Bull Math Biol (2014) 76:2306–2333
DOI 10.1007/s11538-014-0007-y
ORIGINAL ARTICLE
A Multilayer Grow-or-Go Model for GBM: Effects of
Invasive Cells and Anti-Angiogenesis on Growth
Olivier Saut · Jean-Baptiste Lagaert ·
Thierry Colin · Hassan M. Fathallah-Shaykh
Received: 16 April 2013 / Accepted: 25 July 2014 / Published online: 23 August 2014
© Society for Mathematical Biology 2014
Abstract The recent use of anti-angiogenesis (AA) drugs for the treatment of glioblastoma multiforme (GBM) has uncovered unusual tumor responses. Here, we derive a
new mathematical model that takes into account the ability of proliferative cells to
become invasive under hypoxic conditions; model simulations generate the multilayer structure of GBM, namely proliferation, brain invasion, and necrosis. The model
is able to replicate and justify the clinical observation of rebound growth when AA
therapy is discontinued in some patients. The model is interrogated to derive fundamental insights int cancer biology and on the clinical and biological effects of AA
drugs. Invasive cells promote tumor growth, which in the long run exceeds the effects
of angiogenesis alone. Furthermore, AA drugs increase the fraction of invasive cells in
the tumor, which explain progression by fluid-attenuated inversion recovery (FLAIR)
signal and the rebound tumor growth when AA is discontinued.
Keywords Mathematical modeling · Glioblastoma · Partial differential equation ·
Brain tumors · Angiogenesis · Malignant progression
O. Saut · T. Colin
IMB, UMR 5251, University of Bordeaux, 33400 Talence, France
e-mail: [email protected]
T. Colin
e-mail: [email protected]
J.-B. Lagaert
Grenoble-INP/UJF-Grenoble 1/CNRS, LEGI UMR 5519, 38041 Grenoble, France
e-mail: [email protected]
H. M. Fathallah-Shaykh (B)
Departments of Neurology and Mathematics, University of Alabama at Birmingham,
1530 Third Ave S, FOT 1020, Birmingham, AL 35294-3410, USA
e-mail: [email protected]
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1 Introduction
Malignant gliomas comprise a significant number of new cases of brain cancer diagnosed each year. Magnetic resonance imaging (MRI) is the standard in evaluating
GBM growth and invasion. GBM, a grade IV malignant glioma, exhibits a classical
multilayer structure, which consists of a necrotic core (i.e., an area with no living
cells), a rim of proliferative cells, and a margin of infiltrative cells; an example is
shown in Fig. 1 (Louis et al. 2007). The presence of necrosis, which includes dead
cells, is a diagnostic feature of GBM (see green arrow in Fig. 1b and black arrow in
Fig. 1c). Necrosis is typically localized in the center of the tumor and is believed to be
secondary to local hypoxia generated by rapid growth rates that surpass the supply of
oxygen. The enhancing rim is believed to be due to the disruption of the blood brain
barrier, which permits the entry of the contrast agent into the tumor bed (see red arrow
in Fig. 1c); this layer localizes to the rapidly proliferating component of the tumor.
The layer consisting of increased fluid-attenuated inversion recovery (FLAIR) signal
(see black arrow in Fig. 1d) minus the enhancing layer includes edema (water) and
tumor cells with a high invasion but low proliferation rates (Giese 2003; Tonn and
Goldbrunner 2003).
Bevacizumab, a humanized anti-VEGF (vascular endothelial growth factor) monoclonal antibody, has recently gained the approval of the Food and Drug Administration
for use in the treatment of recurrent GBM, based on clinical trials that revealed both
efficacy and favorable side effects. AA therapy has raised questions about the biological and clinical behavior of these tumors (Provence et al. 2011). In particular, the
observation that GBM in some bevacizumab-treated patients recurs by a significant
increase in FLAIR signal has led to the development of the RANO (Response Assessment in Neuro-Oncology working group) criteria (Wen et al. 2010). The results of two
large phase III clinical trials, sponsored by Roche and the Radiation Therapy Oncology
Group, were recently reported; patients with newly diagnosed GBM were randomized
to standard of care with or without bevacizumab (Chinot et al. 2014; Gilbert et al. 2014;
Weller and Yung 2013). Unfortunately, the two trials did not show any beneficial effects
of bevacizumab on overall survival times. However, both trials revealed that patients
treated by bevacizumab experienced a prolongation of progression-free survival times
and better quality of life. The prolongation of progression-free survival times reached
statistical significance in the Roche trial. The fact that bevacizumab does not prolong
overall survival time may be explained by a rebound and rapid tumor growth when
bevacizumab is either discontinued or when the tumor acquires resistance. For example, Fig. 2 illustrates the rebound acceleration in growth of the enhancing/proliferative
layer of a GBM tumor when bevacizumab is discontinued (Zuniga et al. 2009).
Here, we seek answers to the following clinically relevant questions. Why does
bevacizumab prolong progression-free survival times without any significant effects
on the overall survival times of patients diagnosed with GBM (Kreisl et al. 2009;
Friedman et al. 2009; Raizer et al. 2010; Lamszus et al. 2003)? Are invasive cells
as important as angiogenesis for the growth of GBM treated by AA therapy? Does
hypoxia (low oxygen) influence the fraction of invasive cells in a GBM? What causes
rebound tumor growth when AA therapy is discontinued in some patients? The answers
are important as they have profound clinical and biological implications.
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Fig. 1 Multilayer structure of GBM. a shows a GBM at autopsy. b shows the typical pathological features
of GBM, including necrosis (green arrow) and vascular proliferation (black arrow). c is a gadoliniumenhanced T1 MRI showing necrosis (black arrow) and enhancement (red arrow), due to the breakdown of
the blood brain barrier in the part of the tumor that includes rapidly proliferating cells. d is FLAIR MRI
showing increased signal (black arrow) due to edema and infiltrating tumor cells
Tang et al. reported that glioblastoma cell lines exhibit a phenotype of low-oxygeninduced accelerated brain invasion (Tang et al. 2013). In particular, four out of eight
human GBM cell lines show accelerated invasion in low ambient oxygen as measured
by in vitro motility assays and in organotypic brain slice cultures; furthermore, this
phenotype is mediated by activation of c-src and neural Wiskott–Aldrich syndrome
protein (NWASP). Interestingly, the threshold of oxygen that controls the phenotypic
switch is higher than what is typically anticipated for cancer-related hypoxia (i.e., 0.3–
1 %); in fact, the enhancement in motility is observed at 5 % as well as 1 % ambient
oxygen. Keunen et al. studied GBM xenografts in animal brains and showed that
treatment with bevacizumab lowered blood supply but was associated with an increase
in infiltrating tumor cells (Keunen et al. 2011). Furthermore, Plasswilm et al. showed
that hypoxia significantly increases motility of a glioblastoma cell line in an in vivo
chicken model (Plasswilm et al. 2000).
This property of low-oxygen-induced acceleration in brain invasion has been also
called the grow-or-go phenotype, which indicates that GBM cells: 1) exist in either
one of two states, either dividing (P cells) or migrating (I cells), 2) proliferate when
the local environment has sufficient nutrients and oxygen, 3) migrate when insuffi-
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Fig. 2 Rebound growth when bevacizumab is discontinued. a and b show, respectively, the gadoliniumenhanced T1 and FLAIR MRIs of a 63-year-old female patient with GBM at first recurrence after receiving standard radiation with concomitant and adjuvant temozolomide. She was started with bevacizumab
(10 mg/kg every 2 weeks) on March 27, 2012, and irinotecan was added on June 7, 2012. The patient
returned to clinic on August 13, 2012, with continued clinical improvement; c and d are, respectively, the
gadolinium-enhanced T1 and FLAIR MRIs. Observe the improvement in FLAIR and contrast enhancement. Because of the small nodular enhancement (arrow in c), the referring physician determined that the
patient progressed and stopped bevacizumab. She developed acute worsening; on September 6, 2012, the
gadolinium-enhanced T1 e and FLAIR MRIs f showed a significant rebound in tumor size
cient nutrients and oxygen are available, and 4) switch between the proliferative and
invasive phenotypes as a function of the local concentrations of nutrients and oxygen
(Hatzikirou et al. 2010). Here, we seek answers to the aforementioned questions by
constructing and interrogating a new model of GBM, based on the phenotypic switch
of the go-or-grow phenotype, at the scale of clinical MRI.
The main challenge arises from the complexity of GBM and the large number of
factors that influence its growth, including mitosis, angiogenesis, genetic regulation,
biomechanics, nutrient supply, and environment influence. A mathematical model is an
optimal method for tackling multifactorial time-varying (dynamical) questions. Different approaches have been used; the discrete ones model each cell individually (Alarcón
et al. 2003; Drasdo and Höhme 2003; Gerlee and Anderson 2009; Mansury et al. 2002;
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Anderson et al. 2009; Drasdo and Höhme 2003). As each cell is described individually, the microscopic scales are accurately represented (Ramis-Conde et al. 2008),
but some macroscopic aspects (as nutrient and oxygen supply) have to be described
with continuous equations (Cristini and Lowengrub 2010; Roose et al. 2007). The
latter, which classically use partial differential equations (PDEs) on various cellular
densities, better describes these macroscopic aspects (Macklin et al. 2009). They are
also less computationally expensive than discrete models.
To take the invasive properties of GBM cells into account, Swanson et al. have proposed a model of GBM with a reaction–diffusion equation (Swanson et al. 2003; Swanson 2008; Konukoglu et al. 2010a, b). Swanson has also introduced the proliferation–
invasion–hypoxia–necrosis–angiogenesis (PIHNA) model that stipulates the presence
of two glioma cell populations in normoxia and hypoxia and explicitly incorporates the
angiogenic cascade (Swanson et al. 2011). A complex model, developed by Frieboes et
al. (2007), includes mechanical stress, interactions with extracellular matrix, discrete
(and realistic) angiogenesis, and growth-promoting factors. Neither of the models of
Frieboes and Swanson consider the go-or-grow phenotype. In particular, the Swanson
model stipulates that glioma cells in the normoxic state diffuse at the same rate as
those in hypoxic states (Swanson et al. 2011).
We introduce a mathematical model, built at the scale of medical images, that reproduces the multilayer structure of GBM and accounts for the main aspects of glioblastoma growth and motility, like cellular replication, angiogenesis, and the effects of
hypoxia on cellular phenotypes (i.e., proliferative vs. invasive). The equations take
into consideration the grow-or-go phenotype, i.e., cells switch from the proliferative
to the invasive phenotype when their local environment becomes deficient in nutrients
and oxygen (Hatzikirou et al. 2010). The outline of the paper is as follows: We start
by summarizing the assumptions in Sect. 2 and then present a detailed construction of
the mathematical model in Sect. 3 (for population of cells) and in Sect. 4 (for angiogenesis). Section 5 is devoted to the presentation of the numerical results and of the
sensitivity analysis, and Sect. 6 is the discussion. The values of the parameters used
for the simulations are given in the Appendix as well as the description of the penalty
methods (Appendix 7.1) that takes into account the geometry. The numerical schemes,
used for the discretization of the model, are described in Appendix 7.2 After showing
that the numerical results generate the multilayer structure of GBM, we interrogate
the model to investigate the effects of invasive cells and AA therapy on tumor growth
and to explain the unusual behavior of GBM in response to AA therapy. The results
offer insights into the growth of GBM in the presence and absence of angiogenesis
and an apparent explanation of the rapid growth when the tumor acquires resistance
to bevacizumab.
2 Assumptions
The model includes different cell behaviors—such as proliferative, invasive cells and
necrosis—in addition to local angiogenesis and environmental regulation (Lagaert
2011). Here, we introduce the basic assumptions of the model (see Fig. 3); important
details are included in the next section.
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Fig. 3 Cartoon depicting the assumptions. Severe hypoxia leads to necrosis (N). Under hypoxic conditions,
proliferative cells (P) switch to invasive cells (I), which migrate rapidly away from the hypoxic environment.
After traveling in the brain, invasive cells can stop their progression and become proliferative again, leading
to a new kernel of tumor. Rapid proliferation recreates the hypoxic conditions
1. Proliferative cells, whose density is denoted by P, undergo rapid mitosis. These
cells, typically located in the enhancing part of the tumor, consume nutrients (i.e.,
glucose) as evidence by 18F-FDG PET scans (see Tralins et al. 2002).
2. In the context of a glioblastoma, a cell can switch from the proliferative state to
being invasive because of hypoxia (low ambient oxygen).
3. Infiltrative cells, whose density is denoted by I , have the ability to leave the core
of the tumor and to invade the brain. After traveling in the brain, invasive cells can
stop their progression and become proliferative again, leading to a new kernel of
tumor.
4. We consider diffusible environmental factors such as oxygen and nutrients, typically supplied by blood vessels. For the sake of simplicity, we will only consider
one factor, oxygen. The proliferation rate depends on the quality of the environment, reflected by the oxygen concentration: When oxygen is low, the mitosis rate
decreases and eventually vanishes. If the hypoxia is severe, cancer cells P and I
eventually die; this is called necrosis (N), typically located in the center of GBM.
Due to their important mobility, invasive cells are able to flee quickly far from
hypoxic area.
5. We compare two angiogenesis models. The first uses a diffusible VEGF-like molecule (Model I VV E G F ). The second is simpler and more computationally efficient as it includes a smaller number of equations (Model I VI ). Because hypoxia
enhances both angiogenesis and invasion, model I VI couples angiogenesis to I
cells.
3 Mathematical Model
3.1 Modeling the Growth of Gliomas
We aim to maintain a balance between numerical efficiency and our goal of writing a mode that replicates the multilayered structure of GBM. Because we intend to
model GBM at the level of medical images, we have chosen a scale that differs from
approaches that study individual cells and single vessels (Tektonidis et al. 2011).
3.1.1 Different Cellular Types
As mentioned above, the model considers several populations of cells; P cells undergo
rapid mitosis, and I cells have the ability to leave the core of the tumor and to invade the
brain. Our assumption is that, in the context of a glioblastoma, a cell can switch from
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the proliferative state to become invasive mainly because of hypoxia. After traveling
in the brain, invasive cells can stop their progression and become proliferative again,
leading to a new kernel of tumor. If the hypoxia is severe, cells eventually die.
3.1.2 Environment
Environmental conditions play a key role in the control of the cell populations. We
consider diffusible environmental factors such as oxygen and nutrients, typically supplied by blood vessels. For the sake of simplicity, we will only consider one factor,
oxygen. Angiogenesis, a cornerstone of tumor growth, is activated when the oxygen
in its neighborhood (avascular stage) is exhausted.
3.1.3 Mechanical Aspects and Cellular Displacement
In this work, the brain is considered as a bounded and incompressible media. The
cranial cavity is considered as an incompressible media, such that the sum of each
density species is constant. One can rescale these densities in order to normalize their
sum to 1.
For the sake of simplicity, the brain is considered as undeformable. It is possible to
take the effects of the deformations into account by adding viscoelasticity as in Bresch
et al. (2009). This would significantly increase the model complexity, the number of
parameters, and the computational cost without significantly improving the accuracy
of the model.
Cell division causes an increase in volume. As we consider the brain as an incompressible media, any volume variation has to be balanced by a displacement: A dividing
cell will push its neighborhood to allow for this growth (see Fig. 4). The related velocity
field will be denoted by v.
This movement is imposed to all brain components. It does not include any active
cell displacement. In particular, it does not take the infiltration of invasive cells in
healthy tissues, neither their important mobility into account.
3.2 Equation for Tumor Cells Densities
As we consider cell populations, we model their behaviors with partial differential
equations on their densities. For each brain component, a conservation law is written:
∂t d + ∇ · Jd = sources,
(1)
where sources mean birth and death or production/removal, d denotes the species
density and Jd its flux. We choose to write the flux term as an advection term, i.e.,
Jd = dv, with v the global movement velocity as in Ambrosi and Preziosi (2002).
Fig. 4 Global displacement induced by mitosis (1D-example)
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As we have seen, in order to accurately model tumor growth, several types of cancer
cells (proliferative and invasive cells) have to be distinguished and described. Since
glioblastoma is characterized by a necrotic core, the necrotic cells (i.e., exposed to
severe hypoxia) degrade quickly. All these cells follow the global movement (in accordance with Eq. (1)). The invasive cells’ high (active) mobility is modeled by diffusion
and chemotaxis terms. As in Swanson (2008), Konukoglu et al. (2010a), the diffusion
term reproduces the important tumor infiltration in healthy tissues. The chemotaxic
term adds an environment influence leading to active displacement such that the cells
move toward the most optimal environmental conditions where they become proliferative again. Of course, these two terms take the brain structure described in Sect.
3.4.1 into account.
The only movement considered for necrotic cells is the one due to the growth of
volume caused by cellular division (see Ambrosi and Preziosi 2002). As it is a passive
movement, the necrotic cells may move by being pushed by their neighbors. They are
not actively moving contrary to invasive cells. Yet, in the necrotic core, the velocity
corresponding to this movement is almost always vanishing, and in practice, these
cells are not affected by this advection term.
3.2.1 Proliferative Cells
The proliferative cells do not have any active mobility. They only follow the global
movement created by the increase in volume. We assume that their proliferation
depends only on the environment.
Let brain , P(x, t), and I (x, t) denote, respectively, the brain domain (in 2D or
3D), the proliferative, and invasive cells, respectively, at point x ∈ R2 (or R3 ), at time
t ∈ R+ .
The conservation law reads:
∂t P + ∇ · (vP) = m(C)P −
mitosis
αP
transition from P to I
+
βI
transition from I to P
− γ f NP P
necrosis
(2)
where f N P is a threshold function, depending on the oxygen concentration and given
by
f N P = H̃(Chypoxia − C)
(3)
where C is the oxygen concentration, Chypoxia the hypoxia threshold and the function
. R is a stiffness constant (taken
H̃ a smoothed Heaviside function : x → 1+tanh(R.x)
2
arbitrarily to be equal to 500).
The proliferation rate m(C) depends on the quality of the environment, reflected
by the oxygen concentration: When the environment quality is low, this mitosis rate
decreases and eventually vanishes.
No overpopulation threshold is included in the necrosis term (as in Bresch et al.
2010) because when cell density is high enough, oxygen consumption grows and the
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Fig. 5 Mitosis coefficient m as
a function of the oxygen
concentration C
cells become hypoxic. Then, they stop their mitosis and switch to the invasive phenotype, whose cells move away fast. Furthermore, if density of cancer cells continues to
grow, hypoxia becomes severe and cells die.
Environment influence on mitosis The behavior of a tumor cell strongly depends on
the environmental conditions and more precisely on the available quantity of nutrients
(i.e., oxygen). If the oxygen concentration (C) is sufficient, proliferative cells divide
at the maximum rate τ −1 . Mitosis stops when the oxygen concentration falls below
a threshold. We propose the following term based on the grow-or-go phenotype (see
Fig. 5):
⎧
⎪
1/τ
for C > Cinv ,
⎪
⎨
m : C →
C−Cmit
Cinv −Cmit
⎪
⎪
⎩0
/τ for Cinv ≥ C ≥ Cmit ,
for C < Cmit .
where Cinv is an oxygen threshold defined in Sect. 3.2.2 (corresponding to a small
lack of oxygen, bigger than Chypoxia ) and Cmit = (Cinv + Chypoxia )/2. This ensures
that proliferative cell will stop their mitosis before necrosis.
Figure 6 sums up the oxygen influence on the behavior of proliferating cells.
Fig. 6 Oxygen regulation of tumor cells
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3.2.2 Invasive Cells
We assume that they follow some pattern related to the brain structure (Giese 2003;
Tonn and Goldbrunner 2003; Lamszus et al. 2003) and move to more favorable locations, along the gradient of oxygen (Bock et al. 2011; Avni et al. 2011). We choose
to describe their behavior by a diffusion equation as in Konukoglu et al. (2010a).
Because glioma cells are known to track along the white matter tracts, we apply the
diffusion tensor to guide the direction of their migration. For the sake of simplicity, we
will not consider the effects of the alignment of the extracellular matrix on tumor cell
migration (Schedin and Keely 2011). The diffusion tensor reflects the direction of the
diffusion of water along the white matter tracts; our assumptions is that the direction
of migration of the tumor is also along these tracts. A chemotaxis term reproduces
their attraction to more favorable places. As they do not proliferate, the only source
term corresponds to proliferative cells that become invasive. The switching between
proliferative and invasive states is modeled by a random transition rate in our density
equations. This transition is driven by the quality of the environment: In favorable
environment, the transition is from the invasive to the proliferative behavior. If the
environment is not favorable, a transition in the opposite direction is enforced. The
transition rates are more precisely defined in the next paragraph.
Writing the mass balance equation yields:
α P − γ f NI I ,
∂t I + ∇ ·(vI ) − ∇ ·(K I ∇ I ) = − η∇ ·((1 − I )I ∇C) − β I + diffusion
chemotaxis
I →P
P→I
necrosis
(4)
where the function f N I has the same expression as f N P in Eq. (3). The necrosis term
leads to the death of the invasive cells that do not escape the hypoxic zone.
Transition laws between invasive and proliferative cells It is observed that proliferating cells do not invade and migrating cells do not proliferate (Giese et al. 2003).
We assume in this paper that the transition rate α from proliferative cells to invasive
cells is a decreasing function of oxygen concentration. That means that hypoxia is
responsible for the creation of invasive cells or at least that the increase in the number
of invasive cells is related to the hypoxia level (see Fig. 3). We choose to use the same
regularized Heaviside as in Sect. 3.2.1:
α(C) = α0 H̃ (Cinv − C) × 1 − m(C)τ .
(5)
We multiply by the factor (1 − m(C)τ ) to model that a cell either divides itself or
transits to the other type.
The transition rate is assumed as being an increasing function in respect of the
oxygen concentration. We also add a concentration effect (if the density of the invasive
cells is high, transition to proliferative cells will be promoted). The overpopulation is
avoided by forbidding this transition in case of high proliferative cell density.
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This gives:
β(C) = β0 H̃ β − C I ,
(6)
where = P + I is the tumor cell density and β the threshold on tumor cells above
which the transition is inhibited.
3.3 Other Species
As tumor cells may die, the density N of necrotic cells is introduced. This allows
for the emergence of the necrotic core which differentiates a glioblastoma from a
lower-grade glioma.
The brain also contains other constituents such as healthy cells and cerebrospinal
fluid. As they do not play specific key roles in the tumor growth, they are not distinguished one from the others to keep the model as simple as possible. They are all
gathered in a generic compartment we shall call brain matrix and denote by B. As our
model includes some biomechanics, they have to be taken into account: Even passive
components appear in the mass conservation law.
Necrotic cells and brain matrix follow the global movement (Sect. 3.4.2). As invasive cells infiltrate healthy tissues, the ”brain matrix” species (with includes liquid
and healthy cells) has the inverse move to compensate their active mobility. This
translates the incompressible mixing between these two components. This differs
from the description in Bresch et al. (2010) that does not take invasive cells into
account.
The mass conservation laws provide:
∂t N + ∇ · (vN ) = γ f N P P + γ f N I I + γ f N B B,
(7)
∂t B + ∇ · (vB) = −γ f N B B − ∇ ·(K I ∇ I ) + η∇ ·((1 − I )I ∇C).
(8)
3.4 Mechanic
3.4.1 Brain Structure Influence on Diffusion Tensors
Each diffusion tensor is given by the brain structure and the tumor retroaction on this
structure. In accordance with biological knowledge, we make the following assumptions: A healthy brain is a heterogeneous and anisotropic medium. Gray matter is
denser than white matter. Thus, diffusion is more important in white matter (Giese
2003; Swanson 2008; Konukoglu et al. 2010b). As white matter consists mostly of
myelinated axons, it is an anisotropic medium. This implies the existence of preferred
directions of propagation (Lamszus et al. 2003). Tumor core is denser than healthy
tissues. The tumor also reorganizes the tissue; therefore, its core and the necrotic core
are also more isotropic.
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This structure will influence the diffusion of other species like oxygen. We choose
to introduce a standard diffusion tensor K and to take all other diffusion tensors as
proportional to this one: K α = λα K . Thus K I is defined by K I = λ I K .
This standard diffusion tensor is defined by (see 10):
K = (1 − P) K heal + P K tumor ,
(9)
where K heal is the diffusion tensor in the healthy brain tissues and K tumor the diffusion
tensor in the tumor (K tumor ∞ ≤ K heal ∞ ).
Diffusive MRIs allow to obtain the water diffusion tensor in the brain K water . As
this tensor reflects the brain structure, K heal could be taken equal to K water . For the
tumor diffusion tensor, we choose K tumor = λtumor Id, as it is a more isotropic media.
There remains one scalar (λtumor ) to fully determine the diffusion coefficient. Let us
note that diffusion-weighted MRIs may also give it.
3.4.2 Dynamic of the Cells
Global mass conservation As we consider the brain as an incompressible medium
(Sect. 3.1.3), the sum of the densities satisfies:
P+I+N+B=1
(10)
By summing each density Eqs. ((2)–(4) and (7)–(8)), we obtain a divergence law
on the velocity:
∇ · v = m(C)P
This velocity balances volume variations, which are only caused by cell proliferation.
System closure The system is closed thanks to Darcy’s law equation (Ambrosi and
Preziosi 2002):
v = −K v ∇π,
(11)
with K v = λv K a tensor diffusion defined as explained in Sect. 3.4.1. As the scalar
coefficient λv does not have any influence on v, we finally choose to define K v by
K v = K 0 . Here, π denotes the pressure.
Boundary conditions During glioma growth, no brain cell (either tumor or healthy)
leaves the brain. To balance the increase in cell density (due to the tumor growth),
some brain matrix has to disappear (i.e., degraded and removed from the tumor).
We choose to impose a no-flux condition on the brain boundaries. The no-flux
condition matches with a classical Neumann condition in isotropic cases.
This provides the following modified equations on velocity divergence and brain
matrix density laws:
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∂t B + ∇ · (vB) = −γ f N B B − ∇ ·(K I ∇ I ) + η∇ ·((1 − I )I ∇C)
−1 B,
−
m(C)P d
Bd
brain
∇ · v = m(C)P −
(13)
brain
m(C)P d
brain
with B1,brain =
(12)
B
B1,brain
,
(14)
Bd.
brain
The volume created by the growth is subtracted from the right-hand side in the part
occupied by the brain matrix. The no-flux condition ensures the pressure π is well
defined up to an additive constant, and its divergence (the velocity v) is well-defined
and unique.
4 Oxygen, Nutrients, and Angiogenesis
4.1 Nutrients Brought by the Existing Vasculature
A dense network of capillaries and some bigger peripheral blood vessels bring the
nutrients and oxygen supplies to the tumor. The neovasculature is itself an aggregate
of capillaries. Rather than to model every small discrete capillary (this is out of reach
for the time being), we adopt a continuous approach and model the oxygen supply
by a continuous source term. This source term has high values in dense vascularized
locations and smaller values elsewhere.
The oxygen concentration follows a diffusion equation:
∂t C − ∇ · (K C ∇C) = −δ1 P + Falim ,
(15)
K C = λC K ,
(16)
B
Falim = (Cheal − C)F0 1+P+I
+N + (C max − C)Fangio ,
(17)
with for Falim being the diffuse source term and −δ P the consumption by proliferative
cell.
Because FDG position emission tomography (FDG-PET) typically shows that
healthy tissues and the invasive component of the tumor have a much lower nutrient uptake than proliferative cells, we neglect oxygen or nutrient consumption by
healthy tissue and invasive cells.
The production term is limited when the oxygen reaches a given threshold Cmax to
avoid unlimited accumulation far from the tumor. The multiplication by (Cmax − C)
models this oxygen saturation. There is no oxygen exchange through the cranial cavity,
and we impose a no-flux condition in our domain. The source term Falim (17) is itself
split into two parts. The first one, F0 , is related to the existing vasculature, while
the second, Fangio , is given by the angiogenesis process. AA is modeled by setting
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Fangio = 0; furthermore, during AA therapy, the oxygen supply is modeled by the
B
term F0 1+P+I
+N , which decreases at P, I , and N increase.
4.2 Angiogenesis
Angiogenesis is a very difficult phenomenon to model. Keeping an accurate description
is difficult without obtaining a very complex model. Qualitative models are already
well developed but contain too many equations and parameters to be used for clinical
applications. Usually, discrete approaches are used and they are difficult to couple
with one continuous model.
A continuous approach, as proposed by Billy et al. (2009), would be preferred
here. But this model is still too difficult to parametrize and too expensive to be used
for numerical simulations on a 3D real geometry. To keep the model as simple as
possible, a phenomenological angiogenesis model is used (losing the ability to differentiate specific effect of a given anti-angiogenic drugs on neovascularization from
the effect of another one). Endothelial cell proliferation, their migration, and the formation of discrete blood vessel are not described. We assume that hypoxia generates
a diffusible factor (like VEGF) that induces angiogenesis (Carmeliet and Jain 2000);
furthermore, the new blood vessels are the source of oxygen. We model angiogenesis
by two methods; the first couples angiogenesis to the invasive cells. The second method
is traditional in the sense that we use a VEGF-like molecule that is decoupled from
invasive cells. These two methods are compared by numerical results. The advantage
of the first method is its simplicity. An advantage of the second model is that it may
be used to evaluate the dynamics of GBM in the presence of angiogenesis but without
invasive cells (see below).
4.2.1 Coupling Angiogenesis to Invasive Cells
In this model, we configure invasive cells as surrogates for hypoxia (or VEGF) and
proliferative cells for the localization of neovascularization. Recall that in our model,
the density of invasive cells is linked to hypoxia; we therefore decide to define a
numerical angiogenesis factor (ζ ) whose time derivative is proportional to the density
of invasive cells. The main advantage is the simplicity of this approach: Without
adding any new unknown variables, we obtain a numerical quantity that behaves like
the VEGF in Billy et al. (2009). Indeed, invasive cells are created by hypoxia, starting
from the outer part of the tumor, and then undergo a diffusion process.
As these new capillaries appear in the proliferating rim of the tumor, the neovascularization localization is determined as a function of the proliferative cells density
P and therefore the source of oxygen (Fangio ) is given by multiplying this numerical
angiogenesis factor by the density of proliferative cells.
The source term in the equation of oxygen (15) is computed through
Fangio = Fmax P tanh (ζ ) ,
1
ζ
I d −
Pd,
∂t ζ =
τ2 Vbrain τ3 Vbrain (18)
(19)
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O. Saut et al.
with Vbrain the volume (i.e., the measure) of the brain brain . In order to include the
effects of anti-angiogenic drugs, we use Fmax = 0.
This model is of course less complicated than biological models: There is neither
endothelial cell migration, nor maturation time. But it still keeps the main phenomenological behavior. As shown in our numerical simulations, the model is satisfactory
and will allow us to simply but efficiently add anti-angiogenic drugs effects.
4.2.2 Linking Angiogenesis to VEGF-Like Molecule
Here, we decouple angiogenesis from I and link it to a VEGF-like agent, denoted by
χ , which follows a law similar to (4); its evolution is described by
∂t χ − ∇ ·(K χ ∇χ ) + ∇ ·(vχ ) = −βχ χ + αχ P,
(20)
Then, 19 becomes
1
∂t ζ =
τ2 Vbrain
ζ
χ d −
τ
V
3 brain
Pd.
(21)
As the VEGF-like agent decay does not depend on hypoxia, but rather corresponds
to a natural degradation in the tissues, the associated coefficient βχ (C) is considered
as being constant. This decay was neglected so far (and thus the angiogenesis was
overestimated) due to the low rate of transition from invasive cells to proliferative
cells (β0 1). As Eq. (21) only uses the spatial integral of χ , the equation (20) is
replaced by its space integral. This improves the computational time (there is no more
anisotropic diffusion to solve). Then, 20 becomes
∂t χ = −βχ ,0 χ + αχ (C)
Pd,
(22)
and 19 becomes
∂t ζ =
χ
ζ
−
τ2 Vbrain
τ3 Vbrain
Pd.
(23)
5 Results
The details of the numerical methods we have used to discretize the equations of the
model are presented in the Appendix. All our simulations use the brain geometry
provided by the Brainweb model1 (Collins et al. 2002), which provides a realistic and
accurate brain geometry with white and gray matters, ventricles, cranial bones, already
1 http://mouldy.bic.mni.mcgill.ca/brainweb/.
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Effects of Invasive Cells and Anti-Angiogenesis on Growth
2321
Fig. 7 Model generates multilayer structure of GBM. Time evolution (from the left to the right) of invasive
(first row, the max(I ) = 0.012), proliferative cells (second row, max(P) = 1.0), and the necrotic core
(third row, max(N ) = 1). The figure plots simulations of the full model (simulation I V I , see below), for
time t = 17, 35, 52, and 70. The lower densities are shown in blue and the highest in red. To highlight the
invasion, we have plotted on the first row, in gray, isolines of the invasive density for 10 values uniformly
distributed in the range [10−4 , 0.012]. P, I, and N denote the relative density of, respectively, proliferative,
invasive, and necrotic cells
segmented and adapted to a cartesian mesh. It has already been used in different glioma
models (Swanson et al. 2003; Clatz et al. 2005). For simulations, a 2D slice is extracted
from this 3D data.
5.1 Multilayer Structure
Numerical results in Fig. 7 show that the model generates the multilayer structure
of GBM, including the invasive and proliferative cells and the necrotic core. Notice
the rim of proliferative cells, central necrosis, and brain invasion. The highest density
of invasive cells is at the border of the necrotic zone. Furthermore, our findings also
reveal that the density of the new blood vessels is highest at the proliferative rim of
the tumor. These results are consistent with the pathology of GBM.
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2322
Table 1 Description of the
different setups used in the
numerical simulations
O. Saut et al.
I
Exponential growth obtained in context of an
excess of oxygen with no invasion and no
angiogenesis
II
Logistic growth: tumor growth in context of
realistic oxygen supply with no invasion and
no angiogenesis
III
Same as simulation II but with active invasion
but no angiogenesis
I VI
Realistic glioma growth: Invasion and new
angiogenesis model are both active (same
context as simulation III but with new
angiogenesis model)
I VV E G F
Same as I V I but the angiogenic model is
coupled to a VEGF-like molecule instead of
invasive cells
V
Same as I VV E G F but with no invasion
(angiogenesis remains active and decoupled
from invasive cells)
5.2 Model Interrogation
We wish to use the model to better understand the effects of angiogenesis and invasive
cells on the dynamics of GBM growth. We interrogate the model by studying tumor
growth under six different setups, detailed in Table 1, and described more precisely
below. To switch from one to another, only few parameters have to be changed, as
shown in the Appendix (Table 2). More precisely, only one parameter differs between
simulations I and II, between simulations II and III, and between simulations III and
I VI . All simulations are performed on a 181×217 mesh (which matches the brainweb
data).
5.2.1 Simulations I and II: Absence of Invasive Cells and Angiogenesis
In the classical avascular model (like in Bresch et al. 2010), the tumor grows initially
exponentially as long as it is small enough not to be hypoxic. As the oxygen supply
is—at this stage—constant, its growth is finally limited and thus it saturates. The first
two simulations reproduce the effect of different nutrient supplies in a tumor without
any invasive cells and without angiogenesis: As oxygen is the main factor leading
the tumor growth, these simulations show numerically the influence of an excess of
oxygen (simulation I) and, on the contrary, of a lack of oxygen (simulation II). In both
simulations, the transition rate from proliferative to invasive cells is vanishing (see
Appendix). In simulation I, we introduce an excess of oxygen by considering a very
low consumption rate 2 and we obtain logically an exponential growth (Fig. 8). In
simulation II, the oxygen consumption is elevated and therefore the model reproduces
2 This excess could also be obtained with a huge supply in oxygen. As discussed in the Appendix, an
overpopulation threshold equal to 0.99 has been reintroduced.
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Effects of Invasive Cells and Anti-Angiogenesis on Growth
2323
Fig. 8 Evolution of the tumor mass for the different models. a Shown is the evolution of the tumor mass
(i.e., (P + I )) with respect to time for the models detailed in Table 1. b plots the evolution of the tumor
mass in the sole presence of either invasive cells (III) or angiogenesis (V)
a logistical growth (Fig. 8). These two simulations will be used as a reference to
quantify the effect of invasive cells and of angiogenesis on the growth of the tumor.
5.2.2 Tumor Growth by Invasive Cells is as Important as by Angiogenesis
We are now ready to seek answers to the questions posed in the introduction by measuring tumor mass in simulation III, i.e., with invasive cells but without angiogenesis,
and in simulation V, where angiogenesis is the only mechanism considered (see Table
in Appendix). Notice that in order to compare the numerical results, we use the same
parameters for all the simulations whenever possible (see Table in Appendix). The
results, shown in Fig. 8, reveal a novel and key finding that highlights the seminal
contribution of invasive cells to the growth of tumor mass; in particular, the presence of invasive cells (simulation III) generates a growth rate that approaches the one
observed by the sole effects of angiogenesis. Furthermore, though growth seems to be
initially quicker with angiogenesis alone, in the long run through invasive cells alone,
the tumor can reach a higher mass.
5.2.3 Comparing the Full and Simple Angiogenesis Models
Combining both invasive cells with angiogenesis generates faster tumor growth than
the mere effects of angiogenesis or invasive cells alone (compare simulations III/V
to I VI /I VV E G F ). Interestingly, both the simple model I VI and the complex one
I VV E G F yield similar accelerated growth rates.
5.3 Therapeutic and Rebound Effects of AA Drugs
5.3.1 Effects of AA Therapy on the Tumor
We are ready to apply the model to elucidate how GBM reacts to AA therapy (Fig. 9); in
particular, we allow the tumor to grow in the presence of invasive cells and angiogenesis
and then make the environment hypoxic. The results reveal that AA therapy exerts
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O. Saut et al.
Fig. 9 Tumor response to AA drugs. a, b, and d show the evolution of the tumor, proliferative, and necrotic
masses with respect to time. c plots the invasive fraction with respect to tumor mass. The full model and
only invasive cells correspond to simulations I V I and III, respectively (see Table 1). The colored lines show
the behavior of a tumor, initiated by the full model, such that AA therapy was started and discontinued at
times indicated by arrows
Fig. 10 Rebound vascularization. The evolution of vascularization in the brain before the start of AA
therapy in simulation II of Fig. 9 (a), after AA therapy (time = 50) (b), and after stopping AA therapy (time
= 60) (c). The figure plots vascular density (VD), which corresponds to Falim in our model
novel effects on tumor growth because it affects the number of proliferative, invasive,
and necrotic cells. In particular, tumors treated by AA (see Fig. 9a) migrate from the
growth curve of the full model (simulation I VI , see Table 1) to the growth curve that
depicts the dynamics of model III (i.e., with invasive cells and without angiogenesis,
Table 1). This is associated with lower vascular density in the tumor (Fig. 10a, b), a
decrease in the number of proliferative cells (Fig. 9b), and increase in the numbers
of invasive (Fig. 9c) and necrotic cells (Fig. 9d). AA therapy lowers the number of
proliferative cells both by necrosis and by enhancing their transformation toward the
invasive phenotype. Notice that, while the invasive population is small in the presence
of angiogenesis, during AA therapy the fraction of invasive cells is larger than when
angiogenesis is present (Fig. 9b).
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Effects of Invasive Cells and Anti-Angiogenesis on Growth
2325
5.3.2 Rebound Growth When AA is Discontinued
Stopping AA therapy in some GBM patients causes a rebound characterized by rapid
tumor growth (Zuniga et al. 2009). To investigate this phenomenon, we allow the
tumor to grow in the presence of invasive cells and angiogenesis, make the environment hypoxic to simulate AA therapy, and then re-activate angiogenesis (Fig. 9). The
numerical results generate a rebound in both tumor growth and vascularity. In particular, tumor growth accelerates from the curve of simulation III toward that of simulation
I VI (see Fig. 9a). This is associated with an increase in proliferative cells (see Fig.
9b), decrease in invasive cells (see Fig. 9c), and stabilization of necrotic cells (see Fig.
9d). Interestingly, the rebound in the vascular density in the tumor is deep into the
brain (Fig. 10b, c), thus supporting the hypothesis that that invasive cells are the key
determinants of the growth and recurrence of GBM tumors treated by AA.
5.4 Sensitivity Analysis
The aforementioned results highlight the key role of brain invasion in determining
the behavior of malignant gliomas. Because most of the clinical symptoms may be
attributed to the expansion of the enhancing rim, we perform a sensitivity analysis by
computing the time needed for P cells to occupy 1, 2, and 3 % of the brain (see Fig.
11). The findings reveal that tumor growth is very sensitive to small changes in α of
the order of 10−1 (transition rate from P to I ; Fig. 11a); higher values of α enhance
the expansion of P cells (i.e., the brain is invaded by P cells in a shorter time period).
On the other hand, tumor growth is not sensitive to changes in β (transition rate from
I to P; Fig. 11b), as large increments of the order of 100 are needed to slow down
the expansion of P cells. Furthermore, the passive diffusion parameter K I and the
chemotaxis coefficient η are positively correlated with brain invasion, which is more
sensitive to the former (Fig. 11c, f). As anticipated, both a high replication rate (i.e.,
Fig. 11 Sensitivity analysis. Time needed for P cells to occupy 1, 2, and 3 % of the brain in response to
changes in α (transition rate from P to I ; a) β (transition rate from I to P; b) K I (passive diffusion; c)
Fmax (maximal angiogenesis density; d) τ (1 / replication rate; e) and η (chemotaxis coefficient; f)
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O. Saut et al.
small τ ; Fig. 11e) and a robust and effective angiogenesis (i.e., large Fmax , Fig. 11d)
are positively correlated with rapid expansion of the enhancing rim. It is noteworthy
that ineffective angiogenesis (i.e., lower values of Fmax ; Fig. 11d) slows down the
expansion of the enhancing rim at the cost of increasing the fraction of invasive cells
in the tumor (see Fig. 9).
6 Discussion
We have finally obtained a mesoscopic model devoted to the growth of GBM, based on
a development of the work of Bresch et al. (2010). This model generates the multilayer
structure of GBM. The equations take into consideration the grow-or-go phenotype,
i.e., cells switch from the proliferative to the invasive phenotype when their local
environment becomes deficient in nutrients and oxygen (Hatzikirou et al. 2010).
Here, we have chosen to construct a model at the scale of medical images of
human GBM to better understand and explain recent clinical observations from the
use of AA therapy in GBM. In particular, we are interested in generating hypotheses
that explain: 1) the short-lasting therapeutic effects of AA therapy, 2) progression
by FLAIR/invasion, and 3) the rebound growth of some GBM when AA therapy
is discontinued (see Fig. 2). Notice that the model replicates clinical features that
were not part of its hypotheses, including the effects of starting and discontinuing
angiogenesis. In particular, the rebound effect is not explicitly included in the model;
we expected tumor growth to accelerate, when AA therapy is discontinued, but we
did not anticipate that it will rebound to the same level as the simulation that includes
both angiogenesis and invasion.
The results suggest new ideas on tumor growth in vivo and on the use of AA
therapy, applicable to any tumor having the property of grow-or-go. Figure 12 is a
cartoon that illustrates the idea that tumor cells may grow in vivo, in the absence of
angiogenesis, by repeating a cycle of (i) depleting the local oxygen supply provided
by normal existing vasculature, (ii) switching to the invasive phenotype and migrating
toward adjacent brain, and (iii) switching back to proliferating cells, which consume
Fig. 12 Tumor growth in the absence of angiogenesis. Cartoon depicting GBM growth without angiogenesis. After depleting the oxygen and nutrients supplied by existing blood vessels, tumor cells switch from
the proliferative to the invasive phenotype leading to migration toward adjacent brain where the cycle may
be repeated
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Effects of Invasive Cells and Anti-Angiogenesis on Growth
2327
and deplete the oxygen supply of the newly occupied brain. The model also suggests a
justification of the unusual clinical responses of GBM to bevacizumab, which merely
prolongs progression-free survival times. The results reveal that AA exerts its therapeutic effects by killing proliferative cells (see Fig. 9b), which explains the initial clinical
improvement and the prolongation of progression-free survival times. However, AAinduced hypoxia causes GBM-escape by increasing the fraction of invasive cells in
the tumor (see Fig. 9c) leading to the accelerated growth when the tumor acquires
resistance to bevacizumab as invasive cells switch back to the proliferative phenotype.
This acceleration justifies the absence of therapeutic benefits of bevacizumab on overall survival times as it erases the gains. These results also explain the lack of long-term
efficacy of bevacizumab in the treatment of GBM (Kreisl et al. 2009; Friedman et al.
2009; Raizer et al. 2010; Lamszus et al. 2003) and why some bevacizumab-treated
GBM recurs by increase in FLAIR signal/invasion (Wen et al. 2010).
Tektonidis et al. applied lattice-gas cellular automata to model the phenotypic switch
from P to I ; they report that the model, which best explains serial digital images of
glioma spheroids implanted into collagen gel (Stein et al. 2007), assumes densitydependent phenotypic switching and repulsion between tumor cells (Tektonidis et al.
2011). Pham et al. modeled the density-dependent phenotypic switch and found that it
generates complex dynamics similar to those associated with tumor heterogeneity and
invasion (Pham et al. 2012). Hatzikirou et al. apply the lattice-gas cellular automata
model to study the invasive phenotype of GBM; they propose that the grow-or-go
phenotype and low oxygen conditions play a key role in the rapid growth of a GBM
after resection (Hatzikirou et al. 2012). Gerlee et al. propose a stochastic model of
the grow-or-go phenotype such that the motile state is subject to random motion.
They derive two coupled reaction–diffusion equations, which exhibit traveling wave
solutions (Gerlee and Nelander 2012). Because these models are not at the scale of
clinical MRI, they do not yield answers the clinically scale questions listed above.
The clinical scale models of Swanson and colleagues do not consider the go-or-grow
phenotype (see introduction).
The parameters of the model should not be taken literally; at this time, there are no
biological or clinical data that allow us to compute or infer the units to the parameters
(see Table 2). For example, there is no current biological assay that distinguishes
proliferative from invasive cells; hence, the units of the transition rates α and β cannot
be computed. There are no experimental results on the rate of diffusion of oxygen
in human glioma tissue or the rate of diffusion of invasive cells. In addition, there
are no biological experimental results that define the oxygen thresholds. Our aim was
to obtain qualitative results and not fit any particular patient data; the units of the
parameters and their variability between patients are interesting problems for future
clinical and basic experiments.
Finally, we believe the mathematical model has uncovered the following novel biological hypotheses and questions that set the stage for experimentation and validation:
1) GBM grows in the absence of angiogenesis by a cycle of proliferation and invasion
and 2) invasive cells mediate the rapid rebound in growth when the tumor acquires
resistance to bevacizumab. At this time, there are no markers that distinguish proliferative from invasive cells; the results and those of others just discussed highlight the
need of a biological characterization of the switch from proliferative to invasive cells
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and suggest that anti-motility agents may be useful when combined with AA drugs
for the treatment of GBM.
Acknowledgments HFS is supported by R01GM096191 from the National Institutes of Health. TC and
HFS contributed equally to this work.
7 Appendix: Model Parameters and Implementation
The parameters of the simulations of Table 1 are shown in Table 2. For implementation,
the same point of view as in Colin et al. (2013) is chosen. The various numerical
schemes and the main difficulties are presented.
Table 2 Biological parameters used in simulations I, II, III, and I V I
Description
Symbol
Values
I
Division rate
1
τ
III
I VI
7
α
0
β
0.02
Overpopulation threshold
β
0.9
Death rate
γ
0.8
Oxygen thresholds
C0
0.65
Cinv
0.7
Cmax
1.8
Chemotaxis coefficient
η
0.2
Oxygen consumption rate
δ1
5
Oxygen supply
F0
8
Angiogenesis growth velocity parameter
τ2
1/400
Angiogenesis degradation parameter
τ3
1/200
Angiogenesis max. density
Fmax
0
Diffusion parameter of invasive cells
λi
0.2
Diffusion parameters of oxygen
λtumor
0.8
λC
0.4
Transition rate
II
0.15
55
0.8
The parameters are chosen to produce realistic behaviors. Division rate: 1/tau (time−1 ), transition rates
−1
α, β (time−1 ), death rate (time−1 ), chemotaxis coefficient ( distance
time ), oxygen consumption rate: time ,
oxygen supply (time−1 ), angiogenesis growth velocity parameter τ 2 (time), angiogenesis degradation
2
parameter tau3 (time), diffusion parameters ( distance
time )
7.1 Level-Set and Penalty Method
Brain geometry is very complex. Meshing it is difficult and involves a huge number of
points. A level-set formulation is used in order to describe its boundary and to impose
our boundary conditions with penalty methods (Angot et al. 1999). This allows us to
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Effects of Invasive Cells and Anti-Angiogenesis on Growth
2329
use a Cartesian mesh. The distance to the boundary is chosen as a level-set function:
+d x, ∂brain if x ∈ brain ,
l: x →
/ brain ,
−d x, ∂brain if x ∈
where d denotes the classical distance in R2 or R3 .
All the boundary conditions are no-flux conditions on ∂C . It is presented here on
the equation satisfied by the pressure π obtained b taking the divergence of Eq. (11)
and using Eq. (14). This equation together with the boundary condition (K ∇π )·n = 0
(where K is a tensor) is replaced by:
−∇ · K ε ∇π ε = m(C)P −
B
B1,brain
brain
m(C)Pd on ,
with π ε = 0 on ∂, being a box in which brain is embedded. Here, K ε is defined
by:
K ε = χx∈brain K + εχx∈
/ brain .
and χx∈brain denotes the characteristic function of brain . As ε → 0, π ε converge to
π solution of (10):
−∇ · K ∇π = m(C)P −
B
B1,brain
brain
m(C)Pd
on ωbrain with (K ∇π ) · n = 0 on ∂brain .
7.2 Numerical Scheme
Concerning time discretization, a second-order splitting scheme is used. The main
advantage of this strategy is that the resolution of system Eqs. (2), (8), (4), (7), and (15)
on one time step is decomposed into a sequence of well-known problems: diffusion,
convection, and ODEs. In order to benefit from Cartesian meshes, well-known highorder numerical schemes are used to discretize the equations in space (Jiang and Peng
2000; Eymard et al. 2000).
The fluxes are computed by means of centered discretizations. This provides
second-order schemes. We refer to Drblikova and Mikula (2007), Drblikova et al.
(2009) for more details on anisotropic diffusion. For diffusion equation, we use a
Crank–Nicolson discretization in time in order to ensure second-order accuracy. Due
to the diffusion term, the invasive cell density is quite smooth. The chemotaxis part
on invasive cells is solved with a WENO scheme (Jiang and Peng 2000).
The coupling between advection and tumor growth on each density is more difficult
to solve. Indeed, one has to be careful on the advection and mitosis to avoid a loss of
mass: As advection at velocity v is supposed to balance mass variation, they have to
be discretized exactly at he same time. Our ”splitting philosophy” does not give an
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O. Saut et al.
accurate outcome. The mass conservation could be improved using a RK2 scheme on
time to solve Eqs. (2)–(11) as described in Colin et al. (2013).
Here, we choose another strategy.
as a transport term
Let us rewrite
the advection
and a ”divergence” term: ∇ · uv
=
v
·
∇u
+
∇
·
v
u.
By
observing
that both
the transport part—”∂t u + v · ∇u = . . .”—and the other part (source terms and
divergence part - div(v)u) preserve the total density ”P + I + N + B = 1” (Eq. (10)),
we propose the following discretization:
n := m(Cn )Pn ,
1. compute M
Bn
n := −
2. compute the loss B
Bn 1,brain brain Mn d
n − B
n with a finite-volume scheme,
3. solve ∇ · K ∇
πn+1 = M
πn+1 ,
4. compute vn+1 = K ∇
5. compute
n − B
n Pn + M
n ,
n+1 = Pn + t FWeno (Pn ,
P
vn+1 ) − M
n − B
n u n
u n+1 = u n + t FWeno (u n ,
vn+1 ) − M
n − B
n Bn − B
n ,
vn+1 ) − M
Bn+1 = Bn + t FWeno (Bn ,
u = I, N ,
n+1 ,
6. compute Mn+1 := m(Cn ) P
Bn+1
Mn+1 d,
7. compute Bn+1 := − Bn+1 1,brain brain
8. solve ∇ · K ∇πn+1 = Mn+1 − Bn+1
9. compute vn+1 = K ∇πn+1
10. compute
Pn+1
n+1
t
Pn + P
n+1
n+1 , vn+1 ) − Mn+1 − Bn+1 P
+
FWeno ( P
=
2
2
+ Mn+1 ,
t
In+1
In + +
FWeno (
In+1 , vn+1 ) − Mn+1 − Bn+1 In+1 ,
2
2
t
Nn + Nn+1
n+1 ,
n+1 , vn+1 ) − Mn+1 − Bn+1 N
+
FWeno ( N
=
2
2
Bn + Bn+1 t
+
FWeno ( Bn+1 , vn+1 )− Mn+1 −Bn+1 Bn+1 −Bn+1 ,
=
2
2
In+1 =
Nn+1
Bn+1
where FWeno (u, v) denotes the numerical computation of v∇u with an WENO5
Scheme.
This discretization provides a second-order accuracy in time (Heun’s scheme as in
Colin et al. 2013). There does not remain any splitting between advection and mass
variation due to tumor growth and loss of B. The other terms—such as transition
between proliferative and invasive states, necrosis, oxygen concentration—are still
dealt with by a splitting scheme.
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Effects of Invasive Cells and Anti-Angiogenesis on Growth
2331
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