Integrating intracellular dynamics using CompuCell3D and Bionetsolver: applications to multiscale modelling of cancer cell growth and invasion.

Abstract

In this paper we present a multiscale, individual-based simulation environment that integrates CompuCell3D for lattice-based modelling on the cellular level and Bionetsolver for intracellular modelling. CompuCell3D or CC3D provides an implementation of the lattice-based Cellular Potts Model or CPM (also known as the Glazier-Graner-Hogeweg or GGH model) and a Monte Carlo method based on the metropolis algorithm for system evolution. The integration of CC3D for cellular systems with Bionetsolver for subcellular systems enables us to develop a multiscale mathematical model and to study the evolution of cell behaviour due to the dynamics inside of the cells, capturing aspects of cell behaviour and interaction that is not possible using continuum approaches. We then apply this multiscale modelling technique to a model of cancer growth and invasion, based on a previously published model of Ramis-Conde et al. (2008) where individual cell behaviour is driven by a molecular network describing the dynamics of E-cadherin and β-catenin. In this model, which we refer to as the centre-based model, an alternative individual-based modelling technique was used, namely, a lattice-free approach. In many respects, the GGH or CPM methodology and the approach of the centre-based model have the same overall goal, that is to mimic behaviours and interactions of biological cells. Although the mathematical foundations and computational implementations of the two approaches are very different, the results of the presented simulations are compatible with each other, suggesting that by using individual-based approaches we can formulate a natural way of describing complex multi-cell, multiscale models. The ability to easily reproduce results of one modelling approach using an alternative approach is also essential from a model cross-validation standpoint and also helps to identify any modelling artefacts specific to a given computational approach.

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Integrating Intracellular Dynamics Using CompuCell3D
and Bionetsolver: Applications to Multiscale Modelling of
Cancer Cell Growth and Invasion
Vivi Andasari1*, Ryan T. Roper2, Maciej H. Swat3, Mark A. J. Chaplain1
1Division of Mathematics, University of Dundee, Dundee, Scotland, United Kingdom, 2Issaquah, Washington, United States of America, 3Biocomplexity Institute and
Department of Physics, Indiana University, Bloomington, Indiana, United States of Americs
Abstract
In this paper we present a multiscale, individual-based simulation environment that integrates CompuCell3D for lattice-
based modelling on the cellular level and Bionetsolver for intracellular modelling. CompuCell3D or CC3D provides an
implementation of the lattice-based Cellular Potts Model or CPM (also known as the Glazier-Graner-Hogeweg or GGH
model) and a Monte Carlo method based on the metropolis algorithm for system evolution. The integration of CC3D for
cellular systems with Bionetsolver for subcellular systems enables us to develop a multiscale mathematical model and to
study the evolution of cell behaviour due to the dynamics inside of the cells, capturing aspects of cell behaviour and
interaction that is not possible using continuum approaches. We then apply this multiscale modelling technique to a model
of cancer growth and invasion, based on a previously published model of Ramis-Conde et al. (2008) where individual cell
behaviour is driven by a molecular network describing the dynamics of E-cadherin and b-catenin. In this model, which we
refer to as the centre-based model, an alternative individual-based modelling technique was used, namely, a lattice-free
approach. In many respects, the GGH or CPM methodology and the approach of the centre-based model have the same
overall goal, that is to mimic behaviours and interactions of biological cells. Although the mathematical foundations and
computational implementations of the two approaches are very different, the results of the presented simulations are
compatible with each other, suggesting that by using individual-based approaches we can formulate a natural way of
describing complex multi-cell, multiscale models. The ability to easily reproduce results of one modelling approach using an
alternative approach is also essential from a model cross-validation standpoint and also helps to identify any modelling
artefacts specific to a given computational approach.
Citation: Andasari V, Roper RT, Swat MH, Chaplain MAJ (2012) Integrating Intracellular Dynamics Using CompuCell3D and Bionetsolver: Applications to Multiscale
Modelling of Cancer Cell Growth and Invasion. PLoS ONE 7(3): e33726. doi:10.1371/journal.pone.0033726
Editor: Soheil S. Dadras, University of Connecticut Health Center, United States of America
Received October 11, 2011; Accepted February 16, 2012; Published March 26, 2012
Copyright: ? 2012 Andasari et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: Somite grant No. R01 GM076692 ‘‘Multiscale Studies of Segmentation in Verterbrates’’, CC3D grant No. R01 GM077138 ‘‘Development and
Improvement of the Tissue Simulation Environment’’, and ERC (European Research Council) AdG Grant No. 227619 ‘‘From Mutations to Metastases: Multiscale
Mathematical Modelling of Cancer Growth and Spread’’ (http://erc.europa.eu/). The funders had no role in study design, data collection and analysis, decision to
publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: vivi@maths.dundee.ac.uk
Introduction
0.1 About Multiscale Modelling
Computational models of complex biomedical phenomena,
such as tumour development, are becoming an integral part of
building our understanding of underlying cancer biology.
Mathematical models which are generated from biological data
and experiments, e.g., in vivo or in vitro, through phenomenological
observations in real patients help in explaining the mechanisms of
this complex phenomenon. Quantitative, predictive models have
the potential to significantly improve biomedical research by
allowing virtual, in silico modelling.
Experimentalists and theoreticians have agreed that cancer
progression involves processes that interact with one another and
occur at multiple temporal and spatial scales. The time scales
involved vary from nanoseconds to years: signalling events in the
cell typically occur over fractions of a second to a few seconds,
transcriptional events may take hours, cell division and growth and
tissue remodelling require days, tumour doubling times are on the
order of months, and tumour growth occurs over years, etc.
Typical spatial scales range from nanometres for protein-DNA
interactions to centimetres for a the development of a solid tumour
mass, tumour-induced angiogenesis, tissue invasion, etc. These
scales are strongly linked with each other. A phenomenon cannot
be completely considered using a single scale, fully isolated without
taking into account what happens at other smaller or larger scales.
In general, when incorporating different temporal and spatial
scales into mathematical models, there are three commonly used
viewpoints: the subcellular level, the cellular level, and the tissue
level. Or, from a modelling point of view these levels can also be
referred to as the microscopic scale, the mesoscopic scale, and the
macroscopic scale, respectively. Cancer usually starts at the
subcellular level marked by events that occur within the cell, such
as genetic mutations, transduction of chemical signals between
proteins, and a large number of intracellular components that
regulates outward activities at the cellular level such as
uncontrolled cell division, and cell detachment that leads to
epithelial-mesenchymal transition (EMT), etc. The main activities
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of cell populations, such as interactions between tumour cells and
host cells, intravasation and extravasation processes, proliferation,
apoptosis, aggregation and disaggregation properties, are all
viewed from a larger scale, that is the mesoscopic scale. The
macroscopic scale concerns activities that occur at the tissue level
such as cell migration, convection and diffusion of chemical
factors, all of which are typical for continuum processes [1].
During the last decade or so many approaches to multi-cell,
multiscale modelling of cancer growth and treatment therapy have
been developed. For example, see articles by [2–20] for modelling
details and [21–23] for reviews on multiscale modelling. The goal
of each approach is, in the first instance, to be able to replicate
observed experimental results and data. Since the biology of
cancer is very complex, models have to focus on ‘‘first order’’
effects and introduce certain simplifications to make them
computationally feasible. These simplifications often introduce
modelling artefacts i.e., observed model behaviours or side effects
which are due to the particular choice of the mathematical/
computational method. Isolating the source of modelling artefacts
is very difficult and quantifying the impact of such modelling
artefacts on model predictions is a daunting task. Therefore, in
order to identify deficiencies and limitations of modelling methods
currently in use, we have to be able to routinely conduct rigorous
model cross-validation to ensure that predictions of different
modelling approaches for a single biological system are in
agreement, at least qualitatively, with each other and with
experimental data. Since in many situations experimental data is
hard to find or simply unavailable model, the issue of model cross-
validation is even a more important issue.
For mathematical models of biomedical systems to be credible
and usable on a larger scale by a variety of biomedical researchers,
they have to be: a) easy to set up, b) easily reproducible, c)
transparent and open to peer review and challenge, d) publicly
accessible and able to run on multiple operating systems without
the need to recompile, and e) interactive and easily modifiable.
In this paper we present a case study on model cross-validation.
We reproduce a cancer invasion model, originally described in
[14], using a CC3D-based implementation and compare our
simulation results to those of the original paper (in which a centre-
based implementation was used). We document the details of
model building based on the published article, highlight obstacles
in reproducing published results and suggest a streamlined,
systematic approach to cell-based model cross-validation.
0.2 CC3D-Bionetsolver framework for multiscale
simulation
Modelling methodologies that explicitly represent individual
cells are particularly appropriate for modelling and simulation of
cancer invasion. There are important events and physical
phenomena associated with cancer invasion on the single-cell
level that can only be suitably captured in computational
simulations by accounting for individual cell properties and
important aspects of cell-cell interactions, such as changes in
cell-cell contact area.
In modelling the various stages of cancer progression, certain
computational and mathematical methodologies are more suitable
than others. For example, in the case of solid avascular tumour
growth, continuum models are well-suited since they capture bulk
properties of tissues. Instead of explicitly treating individual cells,
collective properties of the whole tumour tissue are modelled, such
as cell density and oxygen concentration. An advantage of such an
approach is that systems with a large number of cells, such as on
the order of 106or higher, can be handled. On the other hand,
explicit representation of individual cells and their properties (e.g.,
locations, radii, morphology, surface area, volume, etc.) can
become computationally burdensome when trying to model on the
order of 104to 106cells. Nevertheless, such individual cell-based
modelling approaches are capable of capturing phenomena and
behaviour in multicellular systems that continuum strategies
cannot capture.
Systematic development of biomedical models may be divided
into the following distinct stages: a) creating a conceptual
biomedical model, b) developing a formal description of the
model based on an established modelling language such as the
Systems Biology Markup Language or SBML, c) translating the
formal language into a set of mathematical representations, for
example, SBML is translated into a set of ordinary differential
equations or ODEs, and d) developing a computational imple-
mentation of c).
‘‘Traditional’’ biomedical model building usually skips interme-
diate stages and jumps from a conceptual model description
directly into low-level code. This is often convenient from the
perspective of a modeller but it greatly impedes model cross-
validation, reuse or sharing. Problem solving environments, such
as CC3D, Mason, or Flame, greatly reduce the amount of effort
necessary to build models which rigorously follow stages a)–d) and
at the same time offer the same level of flexibility in model
construction as low-level programming languages. To build and
run our models we used CC3D - an open source simulation
environment based on the Glazier-Graner-Hogeweg (GGH)
model which allows simulating cell behaviours on an individual
cell basis, where individual cells can interact with each other or
with the underlying medium. Several models of tumour growth
and angiogenesis have already been simulated using CC3D
environment. See, for example, articles by [24–28].
Multiscale models in CC3D-Bionetsolver are described using a
combination of the CompuCell3D Markup Language (CC3DML)
and Python scripting. Such a combined approach allows one to
build complex biomedical models and does not require recompi-
lation when running them. In a typical CC3D simulation ‘‘static’’
aspects of the model, such as lattice size, simulation runtime, list of
cell types, initial conditions or cadherin affinities, are usually
described using CC3DML. We can replace CC3DML with
equivalent Python syntax. The ‘‘dynamic’’ part of the CC3D
model is described using Python scripting. Since Python is a full-
featured programming language, modellers are able to express
complex cell type differentiation rules, couple cell properties to
concentrations of diffusive chemicals or to cell-cell signalling or
parameterise cell adhesive properties in terms of underlying
molecular or gene regulatory networks.
0.3 Comparison of center-model and GGH-model for
multicellular simulation
Here we briefly discuss the main differences and some
similarities between the centre-based model of [14] and our
model based on the GGH model. As indicated, CC3D is a
software application that implements the GGH model, allowing
lattice-based simulation of multicellular systems. Each biological
cell is represented as a set of contiguous sites on a lattice and the
system evolves in time through an energy minimization procedure.
On the other hand, the centre-based model represents each
biological cell in terms of the location of its centre of mass and its
radius. This fundamental distinction between the two methodol-
ogies is illustrated in Fig. 1.
The cells of the centre-based model behave as elastic spheres
and equations describing their behaviour and interactions are
derived on the basis of classical mechanical concepts. The centre-
based model approximates cell-cell contact areas using the radii of
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neighbouring cells and the distance between their centres. In
contrast, the concept of cell neighbour has an explicit represen-
tation in the GGH model since two cells share one or more lattice
edges (for 2D simulations) or faces (3D simulations). Because of
these differences, each modelling approach has relative strengths
and weaknesses with respect to capturing different biophysical
processes and phenomena. On the other hand, the GGH and
centre-based models also have some important similarities. Both
methodologies use continuum, reaction-diffusion equations to
model extracellular chemical fields and they both incorporate cell-
cell adhesion and mechanical constraints on cell shape. In each
case, extracellular chemical fields can both modify and be
modified by cell behaviours or properties such as cell growth
rates, secretion, absorption and chemotaxis.
0.4 An application to multiscale modelling of cancer
growth and invasion
The multiscale model of epithelial-mesenchymal transition
(EMT) developed by [14] incorporates important aspects of E-
cadherin-b-catenin signaling and its coupling to cell-level properties
of intercellular contact and adhesion. This model requires explicit
representation (on a cell-to-cell basis) of localised and spatially
heterogeneous changes in cell-cell adhesion strength and contact
areas. It is at this level of granularity that invasive cancer cells sense
and respond to their environment. In terms of biological processes,
the model of [14] captures cell-contact-dependent recruitment of E-
cadherin and b-catenin to the cell membrane and reincorporation
of both back into the cytoplasm. Computationally, the simulations
incorporated (1) time-varying changes in cell-cell adhesion as a
function of a system of ordinary differential equations (ODEs) for
intracellularreaction kineticsofE-cadherin-b-cateninsignalling and
(2) changes in rate parameter values in the reaction kinetic model as
a function of changing contact areas between neighbouring cells.
Results
We ran three sets of 3D simulations to model: (1) detachment
waves of b-catenin in a thin layer of epithelial cells, described in
subsection 0.5, (2) tumour growth and detachment of cells from a
layer of epithelial cells, and (3) tumour growth and detachment of
cells from a multicellular tumour spheroid, both described in
subsection 0.6.
Initially, all cells were individually created in the shape of a cube
of size 7|7|7 pixels, with gaps of 1 pixel length between them.
From 0 MCS to 20 MCS we allow the cells to grow, during which
time the volumes and surface areas of the cells increase and the
cells become more spherical. During this period of the simulations,
cell-cell contact areas undergo an equilibrating transient that does
not reflect natural phenomena. Thus, we did not start the
numerical integration of the differential equations (corresponding
to the subcellular biochemical networks) until 20 MCS. Keeping in
mind that the subcellular model is sensitive to changes in
intercellular contact areas, if numerical integration occurred
during the initial cell shape changes, unrealistic subcellular
dynamics could occur as an artefact of these changes. Starting
the integration at 20 MCS helped avoid this. All parameter values
used in the computational simulations are listed in Table 1, unless
stated otherwise.
0.5 Detachment Waves of Epithelial Layer Simulations
To simulate detachment waves of b-catenin in a thin layer of
epithelial cells, we performed the simulation on a domain or a
lattice of 264|224|60 pixels in x, y, and z directions,
respectively, with the z-axis being perpendicular to the page. In
the lattice, we place a sheet of cells with 30 cells along the x-axis
(horizontal), 25 cells along the y-axis (vertical), and 1 cell along the
z-axis. As mentioned, initially each cell occupies a cube 7|7|7
pixels and we insert a gap of 1 pixel between each cell, as can be
seen in the top left figure of Fig. 2. The aim of giving a 1-pixel gap
for this simulation is to give space for the cells to grow where cell
volume increases followed by increasing cell surface area until the
cells become spherical and tightly attached to each other, as can be
seen from the top second left figure (MCS 30) of Fig. 2. The initial
target volume for cells is set to 1:2 times the cell volume, making
the average volume of each cell about 410 pixels. We set 1 pixel
equal to 2mm. Therefore one tumour cell has a volume of about
3280mm3. The sheet represents a thin layer of tissue with a volume
of 0:48|0:408|0:014mm3.
In the intracellular model, summarised in Eqs. (15)–(18),
disruption of cell-cell adhesion occurs when there is an increase
in the concentration of free b-catenin in the cytoplasm, that is
when b-catenin concentration exceeds a specified threshold value
as a result of disassociation of E-cadherin-b-catenin complex at the
cell membrane. The threshold we specified for our simulations is
Figure 1. Schematic illustration of a lattice-based representation of cells in the GGH model (left figure) and a lattice-free
representation in the centre-based model (right figure).
doi:10.1371/journal.pone.0033726.g001
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50:0. As explained previously, for cell detachment to occur,
nuclear b-catenin must exceed this threshold value.
Depending upon whether b-catenin is above or below the
critical EMT-MET threshold, the terms A1and A2in Eqs. (11)
and (12) are changed appropriately. In each case, if b-catenin is
below the threshold, the first terms in Eqs. (11) and (12) are used.
On the other hand, if b-catenin is above the threshold, the second
terms in each of the equations are used. However, in the SBML
Table 1. Dimensionless intracellular parameter values for the cell detachment simulations.
Parameter Definition ValueReference
n
E-cadherin-b-catenin binding rate
100
[14]
kz
b-catenin-proteasome downregulated binding rate
1:5
Estimated
k{
b-catenin-proteasome dissociation rate
19
[14]
k2
b-catenin degradation rate in proteasome
1
[14]*
km
b-catenin production rate
14
[14]*
a
E-cadherin-b-catenin dissociation rate
2
[14]
cT
b-catenin threshold value
50
[14]
rc
E-cadherin cytoplasm-surface translocation rate
200
[14]
rd
E-cadherin surface-cytoplasm translocation rate
200
[14]
PT
Proteasome total concentration
21
[14]*
ET
E-cadherin total concentration
100
[14]
*appears in the paper’s correction.
doi:10.1371/journal.pone.0033726.t001
Figure 2. Plots showing a sequence of the disruption of a layer of epithelial cells due to an increase in the b-catenin concentration
inside the cells. After all cells have detached from the layer of cells or from each other (EMT), b-catenin concentrations eventually drop, causing
cells that are close to each other to undergo re-attachment (MET) while other cells that are not close remain as mesenchymal cells. Colours of the cells
correspond to concentration of b-catenin.
doi:10.1371/journal.pone.0033726.g002
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implementation of our subcellular model, we do not actually
implement two separate sets of equations for attached (below-
threshold) and detached (above-threshold) cells. Instead, we
include both terms from Eq. (11) and both terms from Eq. (12)
in the same SBML file. We effectively ‘‘include’’ or ‘‘omit’’ one
term or the other (depending on whether cells are below-threshold
or above-threshold) by either (1) setting a equal to 0 and n equal to
a non-zero value (see n in Table 1) for the case of a below-
threshold cell or (2) setting n equal to 0 and a equal to a non-zero
value (see a in Table 1) for the case of an above-threshold cell.
In our CC3D-Bionetsolver implementation (i.e., our Python
script), the increase of b-catenin concentration above threshold is
deliberately initiated by decreasing the value of kzat a specified
time (70 MCS) from kz~1:5 to kz~1:0. This parameter
influences the association rate of b-catenin with the proteasome.
When kz~1:5, b-catenin-proteasome complex formation is
sufficiently rapid to keep the b-catenin concentration of all cells
well below the threshold of cT~50:0. However, when kzis
decreased to a value of 1:0, b-catenin accumulates in the
cytoplasm as a result of decreased proteasomal degradation.
We check the b-catenin concentration for every cell at each
MCS. If the b-catenin concentration for a cell of type ‘‘Low-
BetaCat’’ increases above a threshold value of 50:0, the cell type is
changed to ‘‘HighBetaCat’’, n is set to 0:0 instead of 100:0 and a is
set to 2:0 instead of 0:0. Similarly, when the b-catenin
concentration of a ‘‘HighBetaCat’’ cell decreases below the
threshold, the value of n for that cell is set to 100:0 and a is set
to 0:0 and the cell type is switched to ‘‘LowBetaCat’’.
In the case of an EMT event (i.e., a cell type change from
‘‘LowBetaCat’’ to ‘‘HighBetaCat’’), changing the values of n and a
as described is equivalent to swapping the expressions in A1and
A2, between the below-threshold (½b?vcT) expressions and the
above-threshold (½b?wcT) expressions. Physically, this corresponds
to (1) a cessation of E-cadherin-b-catenin complex formation in
the membrane (n~0:0) and (2) an accelerated dissociation of E-
cadherin-b-catenin complex (i.e., the dissociation rate parameter
di(t), is increased by a~2:0) to form cytoplasmic (free) E-cadherin
and free b-catenin. Together, the effects of these two phenomena
are (1) an increased concentration of b-catenin in the cytoplasm
and (2) a significantly reduced adhesion strength between the
transformed cell and its neighbouring cells due to the loss of E-
cadherin-b-catenin complex in the membrane.
In Fig. 2 an increase of b-catenin above threshold occurs in
several cells, randomly. When one cell is induced with a high b-
catenin concentration above the threshold cT, the cell becomes
vulnerable to a loss of cell-cell attachment resulting in EMT. The
event propagates outward from this localised event, affecting
neighbouring cells. When a given cell detaches, the neighbouring
cells in turn become vulnerable to EMT because of increased free
b-catenin concentration inside the neighbouring cells. These cells
detach from surrounding cells and the effects propagate through-
out the layer of cells. At 130 MCS, we observe a small group of
cells that start to detach. By 200 MCS, detachment waves have
spread outward to adjacent cells. As time evolves, some cells at
other positions also show detachment waves independently.
Eventually around 500 MCS all cells in the layer have been
affected and have detached from each other. This is the hallmark
of EMT events. Due to the stochastic nature of the GGH model,
regular waves of cell detachment which originate from one cell
and then spread radially and regularly outward as seen in [14] can
not be produced using CompuCell3D. Nevertheless, the results are
qualitatively the same between our implementation and that of
[14]. Also, our aim in this paper is to illustrate differences between
the two approaches. They are different in different ways and we
may biologically conjecture that neither is superior to the other.
In order to see how the concentrations of proteins inside
individual cells vary over time, we wrote functions in our CC3D-
Bionetsolver code to record values to output files of all
concentrations each MCS. In Fig. 3, we plot concentrations of
b-catenin, E-cadherin-b-catenin complex, and b-catenin-protea-
some complex for a typical cell undergoing EMT and MET (see
the simulation results shown in Fig. 2). Because of the stochastic
nature of the GGH model, the concentrations fluctuate in
response to fluctuations in contact area between cells. When the
concentration of b-catenin increases significantly (due to loss of
contact area between cells and complete detachment) the
transition curve (during the period of detachment) becomes
smooth (i.e., fluctuations cease). After the cell regains contact with
other cells, the curve is observed to fluctuate again. The top figure
of Fig. 3 shows the concentrations of b-catenin, E-cadherin-b-
catenin complex, and b-catenin-proteasome complex when
running the simulation up to 5000 MCS. Here we see three
cycles of detachment and attachment, as shown from the repeated
cycles of high and low concentrations of b-catenin (yellowish-green
line).
Once the cells have detached from each other, they are free to
randomly migrate from their original positions. However, in this
simulation we did not apply any source of attractants that should
cause the cells to migrate away from the layer; the cells detach but
stay in their position or slightly move due to the stochastic nature
of CC3D. Allowing the simulation to proceed all the way to 5000
MCS, we observe that b-catenin concentrations in detached cells
gradually decrease back toward the threshold. When the
concentration reaches the threshold, a in the internal model is
again set to zero and n is set to a non-zero value. This alters the
internal kinetics such that b-catenin is no longer rapidly degraded.
Instead, it accumulates inside the cells and is reincorporated into
E-cadherin-b-catenin complex. This process occurs rapidly so that
near-zero concentrations of free b-catenin are observed in some
cells as seen in the bottom left figure of Fig. 3 (bottom figures are
plots of the concentrations to 1000 MCS or for one cycle of
detachment).
The increase of E-cadherin-b-catenin complex increases the
adhesiveness of cells and they undergo mesenchymal-epithelial
transition (MET) resulting in the reattachment of neighbouring
cells. Thus, cells again exhibit an epithelial phenotype, but this
time with an irregular configuration of the cell layer, or loss of
epithelial configuration. This is because of the random migration
of cells away from their original positions that occurred when they
were detached. The results we report here resemble those in Fig. 7
of [14]. We also observe from the simulation results that, after the
first stage of EMT events, a few cells migrate so far that they
cannot reattach to other cells. These cells remain as mesenchymal
cells.
As for cells that cannot reattach after the first detachment
(because they have migrated too far from other cells and thus
remain mesenchymal), the concentrations of the subcellular
proteins immediately reach their own steady states, as shown by
plots of data in Fig. 4.
0.6 Tumour Growth and Invasion
For simulations involving tumour growth, the GGH target
volume is incremented each MCS during growth phases at a
constant rate of 0:02 times the current cell volume and GGH
target surface area is also incremented at a constant rate of 0:02
times the current cell surface area. This results in a doubling of cell
number approximately every 40 MCS. Cell division was set to
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