Cancer disease: integrative modelling approaches

Conference Paper (PDF Available) · May 2006with23 Reads
DOI: 10.1109/ISBI.2006.1625040 · Source: IEEE Xplore
Conference: Biomedical Imaging: Nano to Macro, 2006. 3rd IEEE International Symposium on
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 Kirchoffs 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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