furthermore, as the available DTI scans lack the adequate resolution required to construct 80 more appropriate micro-fibre distributions, in this work we considered a simple case where 81 we set the fibre distributions to be either random or oriented based on whether they are 82 positioned in the grey or white matter, respectively.

83

 83

84 explicitly

 84 capture the underlying tissue structure changes via the micro-fibre rearrangement 85 process and trace how the boundary of the tumour changes over time. Moreover, the cancer 86 cells' movement is not only influenced by the T1 weighted and DTI scans (through an 87 anisotropic diffusion term) but also biased by crucial non-local adhesions such as cell-cell 88 and cell-fibre ECM adhesion the latter of which is naturally interconnected to the tissue 89 structural changes. Hence, our model presents a novel approach that can incorporate 90 several vital processes for tumour development and have not been available in other 91 previous models. This allowed us to produce exciting and valuable results/outcomes 92 which provide further insights into the overall tumour development process.

93

 93

97 of

 97 the tumour inside the brain, as Figures4 and 5do not exhibit changes consistent to 98 the ones observed between the three sub-panels of Figure3. This may be the result of the 99 underlying brain structure and its microscopic fibre representation, which seems to take a 100 leading role in influencing cancer-invasion patterns through the underlying cell-adhesion 101 process (see Eq. (5)), overshadowing this way the diffusion process. More precisely, the 102 simplified fibre representation might not be sufficient for Figure3, where the initial tumour 103 was positioned in the right-middle part of the brain. However, this fibre representation 104 might be enough for Figure4(with tumour positioned in the front-right of the brain) and 105 for Figure5(with tumour positioned in the middle of the brain), where we did not see 106 significant morphological differences between the three sub-panels considered in each of 107 these figures.

  108

149

  Appendix C. Further Details on the MDE micro-scale 150Following[START_REF] Trucu | A Multiscale Moving Boundary Model Arising In Cancer Invasion[END_REF], we briefly detail here the way the MDE micro-dynamics (13) deter-151 mines the macro-boundary of the progressed tumour domain Ω(t 0 + ∆t). To that end, 152 on any arbitrary boundary micro-domain Y ∈ P (t 0 ) we consider an appropriate dyadic 153 cubes decomposition {D k } I , and we denote the barycentre of each D k by y k . Then, a 154 subfamily of small dyadic cubes {D k } J * is sub-sampled by selecting only those dyadic 155 cubes that are furthest away from the boundary point x * Y while being located outside of 156 the tumour domain Ω(t 0 ) and carrying an above average mass of MDEs. This enables us 157 to define the associated direction η Y and displacement magnitude ξ Y of the movement, 158 which are given by 159 η Y(x * Y ) := x * Y + ν ∑ l∈J * D l m(y, τ) dy y lx * Y , ν ∈ [0, ∞),

Table 1 :

 1 modelling point of view but from a medical one as well. For instance, we 115 showed how different model parameters change the evolution of the tumour boundary, 116 this being a vital and desired information in any clinical/medical decision-making process. Such a model with tuned parameters (to specific patients) could help, for example, decide 118 the optimal amount of tissue to resect so that the chance of survival is maximised. Parameter set used for the numerical simulations.Consequently, we transport p move • f (z, t) amount of fibres to the new position z * and the 148 rest (1p move ) • f (z, t) remains at the original position z.

	119
	Finally, as our simulations are able to reproduce known tumour patterns of growth

109 whether these changes in tumour invasion patterns are caused by the lack of directional in-110 formation on the fibre micro-scale level or an anisotropic diffusive cell motility is necessary 111 to better represent the invasion process. A feasible approach would also be to use a new 112 imaging technology called magnetic resonance elastography, but this is beyond the scope 113 of this current work. Nonetheless, these results are not only interesting and important from 114 a mathematical 117

  160for each boundary point ξ Y , the actual relocation of ξ Y only occurs if sufficient but not 161 complete ECM degradation will have occured in the peritumoural region Y \ Ω(t 0 ). To Then, the movement of a boundary point is exercised only when adequate but not complete 164 degradation of the peritunoural ECM occurs, which is characterized by the situation when 165 this transitional probability q(x * Y ) exceeds a certain tissue threshold ω(•, •) (as defined in ∈ (0, 1) controls the optimal level of ECM for cancer invasion and v(x, t) := 168 l(x, t) + F(x, t).

	166
	167	[19]), namely
		ω(β, Y) :=	            	sin sin	π 2 2(1-β) 1-β • sup v(x * Y , t 0 + ∆t) v(ξ, t 0 +∆t) ξ∈∂Ω(t 0 ) ξ∈∂Ω(t 0 ) π v(x * Y , t 0 + ∆t) sup v(ξ, t 0 +∆t)	-β	if otherwise, v(x * Y , t 0 +∆t) sup v(ξ, t 0 +∆t) ξ∈∂Ω(t 0 )	≤ β,
		where β

162

quantify the amount of ECM degradation, we use a transitional probability that we define by

163 q(x * Y ) := Y(x * Y )\Ω(t 0 ) m(y, τ)dy Y(x * Y )

m(y, τ)dy ,

As we mentioned, we see significant differences between the three cases only in Figure 46 3. This either indicates that the anisotropic diffusion tensor provides valuable information 47 only in certain cases or that the initial micro-fibre density differs from the one that produced position z * , we use the so-called reallocation vector ν δY(x) (z, t) which takes into account the 139 rearrangement vector r(δY(x), t), defined in [START_REF] Anderson | A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion[END_REF], the degree of alignment between r(δY(x), t) 140 and the barycentral position vector x dir := zx and also incorporates the level of fibres at 141 position z. Hence, following [START_REF] Shuttleworth | Multiscale Modelling of Fibres Dynamics and Cell Adhesion within Moving Boundary Cancer Invasion[END_REF], we define it as

where f max > 0 is the maximum level of fibres, f * := f (x, t)/ f max is the saturation level