# Cancer disease: integrative modelling approaches

**Abstract**

Cancer is a complex disease in which a variety of phenomena interact over a wide range of spatial and temporal scales. In this article a theoretical framework will be introduced that is capable of linking together such processes to produce a detailed model of vascular tumour growth. The model is formulated as a hybrid cellular automaton and contains submodels that describe subcellular, cellular and tissue level features. Model simulations will be presented to illustrate the effect that coupling between these different elements has on the tumour's evolution and its response to chemotherapy

CANCER DISEASE: INTEGRATIVE MODELLING APPROACHES

Helen M. Byrne, Markus R. Owen

∗

,

Centre for Mathematical Medicine,

School of Mathematical Sciences,

University of Nottingham,

Nottingham NG7 2RD,

England,

Tomas Alarcon

†

and Philip K. Maini,

Centre for Mathematical Biology,

Mathematical Institute,

24-29 St Giles’,

Oxford OX1 3LB,

England.

ABSTRACT

Cancer is a complex disease in which a variety of phenomena

interact over a wide range of spatial and temporal scales. In

this article a theoretical framework will be introduced that is

capable of linking together such processes to produce a de-

tailed model of vascular tumour growth. The model is for-

mulated as a hybrid cellular automaton and contains submod-

els that describe subcellular, cellular and tissue level features.

Model simulations will be presented to illustrate the effect

that coupling between these different elements has on the tu-

mour’s evolution and its response to chemotherapy.

1. INTRODUCTION

Cancer is a complex and insidious disease in which controls

designed to regulate growth and maintain homeostasis be-

come disrupted. It is frequently initiated by genetic mutations

that increase the net rate of cell division and lead to the forma-

tion of a small avascular lesion. Successful angiogenesis (i.e.

the formation and ingrowth of a network of new blood vessels

to the tumour) is needed before the extensive and rapid growth

associated with vascularised tumours can occur. To confound

matters, the processes involved in tumour growth are inter-

linked and act over a wide spectrum of spatial and temporal

scales: the spatial scales of interest range from the subcel-

lular level, to the cellular and macroscopic levels while the

timescales may vary from seconds (or less) for signal trans-

duction pathways to months for tumour doubling times.

The advent of increasingly sophisticated technology means

that it is now possible to collect experimental data associated

with the spatial and temporal scales of interest. This is cre-

ating a demand for new theoretical models that have the ca-

pacity to integrate such data in a meaningful manner and are

∗

The authors gratefully ackno wledge ﬁnancial support provided by the

EPSRC under grants GR/5090067, GR/S72023/01 and AF/00067.

†

The third author is currently based at the Bioinformatics Unit, Depart-

ment of Computer Science, University College London, Gower Street, Lon-

don WC1E 6BT, England.

able to address the fundamental problem of how phenomena

at different spatial scales are coupled.

In this paper we review our recent progress in d eveloping

a mathematical model for studying vascular tumour growth

that is capable of integrating phenomena that act on different

scales [1]. Our theoretical framework extends earlier work

by Gatenby and coworkers [2] and links submodels which

describe processes operating on different spatial scales. In

section 2 we introduce our hybrid cellular automaton, before

presenting numerical results in section 3. The simulations il-

lustrate how the coupling between the submodels inﬂuences

the tumour’s evolution and its response to chemotherapy. We

conclude in section 4 with a summary of our results and a

discussion of possible directions for future research.

2. MODEL FORMULATION

In this section we introduce our multiscale model of vascular

tumour growth. It accounts for a variety of inter-related phe-

nomena that operate on vastly different space and time scales.

We consider a vasculature composed of a regular hexagonal

network embedded in a two-dimensional NxN lattice com-

posed of normal cells, cancer cells and empty space. Progress

through the cell cycle and the production of proteins such

as vascular endothelial growth factor (VEGF) that stimulate

angiogenesis are incorporated at the subcellular level using

ODE models. Cell-cell communication and competition for

resources are included at the cellular level through rules that

deﬁne our cellular automaton. At the tissue scale, reaction-

diffusion equations model the diffusion, production and up-

take of oxygen and VEGF: the vessels are regarded as sources

(sinks) o f oxygen (VEGF) and the cells as sinks (sources)

of oxygen (VEGF). Blood ﬂow and vascular adaptation are

also included at the tissue scale (see Figure 1). We impose

a pressure drop across the vasculature, assuming that blood

ﬂows into the idealised “tissue” through a single inlet ves-

sel and drains out through a single outlet vessel. We use the

Poiseuille approximation and compute the ﬂow rates through,

and pressure drops across, each vessel using Kirchoff’s laws.

8060-7803-9577-8/06/$20.00 ©2006 IEEE ISBI 2006

The vessel radii are updated using a structural adaptation law

similar to that proposed by Pries et al. [3] (for details, see

[1]).

Thus the model is formulated as a hybrid cellular automata,

with different submodels describing behaviour at the subcel-

lular, cellular and macroscopic (or vascular) levels (see Figure

1). Coupling between the different submodels is achieved in

several ways. For example, local oxygen levels which are de-

termined at the macroscale inﬂuence both progress through

the cell cycle and VEGF production at the subcellular level.

Conversely, the intracellular production of VEGF modulates

vascular adaptation at the macroscale and this, in turn, con-

trols oxygen delivery to the tissue. We stress that the submod-

els we use simply illustrate how such a multiscale model can

be assembled: the framework we present is general, with con-

siderable scope for incorporating more realistic (i.e. complex)

submodels. This raises the important issue of how the level

of detail incorporated at each spatial scale inﬂuences the sys-

tem’s behaviour: this will form the basis of future research.

Vascular

Structural

Adaptation

Haematocrit

Distribution

(Oxygen Source)

Blood Flow

CellŦcycleApoptosisVEGF Secretion

Spatial

Oxygen

Distribution

Spatial

VEGF

Distribution

Vascular

Layer

Cellular

Layer

Intracellular

Layer

Spatial Distribution

(Oxygen Sink)

CancerŦNormal

Competition

Fig. 1. Schematic showing the structure of our hybrid cellular

automaton model. Reproduced with permission from [1].

3. NUMERICAL RESULTS

3.1. Vascular adaptation inﬂuences tumour growth

In Figure 2 we present simulations that illustrate the impor-

tance of accurately modelling blood ﬂow through the tissue.

The upper panels correspond to a case for which the ves-

sels undergo structural adaptation and, hence, oxygen is dis-

tributed nonuniformly across the tissue. The lower panels

show how the system evolves when oxygen is distributed uni-

formly throughout the vessels (i.e. blood ﬂow is identical in

all branches of the vasculature). We see that spatial hetero-

geneity has a signiﬁcant effect on the tumour’s dynamics and,

in this case, actually reduces the tumour burden. We note also

that if the oxygen distribution is heterogeneous then the tu-

mour has “ﬁnger-like” protrusions similar to those observed

in invasive cancers. This structure arises here simply because

of the spatial heterogeneity in the nutrient distribution. In-

deed, closer inspection reveals that several parts of the tumour

have almost “broken away”. While this cannot actually hap-

pen in the current model because cell motion is neglected, we

speculate that by allowing cell movement towards nutrient-

rich regions, this may act as a mechanism for metastasis.

10 20 30 40 50 60

10

20

30

40

50

60

(a)

0 20 40 60 80 100

# iterations

0

200

400

600

# cells

(b)

10 20 30 40 50 60

10

20

30

40

50

60

(c)

0 20 40 60 80 100

# iterations

0

1000

2000

3000

4000

# cells

(d)

Fig. 2. Series of images showing the spatial distribution of

cells for growth in inhomogeneous (panel a), and homoge-

neous environments (panels c). In panels (a) and (c) cancer

cells occupy white spaces and vessels occupy a hexagonal ar-

rray denoted by black spaces. The other black spaces denote

”empty spaces”. Panels (b) and (d) show the time evolution of

the number of (cancer) cells for the heterogeneous and homo-

geneous cases, respectively: squares denote the total number

of cancer cells (proliferating + quiescent); diamonds denote

quiescent cells. Reproduced with permission from [1].

3.2. Impact of VEGF on the tumour’s growth dynamics

The simulation p resented in Figures 3 to 5 shows how cou-

pling intracellular and macroscale phenomena can inﬂuence

the dynamics of both the vasculature and the tumour. In con-

trast to the results depicted in Figure 2, where vessel adapta-

tion was independent of VEGF, in Figures 3 to 5 it is regulated

by local VEGF levels. Figures 3 and 4 show how the tumour’s

spatial composition evolves while ﬁgure 5 summarises its dy-

namics. Since there is a single inlet (outlet) to the vasculature

located in the bottom left (top right) h and corner of the tissu e,

the incoming blood ﬂow and haematocrit become diluted as

they pass through the hexagonal lattice. This creates a het-

807

erogeneous oxygen distribution across the domain, with oxy-

gen levels being highest near the inlet and outlet. Over time,

the tumour cells proliferate and spread through the tissue to-

wards oxygen-rich regions. As they increase in number, their

demand for oxygen outstrips that available from the vascula-

ture, and quiescent regions form. These cells produce VEGF

which diffuses through the tissue (see Figures 3 and 4), stim-

ulating vessel adaptation and biasing blood ﬂow towards low

oxygen regions. If the VEGF stimulus is weak then the vascu-

lature does not adapt quickly enough and the quiescent cells

die (this is what happens at early times in ﬁgure 5). VEGF

levels also decline and blood ﬂow to the remaining tumour

cells rises, enabling them to increase in number until the de-

mand for oxygen once again exceeds that being supplied, and

so the cycle repeats, with pronounced oscillations in the num-

ber of quiescent cells (see Figure 5). In order to highlight the

key role played by VEGF in creating these oscillations, also

presented in Figure 5 are the results of a simulation which

was identical in all respects except that vascular adaptation

was independent of VEGF (as per Figure 2). In both cases,

the tumours grow to similar sizes. However, when vascular

adaptation is independent of VEGF the evolution is mono-

tonic, the oscillations in the cell populations disappear and the

number of quiescent cells is consistently much lower. These

results show how coupling between the different spatial scales

can effect not only the tumour’s growth dynamics but also the

proportion o f proliferating and quiescent cells that it contains.

Empty

Normal

Cancer

Quiescient

Vascular

Cells

i=30

Oxygen

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

VEGF

0

1

2

3

4

5

6

x 10

Ŧ3

Radii & Haematocrit

Fig. 3. Series of plots showing how a small tumour intro-

duced into a vascular tissue at t =0has evolved at t =30

(dimensionless time unit). While the oxygen and vessel pro-

ﬁles remain unchanged from their initial conﬁgurations, the

tumour has increased in size and now contains quiescent cells

which p roduce trace amounts of VEGF. Reproduced with per-

mission from [4].

Empty

Normal

Cancer

Quiescient

Vascular

Cells

i=90

Oxygen

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

VEGF

0

1

2

3

4

5

6

x 10

Ŧ3

Radii & Haematocrit

Fig. 4. Series of plots showing how the simulation presented

in ﬁgure 3 has developed at t =90. The tumour continues

to penetrate the tissue region. There are now enough quies-

cent cells to elicit an angiogenic response. As a result, the

vasculature has been remodelled, with blood ﬂow and oxygen

supply (haematocrit) being directed primarily towards the tu-

mour mass. Reproduced with permission from [4].

3.3. Response to chemotherapy

We now investigate how the system’s dynamics change when

a chemotherapeutic drug is introduced. We assume that the

drug is continuously administered to the vessels and, hence,

that its concentration at the inlet vessel is constant. We calcu-

late the drug concentration within and outside the vessels in

a manner similar to that used to determine the oxygen distri-

butions. In particular, once the drug leaves the vessels, it dif-

fuses through the tissue and must be taken up by the normal

and healthy cells before it can act. For simplicity, we assume

that the drug works in the following manner. When a cell

attempts to divide, if the local drug concentration exceeds a

threshold value then the cell fails to divide and is itself killed.

Repeated simulations suggest that, when the drug is used,

three qualitatively different types of behaviour emerge: the

drug is ineffective and the tumour continues to colonise the

tissue, the tumour is reduced in size, or it is completely elim-

inated. Figure 6 summarises the tumour dynamics associated

with the different possible outcomes. The simulations were

obtained by varying a parameter (h

θ

) that measures the rate

at which the drug extravasates. Since increasing h

θ

corre-

sponds to tumour regions with more permeable vessels, our

results suggest that a drug of this type is likely to have greater

speciﬁcity in tumour regions containing immature and leaky

vessels. Alternatively, if we interpret decreasing h

θ

as using

drugs with larger molecular weights then our results suggest

that smaller drugs will be delivered more readily and, hence,

evoke a stronger cell kill than larger, heavier drugs.

808

0

20

40

60

80

100

120

140

160

180

200

0

500

1000

Proliferating tumour cells

0

20

40

60

80

100

120

140

160

180

200

0

100

200

300

400

500

Number of cells

Quiescent tumour cells

0

20

40

60

80

100

120

140

160

180

200

0

500

1000

Time, t

Total tumour cells

Fig. 5. Series of curves showing how, for the simulation in

ﬁgures 3 and 4 the numbers of proliferating (upper panel),

quiescent tumour cells (middle panel) and total number of tu-

mour cells (lower panel) change over time. While the number

of proliferating cells increases steadily, the number of quies-

cent cells undergoes oscillations of increasing amplitude un-

til t ≈ 120. Thereafter, the tumour is sufﬁciently large that

the quiescent cells are never eliminated: quiescent cells that

die are replaced by proliferating cells that become quiescent.

The dot-dashed lines show the evolution of a tumour which is

identical except that its vasculature is not regulated by VEGF.

While both tumours reach similar equilibrium sizes, when

vascular adaptation is independent of VEGF the oscillations

in the cell populations disappear and the number of quiescent

cells is much lower. Reproduced with permission from [4].

4. DISCUSSION

We have presented a hybrid cellular automaton model of vas-

cular tumour growth and shown h ow it may be used to study

the manner in which interactions between subcellular, cellu-

lar and macroscale phenomena affect the tumour’s growth dy-

namics and its response to chemotherapy. We stress that the

submodels we have used to describe the different processes

are highly idealised and chosen s imply to illustrate the poten-

tial value of such a multiscale model as a predictive tool to

test experimental hypotheses and to integrate different types

of experimental data. There is considerable scope for incor-

porating more realistic submodels and specialising the system

to describe speciﬁc tumour types. For example, we are cur-

rently engaged in a large interdisciplinary project which aims

to build a virtual model of the early stages of colorectal cancer

(details at: http://www.integrativebiology.ox.ac.uk ).

Key challenges raised by our simulations that lie at the

heart of such integrative modelling concern the level of de-

tail incorporated at each spatial scale, the mathematical ap-

proaches used and model validation. For example, in this

0

20

40

60

80

100

120

140

160

180

200

0

500

1000

Proliferating tumour cells

h

θ

= 0

h

θ

= 90

h

θ

= 100

0

20

40

60

80

100

120

140

160

180

200

0

100

200

300

400

500

Number of cells

Quiescent tumour cells

0

20

40

60

80

100

120

140

160

180

200

0

500

1000

Time, t

Total tumour cells

Fig. 6. Series of curves showing how the tumour’s over-

all dynamics change when it is treated with two blood-borne

chemotherapeutic agents that differ only in their extravasation

rates. For each simulation, we plot the numbers of proliferat-

ing and quiescent tumour cells and the total number of tumour

cells evolve over time. Key: h

θ

=0(control, drug-free case,

as per ﬁgure 5), solid line; h

θ

=90(moderate drug), dashed

line; h

θ

= 100 (highly permeable and effective drug), dotted

line. Reproduced with permission from [4].

article we chose to use a combination of differential equa-

tions and cellular automata to construct our virtual tumour.

It remains an open question whether the predicted behaviour

would change if we replaced our (subcellular) ODE mod-

els with Boolean networks and/or the cellular automata with

agent-based models.

5. REFERENCES

[1] T. Alarc´on, H.M. Byrne, and P.K. Maini, “A muliple scale

model for tumour growth,” SIAM J. Multiscale Mod. &

Sim., vol. 3, pp. 440–475, 2005.

[2] A.A. Patel, E.T. Gawlinsky, S.K. Lemieux, and R.A.

Gatenby, “Cellular automaton model of early tumour

growth and invasion: the effects of native tissue vascu-

larity and increased anaerobic tumour metabolism,” J.

Theor. Biol., vol. 213, pp. 315–331, 2001.

[3] A.R. Pries, T.W. Secomb, and P. Gaehtgens, “Structural

adaptation and stability of microvascular networks: the-

ory and simulations,” Am. J. Physiol., vol. 275, pp. H349–

H360, 1998.

[4] H.M. Byrne, M.R. Owen, T. Alarc´on, J. Murphy, and

P.K. Maini (in press), “Modelling the response of vas-

cular tumours to chemotherapy: a multiscale approach,”

Math. Mod. Meth. A ppl. Sci., 2005.

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**ABSTRACT:**The mathematical analysis of the tumour growth attracted a lot of interest in the last two decades. However, as of today no generally accepted model for tumour growth exists. This is due partially to the incomplete understanding of the related pathology as well as the extremely complicated procedure that guides the evolution of a tumour. In the present work, we analyse the stability of a spherical tumour for four continuous models of an avascular tumour. Conditions for the stability are stated and the results are implemented numerically. It is observed that the steady-state radii that secure the stability of the tumour are different for each of the four models, although the differences are not very pronounced. © The authors 2015. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. - [Show abstract] [Hide abstract]
**ABSTRACT:**The ellipsoid represents the sphere of the anisotropic space. It provides the appropriate geometrical model for any direction dependent physical quantity. The growth of a tumour does depend on the available tissue surrounding the tumour and therefore it represents a physical case which is realistically modelled by ellipsoidal geometry. Such a model has been analyzed recently by the first author et al. (2012). In the present work, we focus on the stability of the growth of an ellipsoidal tumour. It is shown that, in contrast to the highly symmetric spherical case, where stability is possible to be achieved, there are no conditions that secure the stability of an ellipsoidal tumour. Hence, as in many physical cases, the observed instability is a consequence of the lack of symmetry.

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