# 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
<EMAIL_ADDRESS><EMAIL_ADDRESS>
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 4
deals with hybrid multi-scale and multi-resolution models that merge and
bridge different approaches.

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 tomography (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 transcriptomics. 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 6 and 7
provides 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 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, $\psi(\mathbf{x},t)$, which is a function of
position $\mathbf{x}$ and time $t$, we have for its rate of change with time

$\frac{\partial\psi}{\partial t}=-\nabla\mathbf{J}+S,$ (1)

where $\mathbf{J}$ is the flux of the considered variable, while $S$ is the
sources and sinks for this variable. Thus, Equation (1) constitutes a
conservation law. The exact expression for $\mathbf{J}$ and $S$, 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]

$0=D\nabla^{2}n+S.$ (2)

Here, $D$ denotes a diffusion coefficient, while $n(\mathbf{x})$ is the
relevant nutrient concentration at the location $\mathbf{x}$ within the
considered n-dimensional domain, which is oftentimes denoted by $\Omega$.

Alternatively, let’s consider the (normalized) cancer cell density,
$\rho(\mathbf{x},t)$, at a time $t$ and location $\mathbf{x}$. Many authors
assume that the diffusion of cancer cells can be well-approximated by Fick’s
first law

$\mathbf{J}=-D\nabla\rho,$ (3)

where $D$ denotes 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

$\frac{\partial\rho}{\partial t}=\nabla(D\nabla\rho)+\lambda\rho(1-\rho)$ (4)

where $\lambda$ is the growth rate of the tumour cell population. Other
authors assume exponential tumour growth, substituting the second term on the
right-hand side by $\lambda\rho$ [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

$\frac{\mathrm{d}\rho}{\mathrm{d}t}=S.$ (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].

###  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,
$D(\mathbf{x},t)\in\mathbb{R}^{n\times n}$. To evaluate $D(\mathbf{x},t)$,
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,
$D_{\Pi}(\mathbf{x},t)$, 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 $D_{\Pi}(\mathbf{x},t)$ does not adequately
describe glioma growth, i.e. $D(\mathbf{x},t)\neq D_{\Pi}(\mathbf{x},t)$.
However, based on a transport equation for individual cell movement, Painter &
Hillen [143] establish a relation between $D(\mathbf{x},t)$ and
$D_{\Pi}(\mathbf{x},t)$ 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

$\frac{\partial\rho}{\partial t}=\nabla\nabla(D\rho)+\lambda\rho(1-\rho),$ (6)

where the diffusion tensor $D\in\mathbb{R}^{n\times n}$ is 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, $D(\mathbf{x},t)$ is a symmetric and positive-definite
$3\times 3$ matrix 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 $D$
denoting an $n\times n$ tensor rather than a scalar, Equation (6) includes an
additional advective-type term since
$\nabla\nabla(D\rho)=\nabla(D\nabla\rho)+\nabla(\nabla D\rho)$. 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 6 would correspond to the Fisher’s equation if the
first term on the right-hand side were substituted by $D\nabla^{2}\rho$.
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

$\frac{\partial\rho_{i}}{\partial
t}+\nabla(\mathbf{u}_{i}\rho_{i})=-\nabla\mathbf{J}_{\mathrm{mec},i}+S_{i}$
(7)

where the index $i$ runs over all subpopulations, $\mathbf{u}_{i}$ denotes the
velocity of the considered cell species, and $\mathbf{J}_{\mathrm{mec},i}$ is
the flux that arises from mechanical interactions

$\mathbf{J}_{\mathrm{mec},i}=\mathbf{J}_{i}-\rho_{i}\mathbf{u}_{i}.$ (8)

Note that while Equation (7) appears to differ from the other reaction-
diffusion equations listed above, this 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 $\mathbf{J}_{\mathrm{mec},i}$, 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), $\mathbf{u}_{i}$ is 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 $\mathbf{u}_{i}=-\nabla p+\mathbf{F}_{i}$, where p is the solid
pressure arising from the tumour proliferation, whilst $\mathbf{F}_{i}$
reflects 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

$S_{\mathrm{d}}=\lambda_{\mathrm{A}}\rho_{\mathrm{v}}+\lambda_{\mathrm{N}}\mathcal{H}(n_{\mathrm{N}}-n)\rho_{\mathrm{v}}-\lambda_{\mathrm{C}}\rho_{\mathrm{d}}.$
(9)

Here, the indices ‘d’ and ‘v’ refer to the dead and viable cancer cells,
respectively, while $\lambda_{\mathrm{A}}$, $\lambda_{\mathrm{N}}$, and
$\lambda_{\mathrm{C}}$ denote the rates of apoptosis, necrosis, and the
clearance of dead cells. Moreover, $\mathcal{H}$ is a Heaviside step function,
and $n_{\mathrm{N}}$ 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:

$S_{\mathrm{v}}=-\lambda_{\mathrm{A}}\rho_{\mathrm{v}}-\lambda_{\mathrm{N}}\mathcal{H}(n_{\mathrm{N}}-n)\rho_{\mathrm{v}}+\lambda_{\mathrm{M}}\frac{n}{n_{\infty}}\rho_{\mathrm{v}},$
(10)

where $\lambda_{\mathrm{M}}$ denotes the rate of mitosis, and $n_{\infty}$ 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 properties, 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 $D\nabla^{2}\psi$, 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. 30; 140; 126; 120; 75]. As regards the ECs,
we again encounter logistic or exponential source terms. Meanwhile, the
gradient of the TAF concentration, as well as that of adhesive molecules (e.g.
fibronectin), enter through a flux term encapsulating chemotaxis and
haptotaxis:

$\mathbf{J}=-D\nabla\rho_{\mathrm{EC}}+\sum_{i}\chi_{i}\rho_{\mathrm{EC}}\nabla
c_{i},$ (11)

where $\rho_{\mathrm{EC}}$ denotes the EC cell density, and the sum runs over
all relevant diffusive components, including TAFs. The concentration of and
sensitivity to the $i$th components are denoted as $c_{i}$ and $\chi_{i}$,
respectively [see 126].

Like continuum models for tumour cells, continuum models for ECs cannot
capture the behaviour of individual cells. Some authors, therefore, rely on
agent-based models to study the ECs during angiogenesis [166; 43; 202; 29]. We
refer to Section 3 for a detailed discussion of agent-based models [see also
4]. Table 1 summarises the properties of all continuum models discussed in
this section.

Table 1: Overview of the reviewed continuum models. The references are not exhaustive. orange!20 Model type |  Features |  References
---|---|---
Continuum model
(CM), tumour |  Concept: The tumour cell density can be modelled as a fluid using PDEs based on conservation laws.
Assets: CMs capture macroscopic tumour growth at low computational cost.
Drawbacks: They cannot capture heterogeneity or the behaviour at low cell densities (e.g. at the tumour front). |  Methods: Anisotropic diffusion tensor [143; 53; 176] cf. Sec. 2.1. Interaction energy-based models [199; 62; 203; 32] accounting for lineage [204] cf. Sec. 2.2. Patient-specific studies: [75; 135; 86]
CM, microenvironment |  Concept: Different properties of the environment, such as nutrient flow or vasculature, are modelled on a macroscopic scale.
Assets: They are computed at a low computational cost. Spatial gradients and
time variations, e.g. in nutrient levels, are sufficiently well resolved to
study tumour cells.
Drawbacks: CMs of ECs do not capture the behaviour of individual cells. |  Nutrient flow: [7; 92; 147].
Vasculature: [126; 120]
These models are widely used whether the tumour is represented by a CM or by
agent-based models (cf. Sec. 3 and 4).

##  3 Discrete (agent-based) models

Continuum models are based on the assumption that the behaviour of tumour
cells is well-approximated by macroscopic averages. However, this assumption
breaks down for small cell populations since stochastic events dominate these.
Therefore, continuum models cannot reliably capture the onset of tumour
growth, and they do not give a complete picture of the invasive tumour front
or margin, i.e. the tumour-host interface, where the cancer cell density is
low — although it is possible to address this drawback by linking different
scales, as shown by Trucu et al. [184] (see also Section 4). Due to the
aggressive invasion of healthy tissue by GBM, this shortcoming is especially
problematic for understanding brain tumours. Moreover, continuum models cannot
adequately deal with heterogeneity. To address these issues, one needs to
model cancer on a mesoscopic scale following individual tumour cells or cell
clusters [see also 76]. Mathematical models that do so are referred to by
various names: discrete models, agent-based models, individual-based models,
or cell-based models. In all cases, cells or cell clusters are modelled as
autonomous agents that follow a set of (stochastic) rules prescribing cell
movement, death, division, and growth. By doing so, discrete models capture
the variability among cells that might, for instance, arise as a response to
the microenvironment [e.g. 7]. Note that while discrete models treat cells as
autonomous quantities, continuum models are commonly used to depict the
microenvironment (cf. Section 4.1). PDEs hence still come into play when
dealing with, e.g. nutrient and drug concentrations [e.g. 120; 123].

Discrete models that describe spatial tumour growth or other cell populations,
such as ECs (cf. Section 2.3), fall into two categories: lattice-based and
off-lattice methods. While the former operates on a fixed lattice, the latter
does not impose the same restrictions. Lattice-based models can be divided
into three sub-categories [116; 132]. First, there exists a group of so-called
cellular automata (CA) models that treat all processes, including cell
movement, as stochastic processes (cf. Section 3.1.1). For these models, each
lattice site (e.g. pixel or voxel) can either at most be occupied by a single
cell or a population of a limited size [97; 96; 125; 165]. Each cell or cell
cluster is thus characterised by its position on the grid. Secondly, there
exist so-called lattice gas cellular automata (LGCA). While these models
consider single cells and are conceptually very similar, they differ from
other CA models by attributing a velocity to each cell (cf. Section 3.1.2).
After accounting for stochastic events, these models thus include a
deterministic evaluation of cell propagation at every time step. Thirdly,
there are so-called cellular Potts models (CPM) that phrase cell interactions
within the framework of statistical mechanics (cf. Section 3.1.3). CPMs
attribute several lattice points to each cell, whereby these models capture
cell morphologies. Off-lattice models, also known as lattice-free models, come
in two flavours. Centre-based models (CBM) describe cells as simple particles
whose interactions can be expressed as physical forces. Deformable cell models
(DCM) or vertex models extend this picture by resolving the cells using
several nodes. Like CPMs, DCMs can thereby account for cell morphologies.

One drawback of discrete models is their relatively high computational expense
due to the large number of cells that typically need to be simulated. After
all, tumours reach $10^{5}-10^{6}$ cells per mm3. However, one benefit of
discrete models is their ability to capture emergent properties. The simple
rules that govern the (nonlinear) interactions between individual agents will
naturally give rise to the dynamics of macroscopic tumour growth. As discussed
in Section 3.1.2, reaction-diffusion equations might thus naturally follow
from up-scaling agent-based models [80; 79; 87]. Due to the ability of
discrete models to capture emergent properties, agent-based models can be
calibrated based on patient-specific data and be used to predict macroscopic
parameters [122; 85].

Concerning the implementation of discrete models, it is worth mentioning that
some software tools are publicly available. There is no need for a research
group to start from scratch. For a recent overview of open-source toolkits, we
kindly refer the reader to Table 1 in the paper by Metzcar et al. [132]. In
this connection, we note that one must always select and adapt the numerical
approach based on the problem at hand. For instance, not all problems require
the same geometry. While colon cancer cells and cells in vitro move in two
dimensions, glioblastoma tumour growth in vivo is intrinsically three-
dimensional, which affects the growth pattern [67]. In the following two
sections, we will discuss lattice-based models in more depth (Section 3.1) and
then turn to a detailed discussion of off-lattice models (Section 3.2).

###  3.1 Lattice-based models

In lattice-based models, we follow a population of cells on a rigid grid. If a
regular (e.g. Cartesian) lattice is chosen, artefacts in the established
tumour growth pattern might arise, reflecting the lattice symmetries [116].
While unstructured meshes are harder to implement and combine with existing
PDE solvers, they do not suffer from this shortcoming [18].

####  3.1.1 Cellular automata with one or several cells per lattice site

In this section, we discuss two types of cellular automata (CA) models: those
with at most one and those with multiple cells per lattice site. Both
approaches have their advantages and drawbacks. Below, we will start by
discussing the single-cell CA models and then continue to frameworks that
include many cells per lattice site.

When considering single tumour cells, we follow each of these through a
sequence of time steps. During every time-step ($t\rightarrow t+\Delta t)$, a
cell might migrate to a neighbouring lattice site, it might die, it might
divide, and it might grow, whereby it temporally occupies two sites before
dividing. Each of these events can be modelled as stochastic processes or be
imposed through deterministic conditional statements that encompass our
knowledge about cancer — additional parameters, such as age, mutation, or
phenotype, might be included in these rules [146]. For instance, while cells
might die spontaneously, external factors can make cell death increasingly
likely. Thus, tumours develop a necrotic core due to nutrient deprivation. One
might, therefore, set the probability for individual cell death to depend on
the distance to the tumour front [165], on the local nutrient concentration,
or the local level of toxic metabolites [125].

Tumour cell migration is also affected by the location of the cell within the
tumour. If we consider a cell that is deeply buried within the tumour, all
surrounding lattice sites are already occupied, and there is nowhere to
migrate. This notion also plays a role in cell proliferation. To divide, a
cell deep within the tumour must push its neighbours away to make space for
the daughter cell, which becomes increasingly difficult as the distance to the
tumour edge increases. For simplicity, many authors account for this notion by
introducing a sharp cut-off: If there are no free sites within a certain
distance, the cell will be unable to divide [47]. Other models assume a smooth
decline in proliferation with depth [87; 88]. On a macroscopic level, this
suppression of cell division leads to a constant proliferative rim, below
which we encounter a quiescent cell population surrounding a hypoxic region
with a necrotic core at its centre. In accordance with data, this model
predicts that a tumour spheroid enters linear growth after initially growing
exponentially [83]. As regards cell division, we also note that experiments
suggest that the cell cycle time follows a $\Gamma$-like distribution, e.g. an
Erlang distribution [152]. The rules imposed on the individual cells can
account for a broad range of cell properties. For instance, if the CA model
accounts for the hierarchical nature of tumour cells (GSCs, GCPs and GTPs),
the probability of cell proliferation will also depend on the attributed cell-
type [146].

When solving the system numerically, one must choose a suitable time-step,
$\Delta t$. For this purpose, it is helpful to consider the cell population as
a whole. Every time a cell dies, moves, divides, or grows, the system as a
whole will transition into a new state. The time-step should be chosen
accordingly such that only one event is likely to occur within $\Delta t$, and
thus $\Delta t$ is itself a function of time. To achieve this, one might
distil the temporal dynamics of the system into a master equation and employ
kinetic Monte Carlo algorithms [71; 20; 87]. Moreover, the order in which the
individual cells are updated should be random to avoid grid artefacts [120].

Rather than considering individual cells, one might track cell clusters of a
limited size, keeping track of the number of cells at each lattice site. The
underlying concepts stay the same. The advantage of reducing the resolution in
this manner is the reduced computational cost, which allows for modelling
tumours of centimetre size [152]. Meanwhile, a coarser resolution might lead
to artefacts affecting the predicted tumour growth. Another drawback of both
single- and multiple-cell CA models is the fact that they do not properly
account for cell morphologies. For models that address this issue, see Section
3.1.3.

####  3.1.2 Lattice gas cellular automata models

Like the models in Section 3.1.1, lattice gas cellular automata (LGCA) models
follow a set of stochastic rules to determine whether individual cells die,
grow, or divide. But in addition to specifying the location of the cells on
the lattice, LGCA models attribute a discrete set of velocity channels to each
spatial location. Hence, the cells roam a discretised phase space.

Analogously to migration in the CA models presented above, cells might
stochastically migrate in phase-space: They can move to neighbouring velocity
channels following probabilistic rules. At most, one cell can occupy any given
velocity channel at a given location. This restriction is not to say that
these channels are unique, i.e. at each location, there might be one or more
channels with the same velocity, including channels at which the cell is at
rest.

When using LGCA models, the time evolution of the system is split into two
distinct steps. First, the model allows for a number of stochastic events that
correspond to the number of cells. Afterwards, the model updates the position
of every cell based on their individual velocities complying with momentum and
mass conservation. In this manner, the model alternates between accounting for
probabilistic interactions and deterministic cell movement. These basic
concepts of LGCA models are summarised in Fig. 1.

Figure 1: Schematic overview of the concepts behind lattice gas cellular
automata (LGCA) models. The left panel summarises the cell dynamics that are
handled as stochastic processes: cell migration, death (and lysis), growth,
and cell division. Filled (orange) circles denote occupied states, while there
are no cells in the empty circles. On its own, the left panel corresponds to
the cellular automata models with at most one cell per lattice site that are
discussed in Section 3.1.1. However, as illustrated in the right panel,
several velocity channels are associated with each location, and multiple
cells can occupy the same position. Filled orange circles denote occupied
channels, while empty channels are indicated with empty circles. Here, there
are five velocity channels per location: four channels that lead to migration
and one channel at which the cell is at rest. After determining the stochastic
processes, the model computes the movement of the cells based on the velocity
channels that they occupy.

Within the mean-field approximation, Hatzikirou & Deutsch [80] show that the
Boltzmann equation governs the macroscopic dynamics of LGCA models (The
Boltzmann equation can be used to describe the evolution of thermodynamic
systems that are out of equilibrium. Applications range from quantum mechanics
over cosmology to magnetic resonance imaging [e.g. 46]). Furthermore, through
Taylor expansions and scaling, they arrive at a reaction-diffusion equation
with a logistic growth term, giving a microscopic justification for the
success of the models presented in Section 2.

LGCA models offer an intuitive implementation of cell migration. By comparing
model predictions to experimental data, the group of Deutsch and Hatzikirou
has thus used such models to gain insights into cell-cell interactions and
mechanisms that underlie tumour invasion [34; 182; 79]. In particular, they
focus on the so-called "Go or Grow" (GoG) hypothesis. This hypothesis
addresses the fact that glioma cells exhibit an inverse correlation between
cell migration and proliferation [69]. It states that migration and
proliferation are mutually exclusive events, i.e. that moving cells cannot
divide. This dichotomy stems from shared signalling pathways [see also 72].
Since the transition from benign to malignant tumour growth is coupled with a
transition from a highly proliferative to a highly migrative phenotype, it is
paramount to understand the link between these two phenomena. Specifically,
the group of Deutsch and Hatzikirou investigate the role of hypoxia [see also
133].

####  3.1.3 Cellular Potts models

Cellular Potts models (CPM) define a Hamiltonian function, $H$, to incorporate
cell movement, growth, interactions between adjacent cells, and interactions
between cells and their microenvironment [see 181, for a detailed discussion
of Hamiltonian mechanics ]. Moreover, CPMs attribute multiple lattice sites to
each cell, which allows the models to account for cell morphology. Again, CPMs
do so via the Hamiltonian function. The basic concepts are illustrated in Fig.
2

Figure 2: Schematic overview of cellular Potts models. Each cell (and the ECM)
has a unique identifier. Here, we consider cells 1, 2, and 3 and the ECM
denoted by 0. Each cell occupies several lattice sites. Randomly selected
sites at the cell borders (*) might swap affiliation depending on the energy
change associated with this swap allowing the cells to grow and move.

For simplicity, let’s consider a Hamiltonian of the form [185; 166]

$H=\sum_{(\textbf{x,x'})}E(\tau(\sigma(\textbf{x})),\tau(\sigma(\textbf{x'})))\left(1-\delta\left(\sigma(\textbf{x}),\sigma(\textbf{x'})\right)\right)+\lambda_{\mathrm{V}}\sum_{\sigma}(V(\sigma)-V_{\mathrm{T}})^{2}.$
(12)

The first term on the right-hand side of Equation (12) summarises the contact
energy at the cell interfaces. The sum runs over all pairs of adjacent lattice
sites $(\textbf{x,x'})$. Each cell (and the ECM) has a unique identifier
$\sigma$, i.e. lattice sites with the same value of $\sigma$ are associated
with the same cell (or the ECM). Moreover, each cell has a type $\tau$ [e.g.
GSC, GCP, or GTP, see 65].
$E(\tau(\sigma(\textbf{x})),\tau(\sigma(\textbf{x'})))$ denotes the contact
energy per unit surface area (in 3D) between a cell of type
$\tau(\sigma(\textbf{x}))$ and a cell of type $\tau(\sigma(\textbf{x'}))$. The
Kronecker delta, $\delta\left(\sigma(\textbf{x}),\sigma(\textbf{x'})\right)$,
is needed since we only encounter an interface if
$\sigma(\textbf{x})\neq\sigma(\textbf{x'})$, while there will be an contact
energy associated with cells of the same type. The second term on the right-
hand side of Equation (12) represents a penalty for deviations between the
volume, $V(\sigma)$, of a cell and its target size, $V_{\mathrm{T}}$. Here,
the sum runs over all cells.

At every time-step, the CPM selects a random set of lattice sites at the cell
boundaries and attempts to overwrite their cell affiliations ($\sigma$) with
those of neighbouring sites:
$\sigma(\textbf{x})\rightarrow\sigma(\textbf{x'})$. In this manner, cells grow
and move by high-jacking adjacent sites, whereby they might have to push other
cells aside. Whether or not these proliferation and growth attempts are
successful is decided by a Metropolis algorithm based on the associated change
in the Hamiltonian function [see 73, for a discussion of such algorithms]. The
proposed change in cell affiliation is accepted with the probability

$P(\sigma(\textbf{x})\rightarrow\sigma(\textbf{x'}))=\min\left(1,\exp\left(-\frac{\Delta
H}{\beta}\right)\right),$ (13)

where the Boltzmann temperature, $\beta$, describes the motility of the cells.

One can include additional terms in Equation (12) to add more information
about cell biology. For instance, in analogy to the volume constraint, Ouchi
et al. [142] suggest including constraints on the surface area of the cell (in
3D) [see also 19]. Other authors include chemotaxis, motility, haptotaxis, and
haptokinesis in this manner [163; 43; 179; 115]. One immediate drawback of
CPMs is that they are limited to phenomena and cell properties that can be
formulated as terms in a Hamiltonian.

Meanwhile, when accounted for, cell division and death are treated as
stochastic events or occur when certain conditions are met. For instance, Gao
et al. [65] let cells attempt division when $V(\sigma)>2V_{\mathrm{T}}$ and
impose apoptosis for proliferating glioma cells after a pre-determined number
of division attempts. Other authors impose (probabilistic) conditions on
apoptosis and necrosis based on nutrient levels or overcrowding [166; 19].

###  3.2 Off-lattice models

One of the main drawbacks of lattice-based models is that the lattice
introduces a lower length scale. Off-lattice models overcome this problem at
the expense of increased computational cost. We give a schematic overview of
the two different types of off-lattice models in Fig. 3.

Figure 3: Schematic overview of off-lattice models. The left panel illustrates
spherical cells in a centre-based model (CBM), highlighting forces on cell 1.
Due to the overlap ($\delta_{21}$) between cell 1 and 2, cell 1 is repulsed by
cell 2. Overlapping cells forming a dumbbell also occur during cell division.
$\mathbf{F}_{31}$ exemplifies an adhesive force occurring without direct
contact. In an isotropic environment, the drag force will be proportional to
the velocity, $\mathbf{v}_{1}$. The right panel illustrates a cell made of
multiple nodes in a deformable cell model (DCM). Node $i$ at the cell membrane
is subject to forces from the surrounding nodes ($\mathbf{F}_{ji}$).
Constraints on areas and volumes might lead to additional forces
($\mathbf{F}_{\bigtriangleup i}$), while the movement of the node
($\mathbf{v}_{i}$) gives rise to viscous forces. The cell has a cytoskeleton
(CSK, dashed lines).

####  3.2.1 Center-based models

Centre-based models (CBMs) treat cancer cells as physical particles whose
trajectories can be deduced from their equations of motion [see 181]. The
cells themselves are usually represented as spheres (in 3D) or viscoelastic
ellipsoids that deform when subjected to external forces [41]. During mitosis,
the mother cells are oftentimes modelled as dumbbells, consisting of two
overlapping spherical cells [48].

The equations of motion can be derived within an energy-based picture or by
modelling the relevant interactions as physical forces that act on the
particles. In the latter approach, the models draw on Newton’s second law and
commonly assume that inertial forces and any average acceleration can be
neglected (cf. Brownian dynamics). Put another way, we are dealing with
friction-dominated overdamped motion, which means that friction forces balance
out all the other forces [47; 83; 116]. For cell $i$ on a substrate, we can
then write [cf. 116]

$\Gamma_{i}^{\mathrm{cs}}\frac{\mathrm{d}\mathbf{x}_{i}}{\mathrm{d}t}+\sum_{j\neq
i}\Gamma_{ji}^{\mathrm{cc}}\left(\frac{\mathrm{d}\mathbf{x}_{i}}{\mathrm{d}t}-\frac{\mathrm{d}\mathbf{x}_{j}}{\mathrm{d}t}\right)=\mathbf{F}_{i}^{\mathrm{sub}}+\mathbf{F}_{i}^{\mathrm{mig}}+\sum_{j}(\mathbf{F}_{ji}^{\mathrm{adh}}+\mathbf{F}_{ji}^{\mathrm{rep}})$
(14)

The first term on the left-hand side expresses cell-substrate friction forces,
while the second term deals with friction between cells. Here,
$\Gamma_{ji}^{\mathrm{cs}}$ and $\Gamma_{ji}^{\mathrm{cc}}$ are tensors that
account for anisotropies and inhomogeneities. For instance, for spherical
cells in a homogeneous and isotropic environment,
$\Gamma_{ji}^{\mathrm{cs}}=\gamma\mathbb{I}$, where $\gamma$ is a damping
coefficient, and $\mathbb{I}$ is the identity matrix [47]. Thus, the drag
force that acts on a cell in an isotropic viscous environment is proportional
to the cells’ velocity [127].

On the right-hand side of Equation (14), $F_{i}^{\mathrm{sub}}$ denotes forces
that emerge from adhesion and repulsion between cell $i$ and the substrate.
$F_{i}^{\mathrm{mig}}$ describes the migration forces. For instance,
chemotaxis might introduce a preferred direction for migration [47]. One might
account for the memory of cells, i.e. persistence of a cell to move in a given
direction. Some authors do so using inertial forces. In addition, some authors
include cell polarity giving the cell a preferred direction of motion.
$F_{i}^{\mathrm{mig}}$ becomes a noise term that describes a random movement
in the absence of such forces and assumptions. The last term on the right-hand
side summarises the adhesion and repulsion between cell $i$ and all other
cells. These forces will typically depend upon the distance between the cell
centres or the amount of overlap, $\delta_{ji}$. Contact mechanics offers
different approaches to model these cell-cell interactions, including Hertzian
formulations and the Johnson-Kendall-Roberts (JKR) model of elastic contact
[cf. 48; 80]. Some authors simply assume that all forces involved in cell-cell
interactions are well-approximated by linear springs, generalised linear
springs, or polynomial expressions [e.g. 131; 25; 127, and references
therein].

To exemplify the use of Equation (14), let’s consider spherical cells, whose
movement is dominated by isotropic friction on the substrate and cell-cell
interactions. Meineke et al. [131] and Drasdo [47] use this model to describe
cellular mono-layers in the intestinal crypt and in vitro. For mono-layers, we
might assume that all cells have roughly the same velocity, which leaves us
with the first term on the left-hand side and the last term on the right-hand
side of Equation (14):

$\gamma\frac{\mathrm{d}\mathbf{x}_{i}}{\mathrm{d}t}=\sum_{j}(F_{ji}^{\mathrm{adh}}+F_{ji}^{\mathrm{rep}}),$
(15)

which amounts to a set of coupled ODEs that can be solved using explicit or
implicit schemes, such as a forward Euler scheme [131; 127]. In general,
Equation (14) can be written as a set of linear equations and be solved using
matrix manipulation, for which standard software packages are available.

The computational challenge of CBMs is two-fold: First, all interacting cell
pairs must be identified. Secondly, the relevant interactions take place on
chemical and dynamic time scales that are much shorter than the time scale
that governs tumour growths. Besides their high computational cost, it should
be noted that CBMs might suffer from artefacts when cells are densely packed
[116]. Despite these shortcomings, CBMs have widely and successfully been
applied to study tumour-related issues, such as drug response [e.g. 64; 145].

####  3.2.2 Deformable cell (vertex) models

Deformable cell models (DCMs) are conceptually similar to CBMs: Cell dynamics
are distilled into equations of motion. However, rather than representing each
cell by simple geometrical objects, DCMs depict cells as a number of connected
nodes (cf. Fig. 3). In contrast to CBMs, DCMs hereby capture cell morphology
and intracellular mechanics. The nodes within a cell are connected by
viscoelastic elements that are either part of the cell membrane or the
cytoskeleton. It is essential to include a cytoskeleton since the modelled
cells might buckle without an internal scaffolding structure [24].

Assuming that inertial forces can be neglected, the equation of motion for any
given node takes the same form as that of a single cell in CBMs (cf. Equation
14). Thus, frictional forces balance out all other forces acting on the node
[see also 187]. Node-node interactions can be modelled by letting the
viscoelastic elements act as damped linear springs. To better depict the
behaviour of biopolymers, one might model the elements using nonlinear forces
[117; 138]. Besides node-node interactions, we must introduce forces that
express area and volume constraints on the cell as a whole or on subdivisions
spanned by several nodes. By considering subdivisions of the cell membrane, we
might furthermore introduce forces that express the resistance of the cortex
to bend [138].

Simulating realistic cell dynamics requires a large number of nodes per cell,
and the models can become rather complex. Due to the resulting high
computational cost, it is often only feasible to simulate single cells or
small cell populations on short time scales.

For completeness, it should be mentioned that there is a subgroup of DCMs
called vertex models (VMs). In such models, the cells are represented as
polygons spanned by a grid of nodes, i.e. adjacent cells share vertices and
edges. There is no space between cells. VMs are useful when dealing with
densely packed cells and have thus found applications in tissue mechanics but
have not been applied to GBM [141; 2].

Table 2 gives an overview of all discrete methods discussed above.

Table 2: Overview of the reviewed discrete models. The references are not exhaustive. For other recent reviews, see [116; 192; 78; 132; 195]. orange!20 Model type |  Features |  References
---|---|---
Cellular automata
(CA) models |  Concept: All CA-type models are discrete and lattice-based. Cell migration, death, movement, division, and growth are modelled as stochastic processes. Particularity: Multiple cells or at most one are allowed per lattice site.
Assets: Centimetre-scale simulations are feasible when tracking cell clusters.
Individual cells can be represented.
Drawbacks: They do not represent cell morphology, have a limited spatial resolution and are computationally heavy. For multiple cells per site, the individual cell positions are not tracked. |  One cell per lattice site: [152; 93; 87; 146]
Multiple cells per lattice site: [96; 165]
Bridging scales: Varying number of cell limits per site at different locations
[116].
Lattice gas
cellular automata
(LGCA) models |  Concept: They are CA-type models. There are multiple velocity channels (and hence cells) per lattice site. A deterministic step evaluating movement is included (see Fig. 1).
Assets: Individual cells are represented. Convergence to CM has been shown.
Drawbacks: They do not represent cell morphology, have a limited spatial resolution and are computationally heavy. |  Deutsch, Hatzikirou and collaborators have investigated cell proliferation and glioma invasion [34; 80; 79; 182].
Cellular Potts
models (CPM) |  Concept: They are CA-type models. There are multiple sites per cell. A Hamiltonian function governs stochastic events.
Assets: CPMs draw on the physics behind the cell dynamics and represent cell
morphology.
Drawbacks: They have a limited spatial resolution. Centimetre-scale simulations are not feasible. |  Authors have included a broad range of phenomena through a variety of terms in the Hamiltonian [142; 166; 65; 43; 179].
Centre-based
models (CBM) |  Concept: CBMs model cells as physical particles governed by their equations of motion. No lattice is used. Cells are simple geometrical objects.
Assets: The physics behind the cell dynamics is accounted for. Cells and their
movement are resolved.
Drawbacks: CBMs are computationally heavy. Centimetre-scale simulations are not feasible. Cell morphology is not modelled. |  Cells can be spherical [48; 80] or ellipsoidal [41]. Studies of tumour cell drug response: [145].
Deformable cell
models (DCM) |  Concept: Cells are depicted as a collection of connected nodes described by their equations of motion.
Assets: DCMs model cell morphology and stresses on a subcellular level. There
is no lower length scale since no lattice is used.
Drawbacks: Only simulations of small cell populations or single cells are feasible. |  Single cell adhesion and response to mechanical stress [117; 138]. Tumour cell proliferation [157]. Vertex models [2].

##  4 Hybrid multi-scale models

In the literature, the term hybrid or multi-scale model might denote any
combination of the different methods discussed above. A large variety of
approaches thus fall into this category. Adopting the nomenclature from
Deisboeck et al. [45], we distinguish between three types: composite hybrid
models, adaptive hybrid models, and calibrated models, such as continuum
models with functional parameters. Composite hybrid models track and connect
phenomena across various scales while keeping the scale associated with each
individual aspect fixed (cf. Section 4.1). Thus, all tumour cells are treated
on equal footing.

In contrast, adaptive hybrid models dynamically adjust the local resolution
based on the required level of detail (cf. Section 4.2). That is to say that
not all tumour cells are represented in the same manner. Finally, the third
type of hybrid model denotes, for instance, continuum models, whose parameters
have been calibrated based on agent-based models or biophysical considerations
of microscopic or mesoscopic phenomena (cf. Section 4.3).

###  4.1 Composite hybrid models

When using a discrete model to represent tumour cells, the extracellular
biochemical players, such as oxygen, are commonly modelled as continuous
fluids [150; 151; 149; 103]. Moreover, intracellular changes, i.e. processes
on a sub-cellular scale, might be tracked for each individual cell using ODEs
as discussed in Section 4.1.1. These different microscopic, mesoscopic, and
macroscopic mechanisms are all coupled. For instance, the nutrient levels both
depend on the cell density and impact the probabilities that are used to
predict cell actions. When this interdependence is taken into account, the
discrete tumour cell model becomes one of many components in a larger
framework that spans and links multiple temporal and spatial scales. The
framework as a whole then constitutes a composite hybrid model [cf. 10; 7; 11;
194; 108; 106; 104; 27; 180; 9; 28; 64]. Such hybrid frameworks have been used
to study a wide variety of phenomena, including drug resistance [e.g. 52;
145]. In each case, the focus of the study prescribes the resolution of
different features. For instance, in some studies, discrete models of the
vasculature are combined with continuum models of the tumour to better
understand angiogenesis or drug delivery [e.g. 29]. As another example, May et
al. [128] couple discrete tumour models with biomechanical calculations of
stresses and strains to arrive at a more accurate prediction of the tumour
shape. Finally, we note that composite hybrid models have been employed in
Bayesian inferences to establish posterior probability distributions for
relevant model parameters [88] (see also Section 5).

####  4.1.1 Ordinary differential equations

When dealing with agent-based hybrid models, some authors include sub-cellular
dynamics, modelling molecular networks, pathways, and reactions [10; 11; 210;
105]. Such processes are governed by a system of coupled nonlinear Ordinary
differential equations (ODEs) that represent mass-balance equations and rely
on the stoichiometry of the underlying biochemical reactions.

ODEs thus play a vital role in some composite hybrid models. In general, ODEs
are a common tool in mathematical biology and oncology. For instance, they can
capture global tumour growth patterns [162]. While such models do not
spatially resolve the tumour, they have successfully been used to study the
response to drugs and radiation treatments, recovering experimental
constraints on the overall growth pattern [e.g. 167; 111]. The ODE for tumour
growth take the generic form

$\frac{\mathrm{d}N(t)}{\mathrm{d}t}=S(N)$ (16)

where $N(t)$ is the total cell number or volume, and $S(N)$ denotes an
appropriate source function.

###  4.2 Adaptive hybrid models

Detailed modelling of individual cells is required to capture tumour growth at
low cell numbers. Agent-based models thus give unique insights into glioma
invasion. However, at high cell densities, a lower resolution suffices.
Continuum models realistically capture all relevant aspects of large chunks of
bulk tumours. Adaptive hybrid models take advantage of this notion by
describing different parts of the tumour tissue using distinct modelling
approaches. Thereby, it becomes possible to draw on the strengths of discrete
models when depicting centimetre-sized tumours without the insurmountable
computational cost that would otherwise be involved.

The hybrid model presented by the group of Stolarska and Yangjin [107; 172;
105] treats necrotic tumour zones, quiescent tumour tissue, and the
surrounding microenvironment as continuous fluids. The relevant PDEs are
solved on a regular grid. The cells in the proliferative tumour rim
(100-$200\text{\,}\mathrm{\SIUnitSymbolMicro m}$), on the other hand, are
modelled using a CBM. They are represented as autonomous ellipsoids. At the
boundary between the cell-based and continuous components, interpolation is
used to establish the boundary conditions for the reaction-diffusion equations
and the forces exerted on the individual agents [see also 55, for a related
example from molecular dynamics].

Bearer et al. [14] and Frieboes et al. [61] couple an off-lattice approach
with the continuum models by Wise et al. [199] discussed in Section 2.2. To
study glioma invasion, they include dynamic transitions between the continuum
and cell-based representations based on the microenvironment. In their model,
both mutations and hypoxia might induce the development of a migratory
phenotype [see also 159].

The adaptive hybrid models discussed above allow for mass transfer between the
different components and are carefully constructed to obey laws, such as mass
conservation. However, it is worth noting that the continuum models are pre-
assumed without guaranteeing that the imposed functional form emerges from up-
scaling the cell-based representation (cf Section 4.3). Moreover, due to the
assumptions that enter the continuum and agent-based models, single-cell
measurements might lead to different parameter values than those required for
continuum models to fit patient-specific data [109].

Rather than coupling agent-based and continuum models, other authors [208;
209; 116] combine agent-based models with different resolutions. Such
approaches are known as multi-scale agent-based modelling. For instance, one
might couple CA models that consider single cells with models that deal with
cell clusters at a coarser lattice resolution [see also 160, for a multi-
resolution study of polymers].

###  4.3 Calibrated models

Various authors encode information from cell-based models or biophysical
considerations into heterogeneous and time-varying parameters of continuum
models. This goal can be achieved in different ways. As briefly discussed in
Section 3.1.2, one might up-scale agent-based models using the mean-field
approximation [cf. 80]. Other authors draw on the emergent macroscopic
properties of agent-based models to derive phenomenological relationships [cf.
120; 122].

Each method mentioned above relies on simplifying approximations to derive an
analytical expression, which might limit their predictive power. To circumvent
the need to explicitly state expressions for the macroscopic variables
altogether, Kavousanakis et al. [100] use a form of equation-free modelling
called coarse projective integration [see also 12]. This method invokes the
so-called coarse time-stepper that consists of four phases: lifting,
simulation, restriction, and extrapolation. These four phases are repeated for
every macroscopic time step. In the lifting step, an ensemble of cell
distributions is drawn from the macroscopic tumour cell density. Each cell
distribution is then evolved over a short time span using an agent-based model
— Kavousanakis et al. use a CA model. In the restriction step, the macroscopic
density is updated by computing the ensemble average of the agent-based
simulations. Under the assumption that the macroscopic cell density evolves
more slowly than the microscopic variables, the observed change in the cell
density yields an estimate of its time derivative. Using the forward Euler
method, this information can be employed to project the cell density further
into the future without considering individual cells.

We note that the general idea of calibrating coarse models based on more
resolved representations is not only limited to continuum models. Recently,
Van Liedekerke et al. [186] have thus calibrated the mechanical interaction
forces of CBMs based on DCMs.

##  5 Data-driven modelling

Researchers draw on a wide range of experimental data to inform their
mathematical models of GBM and do so following several different approaches.
One can use data to select the underlying mechanism or inform the model
parameters. For example, as discussed above, some researchers use DTI MRI data
to determine the diffusion properties of cancer cells. Moreover, data allow
for qualitative comparisons and constraints on the simulations. For instance,
most models are set to recover generic properties of cancer, such as spherical
avascular tumour growth. Finally, one can use data to infer model parameter
values using a statistical framework, either maximum likelihood estimation or
a Bayesian approach. Systematic incorporation of imaging and molecular data
into the mathematical models of cancer via statistical inference is a
promising and timely area of future development. This area of development is
bolstered by the latest progress in imaging and omics approaches which are
producing quantitative data on an unprecedented scale. Furthermore, the latest
progress in computational power, simulation-based inference methods, and
machine learning make this task computationally feasible. Several modelling
studies aim to fit specific data from patients, animal models, or in vitro
experiments. Here, we provide a list of illustrative examples and summarise
our perspective on important areas of future research. We also refer the
reader to some recent reviews on the topic [57; 81].

Two decades ago, Stamatakos and collaborators [171; 170] used agent-based
models to investigate the impact of chemotherapy in vivo based on patient-
specific PET, SPECT, T1-weighted MRI, histopathologic and genetic data. They
reduced the computational cost by clustering the cells into macroscopic (1
mm2) regions rather than simulating individual cells. More recently, Gallaher
et al. [64] published a study in which they fit MRI and ex vivo cell tracking
data from rats to off-lattice agent-based models, simulating individual cells.
They accomplish this fit by using random sampling. However, due to the high
computational cost, agent-based models are most commonly found in studies
focusing on small cell populations. For instance, Oraiopoulou et al. [139] use
agent-based models to recreate the invasive morphologies of GBM observed in
vitro. Meanwhile, many papers on agent-based models do not include a fit to
real-world data but rather perform in silico experiments to make qualitative
and quantitative predictions. For instance, based on such analyses, Perez-
Velazquez et al. [145] addressed the development of drug resistance, Kim and
collaborators [106; 104] investigated the key mechanisms behind molecular
switches, and Schmitz et al. studied growth patterns.

The parameters used in state-of-the-art continuum models [109] to capture the
migration and proliferation of GBM are derived from MRI and CT images, and the
application of continuum models to patient-specific data in vivo is widespread
in the literature. Thus, Swanson et al. [177; 178] already used such data to
inform their models twenty years ago [see also 77]. Later studies that build
on their work have included further imaging data, such as PET [190; 75; 135].
Isotropic diffusion reaction equations yield a detailed picture based on T1Gd
and T2 Flair MRI images [35]. Other studies have employed histopathology [62].
The anisotropic continuum models discussed in Section 2.1 likewise built on
MRI data, drawing on DTI images to compute the diffusion matrix [90; 143; 53;
8; 176].

Due to the relatively low computational cost of continuum models, they can be
employed in Bayesian sampling schemes as illustrated by Lipkova et al. [118],
who infer model parameters based on MRI and PET images using a Markov Chain
Monte Carlo algorithm. Lipkova et al. demonstrate that their Bayesian approach
can be used to improve personalised radiotherapy. Ezhov et al. [56] address
the same inverse problem using machine learning. Ezhov et al. construct a
library of 100,000 simulations with different parameter values and use their
atlas to train a neural network. The trained neural network can infer the
patient-specific model parameters based on MRI images within minutes, allowing
for effortless model personalisation. Today, continuum models are thus at the
point where they have the potential to enter clinical settings and contribute
to personalised treatments. While the same does not yet hold for agent-based
models due to their high computational cost, machine learning might likewise
help to overcome this obstacle in the future, as discussed by Joergensen et
al. [94]. Machine learning algorithms would play the same role as they do in
continuum models by circumventing the need to produce further simulations once
a surrogate model or inference algorithm has been trained. Similar strategies
are applied when dealing with agent-based models outside of cancer research
[156].

Like imaging data, molecular data offers unique insights into GBM [21; 130].
In particular, different sequencing techniques, including bulk RNA-seq [23],
scRNA-seq [136; 17; 36; 197; 201; 158] (recently reviewed in [95] and [98])
and spatial RNA-seq [155], have helped researchers to uncover different
aspects of brain tumours. Sequencing data have thus contributed to our
understanding of important cell types, the nature of the invasive tumour
front, the spatial cellular organisation of the tumours, and the molecular
basis of the heterogeneity of GBM. For instance, Ravi et al. [155] recently
identified spatially distinct clusters of genes via spatial transcriptomics,
and Neftel et al. [136] identified four main cellular states among malignant
tumour cells by analysing data at the single-cell level. Furthermore, Neftel
et al. demonstrated that the relative frequency of these four states varies
between GBM samples. On the one hand, such information on the heterogeneity of
GBM is valuable in its own right and can help to inform the underlying
assumptions used in the mathematical models. As exemplified in Section 2.2,
mathematical models thus present a unique tool to investigate the implications
of such information on the tumour phenotype. On the other hand, the vast
number of spatial GBM sequencing data that is becoming available can directly
be used in parameter inference, model selection or validation of model
predictions.

##  6 Clinical application of spatio-temporal modelling in Glioblastoma

Glioblastoma is a biologically complex and dynamic tumour that exists within
the intricate and responsive brain environment. Parcelling and modelling
discreet aspects of these biological processes and systems holds the potential
for numerous direct clinical care applications and has the potential to
address several areas of unmet need.

Mathematical diffusion-tensor modelling is currently widely used in the
context of MRI diffusion-tensor-based tractography for the modelling of white-
matter tracts. Understanding patient-specific anatomy, and anatomico-
pathological relationships, is critical for effective surgical planning and
avoidance of iatrogenic injury [196]. Future refinements in data modelling can
be expected to improve spatial resolution and reliability, which would
directly translate to improved validity of neuro-imaging with direct clinical
applications (cf. Section 5).

A key concern in clinical practice is the prediction of tumour growth
patterns, sites of progression, and the location of tumour recurrence. This
triad is important both for therapeutic planning and prognostication [91].
Mathematical models have the potential to refine these facets of care and
hence to offer highly-tailored conformal radiation dosing and targeted
surgical supra-marginal resection [137; 189]. These therapeutic adaptations
might maximise the prognostic benefits and lead to the destruction of the
tumour while sparing normal neural tissue. Similarly, when combined with the
modelling of growth rates, this information could be assimilated to generate
more accurate prognostic maps with the prediction of anatomico-pathological
functional deficits and overall prognosis.

Recent years have been characterised by increasing recognition of
glioblastoma’s molecular intra- and inter-tumoural heterogeneity and its
functional consequences. This is exemplified in the evolution of the WHO
classification of tumours from a purely histopathological to a fully-
integrated molecular classification [119]. As we have summarised, mathematical
models of glioblastoma hold the potential to illustrate novel prognostic
aspects of tumour biology and the micro-environment not currently captured.
These might include structural metrics, such as tissue density and cellular
linage, or functional ones, including oxygenation, nutrient supply and
metabolism.

The potential for applying spatio-temporal mathematical modelling of
glioblastoma in clinical care is substantial but thus far remains largely
unrealised. As summarised in this review, the foundation of knowledge in this
area is now considerable. The stage is set for those who wish to advance the
field from theoretical models to practical applications.

##  7 Summary and discussion

This paper presents an overview of mathematical models that provide means to
scrutinise and predict the spatial and temporal evolution of tumours. While we
focus on models for glioblastoma and highlight characteristics of this
aggressive cancer type, we also attempt to give a rounded picture of state-of-
the-art approaches across the field of mathematical oncology.

Spatially resolved tumour models fall into two main categories: They either
treat the tumour cell density as a continuous variable (cf. Section 2) or deal
with autonomous agents representing individual cells or cell clusters (cf.
Section 3).

The computational cost associated with continuous models is relatively low.
This property allows for the modelling of the entire tumour volume and the
application to a wide variety of (patient-specific) data. Furthermore,
continuous models can capture many of the characteristics of GBM. We summarise
how continuous models achieve this goal in Table 3. GBM spreads
anisotropically following existing structures in the brain, including white
matter tracks and blood vessels. On a macroscopic level, this behaviour is
well-described by reaction-diffusion equations when including an anisotropic
diffusion term (cf. Section 2.1).

Moreover, by including source terms that couple a set of such reaction-
diffusion equations, continuous models can link different cell sub-populations
and account for interactions with the environment (cf. Section 2.3).
Continuous models are hereby able to encapsulate the hierarchical nature that
is a defining feature of GBM (cf. Section 2.2). The models also show great
versatility: By including additional biologically motivated terms into the
reaction-diffusion equations, continuous models can be painlessly extended to
include phenomena ranging from immune responses to the impact of radiotherapy.

Table 3: Overview of how different model types can deal with properties that define glioblastoma growth. orange!20 Phenomenon |  Continuum models |  Agent-based models
---|---|---
Invasion of healthy tissue |  Diffusion (and advection) terms are included in the PDE. |  Stochastic (and deterministic) rules dictate the migration of individual cells.
Anisotropies in tumour growth |  E.g. anisotropic diffusion tensor is used based on brain structure. |  By allowing for feedback between the cells and environment, one can introduce preferential migration, leading to anisotropic growth along structures.
Cell hierarchy and lineage |  A set of coupled PDEs described subpopulations with different properties. |  Individual cells underlie rules that are affected by cell type.
Mitosis and apoptosis |  E.g. logarithmic or exponential source terms are included in the PDE. These might link different sub-populations and be associated with properties of the environment. |  Both phenomena underlie stochastic or deterministic rules that are invoked for each agent.

Tumour cells must always be understood in relation to their surroundings. In
this connection, aspects, such as nutrition flows, the extracellular matrix,
or the vasculature, are often (if not always) represented using continuous
models. Even when using agent-based models, continuous models thus play a
vital role. However, continuous models have their drawbacks. They do not
account for the stochastic events that dominate at low tumour cell densities.
Consequently, they are not ideal for addressing some questions regarding the
invasive tumour front or tumour recurrence that is a prevalent obstacle in
treating GBM.

At low cell densities, agent-based models have an edge. Such models come in
many different flavours. Some agent-based models borrow from the description
of other phenomena. They thus lean on theoretical models, e.g. from solid-
state physics, posing biological events in terms of physical forces and
potentials (cf. Sections 3.1.3, 3.2.1, and 3.2.2). Other models rely on purely
stochastic rules to encode biological information (cf. Sections 3.1.1 and
3.1.2). Moreover, agent-based models differ in the details that they
encompass. While some models address aspects of cell morphology, others do
not. While some models define a lower threshold for their spatial resolution
by employing a lattice on which the agents move, others do not.

Like continuous models, agent-based models are able to capture the essential
properties of GBM (see Table 3). Anisotropic diffusion along existing
structures can be included by altering the rules that govern cell migration
based on the environment. Moreover, by altering these rules based on the cell
type, the hierarchical organisation of GBM can be incorporated into the model.

The high computational expense of agent-based models constitutes their main
drawback. It can be a significant stumbling block on the way to unleashing
their full potential. However, this shortcoming can be partly circumvented.
One might thus combine different approaches into a hybrid model bridging
across the scales involved (cf. Sections 4.1 and 4.2). Alternatively, one
might study small cell populations to derive emergent macroscopic properties
that can be used to calibrate models on a coarser scale (cf. Section 4.3).
Indeed, one of the main assets of agent-based models is their ability to link
an understanding of individual cells to macroscopic events. To further lower
the computational cost when confronting the models with data, one might use
sophisticated statistical techniques or machine learning methods that
demonstrably reduce the number of models that need to be constructed [see also
121; 94], as discussed in Section 5.

Each of the available mathematical models has its strength and weaknesses.
Which model to employ will hence be dictated by the research question and the
available data. First, some approaches are better suited for depicting certain
phenomena than other methods are. One might drastically simplify the work
associated with the practical implementation by choosing the method wisely.
Secondly, not all models are equally informative. The question is whether the
additional information can be exploited and for what purpose [see also 51]. On
the one hand, the chosen model sets limits on the insights that one might
gain. This notion favours more complex models. On the other hand, one should
avoid cracking nuts with a sledgehammer. To exemplify this, it is thus worth
noting that non-spatial models make full use of certain types of (patient-
specific) data whilst being unable to address any questions on tumour
morphology. They are undoubtedly very valuable tools in their own right. The
same holds for each spatio-temporal cancer model presented above. With this in
mind, we hope that our review provides a useful guide and helps the reader
select a suitable mathematical model for their research.

Mathematical models are only useful if they describe and relate to
experimental data. Our review thus ends with a summary and discussion of the
application of mathematical models to experimental data and its clinical
relevance. A wealth of imaging and molecular data is becoming available,
particularly due to advances in single-cell and spatial transcriptomics. At
the same time, researchers have begun to implement spatial models into
statistical frameworks to perform systematic inference based on these data.
While few such inference analyses have yet been published, advances in machine
learning and computing power promise to further enable the systematic
incorporation of imaging and molecular data into the spatial models of GBM. We
thus envision a greater impact of mathematical modelling in unravelling the
basic biology of GBM and its applications in clinical treatment in the near
future.

## Acknowledgement

This work was supported by The Oli Hilsdon Foundation through The Brain Tumour
Charity, grant number (GN-000595), in connection with the program “Mapping the
spatio-temporal heterogeneity of glioblastoma invasion". Work at the Cancer
Institute Genomics Translational Technology Platform is supported by the CRUK
City of London Centre Award [C7893/A26233].

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