Data-driven spatio-temporal modelling of glioblastoma Andreas Christ Sølvsten Jørgensen1 Ciaran Scott Hill23 Marc Sturrock4 Wenhao Tang1

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Data-driven spatio-temporal modelling of

glioblastoma

Andreas Christ Sølvsten Jørgensen1∗, Ciaran Scott Hill2,3, Marc Sturrock4, Wenhao Tang1,

Saketh R. Karamched5, Dunja Gorup5, Mark F. Lythgoe5, Simona Parrinello3, Samuel

Marguerat6, Vahid Shahrezaei1∗

1Department of Mathematics, Faculty of Natural Sciences, Imperial College London, London, UK

2Department of Neurosurgery, The National Hospital for Neurology and Neurosurgery, London, UK

3Samantha Dickson Brain Cancer Unit, UCL Cancer Institute, London WC1E 6DD, UK

4Department of Physiology and Medical Physics, Royal College of Surgeons in Ireland, Dublin, Ireland

5Division of Medicine, Centre for Advanced Biomedical Imaging, University College London (UCL), London, UK

6Genomics Translational Technology Platform, UCL Cancer Institute, University College London, London, UK

Correspondence*: Andreas Christ Sølvsten Jørgensen and Vahid Shahrezaei

a.joergensen@imperial.ac.uk, v.shahrezaei@imperial.ac.uk

Abstract: Mathematical oncology provides unique and invaluable insights into tumour growth on both the

microscopic and macroscopic levels. This review presents state-of-the-art modelling techniques and focuses on

their role in understanding glioblastoma, a malignant form of brain cancer. For each approach, we summarise

the scope, drawbacks, and assets. We highlight the potential clinical applications of each modelling technique

and discuss the connections between the mathematical models and the molecular and imaging data used to

inform them. By doing so, we aim to prime cancer researchers with current and emerging computational tools for

understanding tumour progression. Finally, by providing an in-depth picture of the diﬀerent modelling techniques,

we also aim to assist researchers who seek to build and develop their own models and the associated inference

frameworks.

1 Introduction

Glioblastoma (GBM) is a malignant hierarchically organised brain cancer. It is both the most common and most

aggressive type of primary brain cancer in adults [44]. Not only does the diﬀusive invasion of glioma cancer cells

into healthy tissue impede complete resection, but GBM harbours a subpopulation of highly therapy-resistant

stem-like cells [113]. Tumour recurrence is inevitable, resulting in a median survival time of 15 months for patients

despite maximal treatment [191]. The current gold standard of treatment is the Stupp protocol, which consists

of maximal safe surgical resection, followed by radiotherapy and chemotherapy with temozolomide, an alkylating

agent [174;173].

Mathematical cancer models have provided a deeper understanding of this immensely complex disease by

unveiling the underlying mechanisms and oﬀering quantitative insights [6;3;5;37]. Such models have entered all

areas of GBM research, ranging from the classiﬁcation and detection of brain tumours to therapy [207;59].

This review provides the reader with an overview of existing mathematical and computational models that

aim to simulate spatially resolved tumour growth. We discuss three main paradigms that have emerged for such

in silico experiments. In Section 2, we introduce so-called continuum models that treat variables, such as the

tumour cell density, as continuous macroscopic quantities based on conservation laws. Alternatively, one might

represent each cell as an individual agent. Such discrete models are discussed in Section 3. Section 4deals with

hybrid multi-scale and multi-resolution models that merge and bridge diﬀerent approaches.

arXiv:2210.01537v1 [q-bio.QM] 4 Oct 2022

Jørgensen et al. Modelling Glioblastoma

Each of the three methods has its advantages and shortcomings. One must choose between them based on

computational limitations and the level of detail required to answer the research questions of interest. With this

review, we aim to assist researchers in choosing between the diﬀerent methods by highlighting the drawbacks and

assets of each approach and by showing how the diﬀerent methods can complement each other. Moreover, we

summarise the main concepts and the key mathematical expressions that lie at the core of each approach. We

hereby aim to strike a balance between providing a brief overview and showing the mathematics involved.

Mathematical models of GBM draw on a wide variety of molecular and imaging data: histopathological data,

computerised tomography (CT), positron emission tomography (PET), single-photon emission computerised to-

mography (SPECT), and magnetic resonance imaging (MRI), such as T1 weighted (+/- Gadolinium contrast),

T2 weighted, T2-FLAIR, and diﬀusion-weighted imaging (DWI), and most recently spatial and single cell tran-

scriptomics. The models have thus been employed to shed light on patient-speciﬁc data in vivo and ex vivo as

well as on data from animal models and in vitro experiments. These analyses have provided invaluable insights

and have deepened our understanding of glioma on a molecular and structural level. We address this issue in

more detail in Section 5. The section also discusses the computational challenges posed by systematic inference

of model parameters. Finally, Sections 6and 7provides a short overview of some clinical applications of these

models and an overall summary, respectively.

While we discuss the diﬀerent methods in light of GBM, we note that the same models are applied to other

types of cancer. Indeed, the models build on concepts that are broadly used to study tissues. We do, therefore,

not only cite sources that deal with GBM but occasionally refer the reader to illustrative papers from other areas

of oncology and biology (see also [89;99]). It is worth noting that the models are, in a broader sense, actually

widely used across scientiﬁc disciplines. Throughout the paper, we thus present ideas and concepts that are also

employed in other ﬁelds ranging from statistical mechanics to solid-state physics. For instance, the cellular Potts

model presented in Section 3.1.3 builds on the so-called Ising model used to describe ferromagnetism. We hence

encourage the reader to think outside the box when exploring the literature, and, in this spirit, we provide a few

citations to areas outside the realm of biology.

Before commencing, we would like to point the reader towards other reviews and papers for further details.

Lowengrub et al. [120] provide a detailed account of continuum models. For an elaborate discussion on discrete

models, we refer the reader to Van Liedekerke et al. [116]. Both Metzcar et al. [132] and Weeransinghe et al.

[195] give a brief overview of the topic and include a list of recent references. For insights into hybrid multi-scale

modelling, we recommend Deisboeck et al. [45] and Chamseddine & Rejniak [28]. Falco et al. [58] present a

concise overview highlighting their clinical implications. Ellis et al. [50] focus on mathematical models that

intratumour heterogeneity and tumour recurrence based on next-generation sequencing techniques. Alfonso et

al. [1] highlight the challenges that mathematical models face when dealing with glioma invasion. Finally, for an

overview of the biology of the GBM from a clinical perspective, we refer the reader to the recent reviews by Finch

et al. and McKinnon et al. [59;129].

2 Continuum models

When dealing with cancer treatment, we face questions related to tumour size, shape and composition. These

questions all address cancer on a macroscopic scale. While macroscopic tumour dynamics emerge from interactions

on a cellular level, it is possible to construct informative mathematical models without tracking individual cancer

cells. Instead, the tumour and its environment can be represented as continuous variables that are governed by

partial diﬀerential equations (PDE). Such continuum models capture many aspects of cancer in vitro,in vivo,

and in patients. They can account for the impact of heterogeneous brain tissue on tumour growth, for diﬀerent

invasive tumour morphologies, and to some extent, even for potential tumour recurrence [e.g. 62;176]. Moreover,

they have successfully been applied in studies on the impact of chemotherapy and repeated immuno-suppression

treatment [178;63]. Continuum models have thus been employed in diagnosis and treatment planning based on

patient-speciﬁc data [e.g. 177;75;35;135;86].

However, PDEs smooth out small-scale ﬂuctuations, which implies that continuum models do not apply

to small cell populations, such as those found in the tumour margin. For such small cell numbers, stochastic

events play a crucial role, and the applicability and predictive power of continuum models are limited. To

Jørgensen et al. Modelling Glioblastoma

properly understand cancer invasion, we must track individual cells. But to do so comes at a high or currently

insurmountable computational cost (see Section 3). Thus the use of continuum models represents a trade-oﬀ that

enables and supports scalability and mathematical insights. As a result, continuum models are widely used in

the community [193;120;195].

This section introduces the basic mathematical concepts of continuum models and their biological motivation.

Any such model in the literature builds on reaction-diﬀusion equations, describing variables such as the tumour cell

density, the tumour volume fraction, the nutrient (oxygen, glucose) concentration, the neovasculature, the enzyme

concentration, or other properties of the extracellular matrix (ECM) [e.g. 211;63;42;148;66]. Considering any

such variable, ψ(x, t), which is a function of position xand time t, we have for its rate of change with time

∂ψ

∂t =−∇J+S, (1)

where Jis the ﬂux of the considered variable, while Sis the sources and sinks for this variable. Thus, Equation (1)

constitutes a conservation law. The exact expression for Jand S, as well as the boundary conditions, will depend

on the variable in question. For instance, when dealing with the quasi-steady diﬀusion of nutrients, Equation (1)

generally takes the form [39;168;38;124;25]

0 = D∇2n+S. (2)

Here, Ddenotes a diﬀusion coeﬃcient, while n(x)is the relevant nutrient concentration at the location xwithin

the considered n-dimensional domain, which is oftentimes denoted by Ω.

Alternatively, let’s consider the (normalized) cancer cell density, ρ(x, t), at a time tand location x. Many

authors assume that the diﬀusion of cancer cells can be well-approximated by Fick’s ﬁrst law

J=−D∇ρ, (3)

where Ddenotes a diﬀusion coeﬃcient, which we discuss in detail in Section 2.1. As regards the sources and sinks

of the cancer cell density, it is commonly assumed that cell proliferation, i.e. tumour growth, is well-described by

a logistic growth term [e.g. 175;161]. So, Equation (1) takes the form

∂ρ

∂t =∇(D∇ρ) + λρ(1 −ρ)(4)

where λis the growth rate of the tumour cell population. Other authors assume exponential tumour growth,

substituting the second term on the right-hand side by λρ [e.g. 177]. Of course, more than one term might be

necessary to summarise the relevant sources and sinks. Several more complex terms are, for instance, needed

when considering diﬀerentiation between diﬀerent interdependent tumour sub-populations (cf. Section 2.2).

It is worth stressing that Equation (1) cannot stand on its own. Other relations and constraints, including

(Neumann) boundary conditions, are needed. One might, for instance, naturally require that there is no ﬂux at

the boundary of the brain domain, i.e. that the tumour doesn’t penetrate the patient’s skull [e.g. 68;143;161].

This being said, Equation (1) is the backbone of any continuum model that spatially resolves the tumour (see

also Section 4.1.1).

While we focus on spatio-temporal cancer models in this review, it is also worth noting that Equation (1) can

be seen as a natural extension to non-spatial models for tumour growth. Indeed, if we drop any spatial dependence

in Equation (1), including the diﬀusion term, we end up with an ordinary diﬀerential equation (ODE) for the

tumour cell density on the form dρ

dt=S. (5)

Depending on the source term, S, Equation (5) might describe exponential, logistic or Gompertz tumour growth

laws that serve as the foundation for non-spatial cancer models. Building on Equation (5), one might thus

construct a sophisticated network of coupled ODEs (or delay diﬀerential equations) that might diﬀerentiate

between diﬀerent tumour cell subpopulations or account for immune responses or the eﬀect of cancer treatment

[e.g. 49;134;13;206, and references therein].

Jørgensen et al. Modelling Glioblastoma

2.1 Anisotropic Diﬀusion

By using a scalar for the diﬀusion coeﬃcient in Equation (4), we assume isotropic tumour growth. However, GBM

spreads anisotropically, primarily expanding along pre-existing structures, such as blood vessels and white matter

tracks [164;70;153;110;74;40;22]. To take the heterogeneous structure of the brain into account, Swanson et al.

[177] hence proposed to adopt diﬀerent values for the diﬀusion coeﬃcients in grey and white matter. Concretely,

based on CT scans by Tracqui et al. [183], Swanson et al. [177] found the diﬀusion coeﬃcient in white matter to

be more than ﬁve times larger than in grey matter.

Taking the idea of heterogeneous diﬀusion further by including anisotropy, other authors [e.g. 90;84;110;143;

9] substitute the diﬀusion coeﬃcient with an n-dimensional diﬀusion tensor, D(x, t)∈Rn×n. To evaluate D(x, t),

Painter & Hillen [143] deploy diﬀusion tensor imaging (DTI) data. DTI is a magnetic resonance imaging (MRI)

technique that measures the anisotropic diﬀusion of water molecules and hereby maps highly structured tissue.

This technique provides the diﬀusion tensor for water molecules, DΠ(x, t), throughout the brain. Of course, due

to the size diﬀerence, the movement of cancer cells is more restricted than that of water molecules, which means

that DΠ(x, t)does not adequately describe glioma growth, i.e. D(x, t)6=DΠ(x, t). However, based on a transport

equation for individual cell movement, Painter & Hillen [143] establish a relation between D(x, t)and DΠ(x, t)

expressed in terms of the Fractional Anisotropy (FA) that is commonly used to quantify DTI data [16]. They do

so based on a set of simplifying assumptions and parabolic scaling to a macroscopic model [see 82, for further

details]. Their ﬁnal macroscopic model takes the form

∂ρ

∂t =∇∇(Dρ) + λρ(1 −ρ),(6)

where the diﬀusion tensor D∈Rn×nis symmetric and positive-deﬁnite, as it is related to the variance-covariance

matrix of the probability distribution function that describes the velocity changes of individual cells [see 82;143].

In other words, when dealing with three-dimension data, D(x, t)is a symmetric and positive-deﬁnite 3×3matrix

that incorporates the impact of the local environment on cell migration. The model by Painter & Hillen [143] has

been extended and applied by other authors [53;54;176].

Note that Equation (6) is subtly diﬀerent from Equation (4). Apart from Ddenoting an n×ntensor rather

than a scalar, Equation (6) includes an additional advective-type term since ∇∇(Dρ) = ∇(D∇ρ) + ∇(∇Dρ).

Models of the form of Equation (6) are referred to as Fokker-Planck models, while Equation (4) is an example of

a Fickian model. The additional advection term of the Fokker-Planck model has a demonstrable impact on the

solution [see also 15]. We also note that Equation 6would correspond to the Fisher’s equation if the ﬁrst term

on the right-hand side were substituted by D∇2ρ. Indeed, some authors employ a diﬀusion term of this kind to

describe the cancer cell density [114].

2.2 Mechanical interactions, cell types, lineage, and feedback

Tumours are hierarchically organised. GBMs harbour stem-like cells (GSC) as well as proliferating (GCP) and

diﬀerentiating (GTP) subpopulations [cf. 169;188;113;22]. Since these three cell types exhibit very diﬀerent

behaviours, it is insightful to diﬀerentiate between these subpopulations, even when dealing with continuum

models.

Models that distinguish between viable and necrotic tumour tissue are the ﬁrst step in this direction. Examples

of such models can be found in the papers by Wise et al. [199;198] and Frieboes et al. [26;60]. Their work is

based on reaction-diﬀusion equations of the form

∂ρi

∂t +∇(uiρi) = −∇Jmec,i +Si(7)

where the index iruns over all subpopulations, uidenotes the velocity of the considered cell species, and Jmec,i

is the ﬂux that arises from mechanical interactions

Jmec,i =Ji−ρiui.(8)

Note that while Equation (7) appears to diﬀer from the other reaction-diﬀusion equations listed above, this

Jørgensen et al. Modelling Glioblastoma

is merely a matter of notation. It’s a Fickian model that can be derived by inserting Equations (3) and (8) into

Equation (1). The advantage of phrasing the problem in this manner is that the mechanical ﬂux reﬂects the

mechanical interaction energy that can be obtained from an understanding of the underlying cell biology [see also

101;200;102;62]. By introducing Jmec,i , it is thus possible to inform the model about the properties of the

tumour and host without the necessity of constructing a suitable diﬀusion tensor.

In papers that employ the Equation (8), uiis computed by imposing relations similar to Darcy’s law that

links the velocity to a gradient in pressure [e.g. 62]. For glioma, the relevant expression often takes the form

ui=−∇p+Fi, where p is the solid pressure arising from the tumour proliferation, whilst Fireﬂects mechanical

interactions.

We exemplify the source functions that enter Equation (7) by listing the relevant terms for the necrotic tissue

according to Wise et al. [199], for which

Sd=λAρv+λNH(nN−n)ρv−λCρd.(9)

Here, the indices ‘d’ and ‘v’ refer to the dead and viable cancer cells, respectively, while λA,λN, and λCdenote

the rates of apoptosis, necrosis, and the clearance of dead cells. Moreover, His a Heaviside step function, and nN

is a viability limit for the nutrient concentration below which cells die. For comparison, the source function for

the viable tissue takes a similar form:

Sv=−λAρv−λNH(nN−n)ρv+λM

n∞

ρv,(10)

where λMdenotes the rate of mitosis, and n∞is the far-ﬁeld nutrient level.

By distinguishing between GSCs, GCPs, and GTPs, Kunche et al. [112] and Yan et al. [204;205;203] have

incorporated the GBM lineage and hereby taken the discussed models one step further. Kunche et al. [112]

use this to investigate feedback regulation of cell lineage progression. Furthermore, while previous papers only

consider adhesion when computing the mechanical interactions, Chen et al. [33;31;32] include the impact of

elastic membranes and the implications of the calciﬁcation of dead tumour cells [144].

2.3 Modelling the macroscopic environment

Understanding the microenvironment is essential for understanding GBM since the brain region and other prop-

erties, such as the patient’s age, have been shown to play a key role in tumour development and heterogeneity

[e.g. 22;154].

The concentrations of diﬀerent chemicals, including (but not limited to) nutrients, drugs, matrix-degrading

enzymes, and ECM macromolecules, are commonly modelled using reaction-diﬀusion equations. The associated

diﬀusion is often in the form D∇2ψ, but other second-order spatial derivatives can be found in the literature

[7;92;147]. The source terms reﬂect the processes at play. For instance, the rate of oxygen consumption is often

assumed to be proportional to the local oxygen concentration and might be proportional to the local cell density.

Overall, the use of continuum models to model, say, nutrient ﬂows is justiﬁed through the diﬀerent scales that

deﬁne cancer. Continuum models reliably capture the spatial gradients and temporal variation of nutrients in

a way that allows us to assess the behaviour of tumour cells. For some purposes, it might even be adequate to

assume that the nutrient levels stay constant over time since cell proliferation takes place on a much longer time

scale than nutrient diﬀusion (cf. Equation 2).

Initially, tumours do not possess their own vasculature but rather rely on the diﬀusion of nutrients and waste

products. During this so-called avascular phase, the tumour may lack some features of malignancy and appear

more benign. But beyond a certain size (1-3 mm in radius), the nutrient inﬂow can no longer sustain the growing

cell population [126]. Hypoxia, i.e. oxygen shortage, sets in. Without stimulating angiogenesis, i.e. recruiting

vasculature, tumour growth will stagnate. However, in response to the hypoxic conditions, the cancer cells

release tumour angiogenic factors (TAFs), which stimulates the migration and proliferation of endothelial cells

(ECs). New blood vessels sprout towards the tumour, and tumour growth resumes. Understanding the transition

from avascular to vascular tumour growth is essential since it is a critical step toward malignancy. To study

angiogenesis, many authors model both the diﬀusion of TAFS and ECs using reaction-diﬀusion equations [e.g.

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Data-drivenspatio-temporalmodellingofglioblastomaAndreasChristSølvstenJørgensen1,CiaranScottHill2;3,MarcSturrock4,WenhaoTang1,SakethR.Karamched5,DunjaGorup5,MarkF.Lythgoe5,SimonaParrinello3,SamuelMarguerat6,VahidShahrezaei11DepartmentofMathematics,FacultyofNaturalSciences,ImperialCollegeLondon,Lon...

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Data-driven spatio-temporal modelling of glioblastoma Andreas Christ Sølvsten Jørgensen1 Ciaran Scott Hill23 Marc Sturrock4 Wenhao Tang1

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