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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 frequentlyinitiated by genetic mutations

thatincreasethenet rateofcell divisionandleadto theforma-

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

the formationand ingrowthof a networkof new bloodvessels

tothetumour)is neededbeforetheextensiveandrapidgrowth

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.

Theadventofincreasinglysophisticatedtechnologymeans

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 acknowledge 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 developing

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-

nomenathat operate on vastly differentspace and time scales.

We consider a vasculature composed of a regular hexagonal

network embedded in a two-dimensional NxN lattice com-

posed of normalcells, 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 oxygenand VEGF: the vessels are regardedas sources

(sinks) of 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 approximationand computethe flow rates through,

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

8060-7803-9577-8/06/$20.00 ©2006 IEEEISBI 2006

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The vessel radii are updated using a structural adaptation law

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

[1]).

Thusthemodelis formulatedasahybridcellularautomata,

with different submodels describing behaviour at the subcel-

lular,cellularandmacroscopic(orvascular)levels(seeFigure

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 oxygendeliveryto the tissue. We stress that the submod-

els we use simply illustrate how such a multiscale model can

be assembled: the frameworkwe present is general, with con-

siderablescopeforincorporatingmorerealistic(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.

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Vascular

Structural

Adaptation

Haematocrit

Distribution

(Oxygen Source)

Blood Flow

Cell?cycleApoptosisVEGF Secretion

Spatial

Oxygen

Distribution

Spatial

VEGF

Distribution

Vascular

Layer

Cellular

Layer

Intracellular

Layer

Spatial Distribution

(Oxygen Sink)

Cancer?Normal

Competition

Fig. 1. Schematic showing the structure of our hybridcellular

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’sdynamics and,

in this case, actually reduces the tumourburden. 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,closerinspectionrevealsthatseveralparts ofthetumour

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.

102030405060

10

20

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(a)

020406080 100

# iterations

(b)

0

200

400

600

# cells

10203040 5060

10

20

30

40

50

60

(c)

020406080 100

# iterations

(d)

0

1000

2000

3000

4000

# cells

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

”emptyspaces”. Panels (b)and(d) show the time evolutionof

the numberof (cancer) cells for the heterogeneousand 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 presented 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-

tionwas independentofVEGF,in Figures3to5it is regulated

bylocal VEGF levels. Figures3 and4 showhowthe 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) hand corner of the tissue,

the incoming blood flow and haematocrit become diluted as

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

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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

oxygenregions. Ifthe VEGFstimulus is weak thenthevascu-

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 pronouncedoscillations 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,theoscillationsin thecell populationsdisappearandthe

number of quiescent cells is consistently much lower. These

resultsshowhowcouplingbetweenthedifferentspatial scales

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

proportionof proliferatingand quiescentcells 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

?3

Radii & Haematocrit

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

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

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

files remain unchanged from their initial configurations, the

tumourhas increased in size and now contains quiescent cells

whichproducetrace amountsof VEGF. Reproducedwith 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

?3

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

vasculaturehas 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 concentrationat 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 (hθ) that measures the rate

at which the drug extravasates. Since increasing hθ 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 hθas 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.

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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-dashedlines show the evolution of a tumour which is

identical except that its vasculatureis not regulatedby 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 how it may be used to study

the manner in which interactions between subcellular, cellu-

lar andmacroscalephenomenaaffect the tumour’sgrowthdy-

namics and its response to chemotherapy. We stress that the

submodels we have used to describe the different processes

are highly idealised and chosen simply 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-

poratingmorerealistic submodelsandspecialising thesystem

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

rently engaged in a large interdisciplinary project which aims

tobuildavirtualmodeloftheearlystagesofcolorectalcancer

(detailsat: 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

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120

140

160

180

200

0

500

1000

Proliferating tumour cells

hθ = 0

hθ = 90

hθ = 100

0

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Number of cells

Quiescent tumour cells

0

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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

chemotherapeuticagentsthat differonlyin their extravasation

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

ingandquiescenttumourcells andthe totalnumberoftumour

cells evolve over time. Key: hθ= 0 (control, drug-free case,

as per figure 5), solid line; hθ= 90 (moderate drug), dashed

line; hθ= 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,andP.K.Maini, “Amuliplescale

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-

oryandsimulations,” 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. Appl. Sci., 2005.

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