Journal of Theoretical Biology 368 (2015) 113–121

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Journal of Theoretical Biology

journal homepage: www.elsevier.com/locate/yjtbi

Mathematical model of brain tumour with glia–neuron interactions

and chemotherapy treatment
Kelly C. Iarosz a,n, Fernando S. Borges b, Antonio M. Batista a,b,c, Murilo S. Baptista a,
Regiane A.N. Siqueira b, Ricardo L. Viana d, Sergio R. Lopes d
a
Institute for Complex Systems and Mathematical Biology, University of Aberdeen, AB24 3UE Aberdeen, UK
b
Pós-Graduação em Ciências/Física, Universidade Estadual de Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
c
Departamento de Matemática e Estatística, Universidade Estadual de Ponta Grossa, 84030-900 Ponta Grossa, PR, Brazil
d
Departamento de Física, Universidade Federal do Paraná, 81531-990 Curitiba, PR, Brazil

H I G H L I G H T S

We have analysed a mathematical model of brain tumour.

We study a system of coupled differential equations.

We consider in our model the interactions among glial cells, glioma, neurons, and chemotherapy.

Glioma is studied aiming to identify values of the parameter for which the inhibition of the glioma cells is obtained with a minimal loss of healthy cells.

We present a model of glioma with glia–neuron interactions.

art ic l e i nf o a b s t r a c t

Article history: In recent years, it became clear that a better understanding of the interactions among the main elements
Received 2 June 2014 involved in the cancer network is necessary for the treatment of cancer and the suppression of cancer
Received in revised form growth. In this work we propose a system of coupled differential equations that model brain tumour
9 December 2014
under treatment by chemotherapy, which considers interactions among the glial cells, the glioma, the
Accepted 7 January 2015
Available online 14 January 2015
neurons, and the chemotherapeutic agents. We study the conditions for the glioma growth to be
eliminated, and identify values of the parameters for which the inhibition of the glioma growth is
Keywords: obtained with a minimal loss of healthy cells.
Brain & 2015 Elsevier Ltd. All rights reserved.
Glioma
Chemotherapy

1. Introduction The most common malignant intrinsic primary tumours of the

adult human brain are the gliomas (Inaba et al., 2011). Gliomas are
Cells growth is a phenomenon that has been studied in the tumours of the neoplastic glial cells. They are classified by the
fields of mathematics, biology, and physics (Adam and Bellomo, World Health Organisation as oligodendroglioma, astrocytoma,
1996; Wolpert et al., 2002; Hirt et al., 2014). Unregulated cells mixed oligoastrocytoma, and ependymoma (Louis et al., 2007;
growth may be associated with a wide group of diseases, where Goodenberger and Jenkins, 2012). Glioma causes regional effects
cells become a lump or cause illness. As a result, several growth by invasion, compression, and destruction of brain parenchyma,
models related to tumours have appeared in the literature arterial and venous hypoxia (Cuddapah et al., 2014). There is
(Menchón and Condat, 2008; Aroesty et al., 1973), such as models release and recruitment of cellular mediators which disrupt
for the metastasis (Pinho et al., 2002), the lack of nutrients normal parenchymal function (Ye and Sontheimer, 1999). Glioma
(Scaleranpdi et al., 1999), the competition for resources, and the cells migrate along blood vessels, displacing the junction among
cytotoxic activity produced by the immune response (Cattani and glial cells and blood vessels. This way, the glioma cells can extract
Ciancio, 2008; Wheldon, 1988). nutrients from the bloodstream. Displacements produce disrup-
tions of glial functions, compromising adequate delivery of glucose
and oxygen to neurons (Cuddapah et al., 2014). Moreover, disrup-
n
Corresponding author. tions of glial cells affect the neurons, because they are responsible
E-mail address: kiarosz@gmail.com (K.C. Iarosz). for delivering nutrients, to provide structural support to them

http://dx.doi.org/10.1016/j.jtbi.2015.01.006
0022-5193/& 2015 Elsevier Ltd. All rights reserved.
114 K.C. Iarosz et al. / Journal of Theoretical Biology 368 (2015) 113–121

(Glees, 1955), and to control the biochemical compositions of the Logistic Growth
fluid surrounding the neurons. The neurons are mainly responsible
for the information processing from external and internal envir-
onments (Otis and Sofronie, 2008; Fieldes, 2006; Shaham, 2005). Influence Competition
Glial Cells
However, glial cells are also responsible for the processing of
information by mediating the neural signal. Neurons and their
synapses fail to function without glial cells.
Mathematical modelling of glioma is an extensively explored
area with a large variety of mathematical models exploring multiple
complexities. An approach to modelling glioma is to use differential Neurons Glioma Cells
equations for the total of cells. In this case, the model ignores the
spatial aspects. Kronik et al. (2008) proposed a mathematical model
using differential equations for glioma and the immune system
Logistic Growth
interactions. They incorporated studies about improved immu- Chemotherapy Agent
notherapy schedules and interventions which can lead to a cure
of glioma. There are models that consider the spatio-temporal Fig. 1. Schematic representation of the agents (in grey coloured boxes), and their
evolution, such as partial differential equations (Harpold et al., interactions (links) considered in our model.
2007) and cellular automaton (Alarcón et al., 2003) since the
evolution of glioma critically depends on spatial geometry. according to schematic representation which is shown in Fig. 1.
In this paper, we propose a mathematical model using differential Our mathematical model describes the cells concentration, and the
equations for the growth of glioma, where the glioma cells attack the concentration of chemotherapeutic agent. Due to mixed effects we
glial cells (Bulstrode et al., 2012). Glioma rises from glial cells (Weille, leave out spatial considerations, and our model is a new approach
2014), and glioma cells never return to be glial cells, resulting in to modelling the dynamic evolution of the cells concentration in a
invasion and destruction of surrounding healthy tissue (Alberts et al., brain tumour with glia–neuron interactions.
1994; Hahn and Weinberg, 2002). In our model, we consider Our model is described by
interactions among glial cells, neurons, glioma cells, and the che-

dGðtÞ GðtÞ P 1 GðtÞQ ðtÞ
motherapeutic agent. The novelty of our model was to introduce the ¼ Ω1 GðtÞ 1   Ψ 1 GðtÞCðtÞ  ; ð1Þ
dt K A1 þ GðtÞ
interaction between glial cells and neurons. This interaction is
biologically relevant since glial cells make crucial contributions to

dCðtÞ CðtÞ P 2 CðtÞQ ðtÞ
the formation, operation and adaptation of neural cells. Glial cells are ¼ Ω2 CðtÞ 1   Ψ 2 GðtÞCðtÞ  ; ð2Þ
dt K A2 þ CðtÞ
essential for neuronal survival, once their removal causes neuronal
death (Allen and Barres, 2009). With this in mind, the main features dNðtÞ P 3 NðtÞQ ðtÞ
_
¼ ψ GðtÞHð _
 GÞNðtÞ  ; ð3Þ
of our model are: (i) treatment will likely preserve glial cells, (ii) dt A3 þ NðtÞ
glioma can be eliminated, but not without also destroying neurons. If
the treatment is ceased without the complete elimination of glioma dQ ðtÞ
cells, concentration of glioma cells increases, (iii) there is an optimal ¼ Φ  ζ Q ðtÞ; ð4Þ
dt
duration for the treatment that reduces significantly the number of
where G represents the glial cells concentration (in kg/m3), C rep-
glioma cells by preserving the levels of glial cells and minimising the
resents the glioma cells concentration (in kg/m3), N the neurons cells
impact on the neural populations.
concentration (in kg/m3), Q is the concentration of the chemother-
A major impediment to chemotherapy delivery for the glioma
apeutic agent (in mg/m2), and H(x) is the Heaviside function, defined
is the blood brain barrier (BBB). The BBB is a unique physiological
as
structure that regulates the movement of ions, molecules, cells
8
between the brain tissue and the blood (Gao and Li, 2014). It is < 0; x o 0;
>
necessary to deliver anti-glioma drugs across the intact BBB to HðxÞ ¼ 12; x ¼ 0; ð5Þ
obtain an efficient treatment of glioma (Srimanee et al., 2014). >
: 1; x 4 0:
There are chemotherapeutic agents that are capable of penetrating
the BBB (Friedman et al., 2000). Yang et al. (2014) showed blood- Table 1 shows the parameters that we consider. In Eqs. (1) and (2), the
brain barrier disruption through ultrasound for targeted drug first term is the logistic growth, the second term is the interaction
delivery. Moreover, phenotypic heterogeneity of glioma contri- between glial and glioma cells. This term is due to microglia cells, that
butes to failure of chemotherapy (Burrel et al., 2013). Gerlee and are a type of glia, which act creating an active immune defense. They
Nelander (2012) studied the impact of phenotypic switching on have the ability to generate innate and adaptive immune responses
glioma growth and invasion. (Yang et al., 2010). The glioma is attacked by microglia, and as a result
the glioma cells discharge immune suppressive factor to defend it by
paralysing the immune effector mechanism (Ghosh and Chaudhuri,
2. Brain tumour model 2010). The last term of Eqs. (1) and (2) is the effect of the
chemotherapeutic agent. We consider that the chemotherapy kills
Fig. 1 shows a diagram illustrating the many agents, and their the cells with different intensities according to the Holling type
interactions being considered in our model. The glioma cells only 2 killing functions. Holling (1965) suggested kinds of functional
attack the glial cells. Neurons are not attacked by glioma cells, and responses to model phenomena of predation. Holling found that the
they interact with glial cells. The chemotherapeutic agent behaves predator has a Holling 2 functional response by taking into account
as a predator acting on all cells (Schuette, 2004). the time a predator takes to handle the prey it has captured (Pei et al.,
There have been relevant studies that model the time and 2005). The first term of Eq. (3) is related with the decrease in the
space evolution of gliomas. However, as mixed effect modelling neural population due to glial cells death, and the second term is the
techniques cannot be yet applied to spatiotemporal equations interaction with the chemotherapeutic agent. Eq. (4) describes the
(Ribba et al., 2012), we have considered differential equations dynamics of the chemotherapeutic agent, presenting an exponential
aiming to yield a simplified description of the biological process decay in concentration. The agent rate ζ in this equation is associated
 K.C. Iarosz et al. / Journal of Theoretical Biology 368 (2015) 113–121 115

Table 1
Description of the parameters according to the literature.

Description Parameter Values Comment

Proliferation rate Ω1 for GCs 0.0068 day  1 Ω1 o Ω2 (Pinho et al., 2013)

Ω2 for CCs 0.012 day  1 Spratt and Spratt (1964)

Loss influences ψ for N due GCs 0–0.02 Pinho et al. (2013)

Interaction coefficients P1 for GCs 2.4  10  5 m2(mg day)  1 Pinho et al. (2013)
P2 for CCs 2.4  10  2 m2(mg day)  1 P 2 4P 1 (Rzeski et al., 2004)
P3 for N 2.4  10  5 m2(mg day)  1 P3 ¼ P1

Chemotherapy agent rate Φ for infusion 0–150 mg(m2 day)  1 Daily doses (Stupp et al., 2005)
ζ 0.2 day  1 Borges et al. (2014) and Said et al. (2007)

Holling type 2 A1 ; A2 ; A3 510 A1 ¼ A2 ¼ A3 ¼ K i

Competition coefficients Ψ1 between GCs and CCs 3.6  10  5 day  1 Cancer hypothesis (Pinho et al., 2013)
Ψ2 between CCs and GCs 3.6  10  6 day  1 Ψ2 oΨ1

Carrying capacity K1; K2; K3 510 kg/m3 Azevedo et al. (2009)

with the decrease of chemotherapy that is delivered to the cells. As a with

result of the washout, and mainly by the blood brain barrier which β1 ¼ Ψ 1 K 2 ; β2 ¼ Ψ 2 K 1 ; α ¼ ψ K 1;
prevents entry of chemotherapeutic agents into the brain.
A1 A2 A3
Glial cells are required, both in vitro and in vivo, for the survival a1 ¼ ; a2 ¼ ; a3 ¼ ;
K1 K2 K3
of the neurons with which they interact. Removal of glial cells
P1 P2 P3
results in neuronal death, unless specific survival factors are added p1 ¼ ; p2 ¼ ; p3 ¼ ; ð8Þ
(Meyer-Franke et al., 1995). There are studies showing that neuron K1 K2 K3
death in the absence of glial is best fit by an exponential decay. where Table 2 exhibits the values of the normalised parameters.
Clarke et al. (2000) verified exponential decay in the kinetics of The normalised model provides variables that reflect the relative
neuronal death in 12 models of photo-receptor degeneration, density of a population of cells with respect to the mass density of
hippocampal neurons undergoing excitotoxic cell death, a mouse the brain approximately K ¼510 kg/m3. For example, n ¼0.3 would
model of cerebellar degeneration, and Parkinson's and Hunting- mean 30% of total number of neurons that an individual could
ton's diseases. Therefore, our main contribution in this work is to have. This normalised model allows to make comparative analysis
model the dependence of neurons on the glial cells, described by when the model is used to simulate glioma grown in different
_
the term ψ GðtÞHð _
 GÞNðtÞ, in Eq. (3). This term represents an individuals. A healthy individual, in this normalised model, would
exponentially decaying function of N(t), given by NðtÞ ¼ expðΛGðtÞÞ, be described by the state variables g ¼1, c ¼0, and n¼ 1.
where Λ ¼ ψ HðÞ. When the glial concentration decreases, G_ Firstly, we check the behaviour of the glioma without the
becomes negative, making this term to contribute negatively to infusion of a chemotherapeutic agent. Since there is no treatment
N(t), leading to a decrease in the neuron concentration. Whereas, the glioma cells kill the glial cells (Fig. 2a), while the glioma cells
the term is null if the rate of glial concentration, G, _ is null or grow (Fig. 2b). Without the glial support, the neurons die (Fig. 2c).
positive. Therefore, a decrease in the glial concentration causes For t¼500 there are around 34% of glial cells, and approximately
death of neurons, whereas an increase does not contribute to a 27% of neurons. Due to the logistic growth of the glial cells, they
change in the neural population. resist longer than the neurons from the attack of glioma. As a
Recent years have witnessed a surge in studies about glial– result, without chemotherapy (Fig. 2d) the glioma cells are going
neuronal interaction, providing insights into the role of glial cells to kill all the cells (Fig. 2a and c).
in neuronal function (Tiwari et al., 2014; Nakae et al., 2014). With Then, we use the infusion of a chemotherapeutic agent in order
this in mind, we consider in this work the glial–neuronal interac- to suppress the glioma growth. Fig. 3 shows a case where our
tion due to the fact that the glioma is originated from the glial cells model exhibits a suppressed state. In the interval 0 o t o 50 the
which are active participants in the nervous system. We believe glial cells concentration decreases (Fig. 3a) due to the attack of
that our model is a first step towards advanced modelling glioma and the presence of the chemotherapeutic agent, as well
techniques treating brain tumour taking into consideration glia– as, for t 4 50 we have c o 0:005 (Fig. 3b). Fig. 3c exhibits a fast
neuronal interactions, which in turn may have relevant impact decay of the neurons due to the effect of the chemotherapeutic
both on tumour research and on clinical practice. agent on the neuron population and the decay of glial cells. Glial
Introducing the normalised variables cells recover their normal concentration level which slows down
G C N the decay of neuron cells. If the chemotherapeutic agent is not
g¼ ; c¼ ; n¼ ; ð6Þ suspended it will eventually kill all neurons, since in our model we
K1 K2 K3
are not considering the neurogenesis. The chemotherapeutics
where Ki is the carrying capacity of the glial, glioma, and neural inducing neurons death is a side effect known as neurotoxicity.
cells, respectively, we obtain the normalised mathematical model Lomustine, cisplatin, topotecan, and vincristine are antitumour
dgðtÞ p gðtÞQ ðtÞ agents that induce cell death. Wick et al. (2004, 2009) analysed the
¼ Ω1 gðtÞð1  gðtÞÞ  β 1 gðtÞcðtÞ  1 ; effect of these drugs on the neurons. They verified which drugs
dt a1 þ gðtÞ
lead to cell death in cerebellar granule neurons in a concentration
dcðtÞ
¼ Ω2 cðtÞð1  cðtÞÞ  β 2 gðtÞcðtÞ dependent manner.
dt
p cðtÞQ ðtÞ dnðtÞ p nðtÞQ ðtÞ
 2 ; ¼ αgðtÞHð
_ _
 gÞnðtÞ  3 ; 3. Local stability
a2 þ cðtÞ dt a3 þ nðtÞ
dQ ðtÞ The study of the local stability is important to verify if the
¼ Φ  ζ Q ðtÞ; ð7Þ
dt suppression of glioma is stable or unstable, or to understand
116 K.C. Iarosz et al. / Journal of Theoretical Biology 368 (2015) 113–121

whether a non-desired state is stable. The model has some λð0Þ

4 ¼  ζ: ð12Þ
equilibria points Eðg; c; n; Q Þ. They are solution of the system
gðtÞ _ ¼ 0, Q_ ðtÞ ¼ 0. We consider the equilibria
_ ¼ 0, c_ ðtÞ ¼ 0, nðtÞ Through the sign of the real part of each eigenvalue we can check
points physiologically feasible. the stability of the equilibrium. In a hyperbolic equilibrium, if
Now, we analyse the local stability for an undesirable equili- the real part of each eigenvalue is strictly negative, then the
1
brium, where this equilibrium is E0 ð0; 0; 0; Φζ Þ. The eigenvalues equilibrium is locally asymptotically stable. If positive, then the
of the Jacobian matrix are equilibrium is unstable. In order to ensure the stability of
1
E0 ð0; 0; 0; Φζ Þ it is necessary that
p1 Φ
λð0Þ
1 ¼ Ω1  ; ð9Þ
ζ a1 Ω1 a1 ζ
Φ4 ; ð13Þ
p1
ð0Þ p Φ
λ ¼ Ω2  2 ; ð10Þ
2
ζ a2 and

p3 Φ
Ω2 a2 ζ
λð0Þ Φ4 ; ð14Þ
3 ¼  ; ð11Þ p2
ζ a3
ð0Þ ð0Þ
where these results are obtained through λ1 o 0, and λ2 o 0. The
values of the normalised parameters are positive, then the
ð0Þ ð0Þ
Table 2 eigenvalues λ3 and λ4 are negative. We consider a1 ¼ a2 ¼ 1,
Values of the normalised parameters. Ω1 ¼ 0:0068, Ω2 ¼ 0:012, p1 ¼ 4:7  10  8 , p2 ¼ 4:7  10  5 , and
ζ ¼ 0:2 (Table 1). With these values we obtain that E0 is linearly
Parameters Values
asymptotically stable for Φ 4 28 936:17. In other words, if Φ 4
β1 1.8  10  2 day  1
28 936:17 the chemotherapeutic agent kills all cells, they will
β2 1.8  10  3 day  1 never recover. Stability of the non-cells state is however granted
α 0.0–10.0 for a very large atypical value of the infusion rate Φ.
a1 ¼ a2 ¼ a3 1.0 We also consider the equilibrium E1 ðg; 0; n; Q Þ, representing the
p1 ¼ p3 4.7  10  8 m2(mg day)  1
complete elimination of glioma cells in the normalised model, but
p2 4.7  10  5 m2(mg day)  1
preserving glial and neuron cells. This equilibrium is obtained by

1.0 1.0

0.8 0.8

0.6 0.6
g

0.4 0.4

0.2 0.2

0 0
0 100 200 300 400 500 0 100 200 300 400 500
t t

1.0 1

0.8

0.6

0
Q
c

0.4

0.2

0 -1
0 100 200 300 400 500 0 100 200 300 400 500
t t
Fig. 2. Temporal evolution of the concentration of (a) glial cells, (b) glioma cells, (c) neurons and (d) chemotherapeutic agent (Φ¼ 0). We consider gð0Þ ¼ 0:99, cð0Þ ¼ 0:01,
nð0Þ ¼ 0:99, Q ð0Þ ¼ 0:0, and parameters according to Table 2.
 K.C. Iarosz et al. / Journal of Theoretical Biology 368 (2015) 113–121 117

1.00 1.00

0.99 0.99
g

n
0.98 0.98

0.97 0.97
0 100 200 300 400 500 0 100 200 300 400 500
t t

0.02 600

400

0.01
Q
c

200

0 0
0 100 200 300 400 500 0 100 200 300 400 500
t t
Fig. 3. Temporal evolution of the concentration of (a) glial cells, (b) glioma cells, (c) neurons and (d) chemotherapy, continuous treatment Φ ¼ 100. We consider gð0Þ ¼ 0:99,
cð0Þ ¼ 0:01, nð0Þ ¼ 0:99, Q ð0Þ ¼ 0:0 and parameters according to Table 2.

p2 Φ
the solution of λð1Þ
2 ¼ Ω2  β 2 g  ; ð19Þ
ζ a2
p1 gQ
Ω1 gð1  gÞ  ¼ 0; p3 Φ
a1 þ g λð1Þ
3 ¼  ; ð20Þ
ζ a3
p3 nQ
 ¼ 0;
a3 þ n
λð1Þ
4 ¼  ζ: ð21Þ
Φ  ζ Q ¼ 0; ð15Þ 1
In order to ensure the stability of E1 ðg; 0; 0; Φζ Þ it is necessary
1
where we obtain n ¼ 0 and Q ¼ Φζ . Thus, the equilibrium that
1
E1 ðg; 0; n; Q Þ is given by E1 ðg; 0; 0; Φζ Þ, meaning that all neurons
1 Ω1 ð1  2gÞða1  gÞ2
are also eliminated. p1 Φζ 4 ; ð22Þ
The first equation of (15) can be rewritten as a1

p1 Φ and
g 2 þ ða1  1Þg  a1 þ ¼ 0; ð16Þ
Ω1 ζ p2 Φζ
1
4 a2 ðΩ2  β2 gÞ; ð23Þ
with solution ð1Þ ð1Þ
n o where these results are obtained through λ and λ1 o0 The 2 o 0.
g ¼ 12 1  a1 7 ½ða1  1Þ2 þ4ða1  p1 Φ=Ω1 ζ Þ1=2 : ð17Þ values of the dimensionless parameters are positives, then the
ð1Þ ð1Þ
eigenvalues λ3 , and λ4 are negatives.
In this way, we verify that g has a null solution when p1 Φ= For a1 ¼1.0 (Table 2) Eq. (22) is satisfied for all g Z0:5.
Ω1 ζ ¼ a1 , and a real, positive and not null solution when Considering a2 ¼ 1.0, Ω2 ¼ 0:012, p2 ¼ 4:7  10  5 , ζ ¼0.2, and
p1 Φ=Ω1 ζ oa1 . Using the parameters of Table 2, g has a real, β2 ¼ 1:8  10  3 (Table 2) in Eq. (23) we have Φ 4 51:064
1
positive and non-null solution when Φ o 28 936:17.  7:660g. As a result, for E1 ðg; 0; 0; Φζ Þ the system presents an
Calculating a lower band for the value of Φ for which the asymptotically stable equilibrium for Φ 4 43:189. Therefore, for
1
equilibrium E1 ðg; 0; 0; Φζ Þ is stable, we determine the stability realistic values of the infusion rate 43:189 o Φ o 28979:255, we
of this equilibrium. The eigenvalues of the Jacobian matrix are should expect that glioma can be eliminated, i. e., c r 10  11 . Doing
similar analyses in the non-normalised system, we observe that
p1 a1 Φ the equilibrium Eðg; 0; n; Q Þ is also stable for 43:189 o Φ o
λ1ð1Þ ¼ Ω1 ð1  2g Þ  ; ð18Þ
ζ ða1 þ gÞ2 28 979:255.
118 K.C. Iarosz et al. / Journal of Theoretical Biology 368 (2015) 113–121

We construct the parameter space shown in Fig. 4 to obtain a parameters causes a large rate of death of the glias (large g). _ The
picture of the stability according to parameters related with the chemotherapy rate Φ also contributes to this low level of n.
chemotherapy. We can observe three regions. Region I represents Since that glial cells provide support functions for the neurons,
parameter in which the glioma cells kill the glial cells and we also analyse the concentration of the glial cells with the
neurons, region II represents parameters for which the equilibrium chemotherapy treatment. For the parameter values shown in
1
E1 ðg; 0; 0; Φζ Þ is locally stable, and region III represents para- Fig. 5, the percentage of glial cells remains larger than 95%.
1
meters for which the equilibrium E0 ð0; 0; 0; Φζ Þ is locally stable. Eqs. (7) show that the glial cells equation does not depend on
The region II shows that glioma can be eliminated without the the parameter α, but it depends on the parameter Φ due to Q.
elimination of the glial cells. However, the longer the duration of When Φ increases, we verify that c decreases. However, there is no
treatment, the larger the decrease in the neural population. It is significant variation in g due to the lifetime glioma τ according to
therefore vital to understand what are the optimal parameters for the chemotherapy agent rate. In other words, increasing the value
which c r10  11 is achieved in the shortest time. of Φ the lifetime of glioma quickly decreases, and in this time
A strongly desired equilibrium is E2 ðg; 0; n; 0Þ. In this case we interval the glial cells concentration does not have a significant
have g ¼ 1 and n has a constant value. The eigenvalues are alteration, due to the fact that the glial cells are able to recover to
λ1 ¼  Ω1 , λ2 ¼ Ω2  β2 , λ3 ¼ 0, and λ4 ¼  ζ . Using the parameters their initial state.
given in Table 2 we obtain negative values for λ1 and λ4, λ2 has a Fig. 6 shows the time τ to achieve suppression of glioma as a
σ
positive value, and λ3 presents a null value. This equilibrium is an function of Φ. There is a power-law relation of the type τ p Φ ,
unstable saddle point. Then, if treatment is ceased without the with σ ¼  12.36 for Φ r 60 mgðm2 dayÞ  1 and σ ¼  1.41 for
complete elimination of the glioma cells (c¼ 0), the glioma con- Φ Z 60 mgðm2 dayÞ  1 . This power-law shows that a significant
centration in our model increases. decrease in τ happens if Φ r 60 mgðm2 dayÞ  1 , whereas little
modification in τ happens if Φ 4 60 mgðm2 dayÞ  1 . Therefore,
the optimal way of reducing the time of treatment by using the
4. Glioma elimination minimal amount of Φ is obtained if Φ  60 mgðm2 dayÞ  1 . Look-
ing at Fig. 5, neurons will also be significantly preserved if α r 2.
Here, we study the performance of our model to understand The same scalings are obtained if another α if considered. The
what are the conditions such that glioma concentration in the chemotherapeutic agent which provides the quickest is Φ r
normalised model reaches levels related to no glioma (c r 10  11 ), 60 mgðm2 dayÞ  1 . Treatment will cause less impact on neural
while glial and neuron cell concentrations are kept high. population if the individual being treated (characterises by a
Fig. 3 shows a case for eliminated glioma. However, the particular α) is provided with an infusion rate Φ such that the
neurons concentration is decreased by 1.5% when the tumour point (Φ,α) falls in the yellow region in Fig. 5.
has significantly decreased. In this case, glioma is eliminated, but a Often chemotherapy treatments are delivered in cycles, where
significant population of neurons are damaged. For this reason we drugs are repeatedly applied for a short time. In the case of glioma
optimise the values of the chemotherapeutic agents in order not and temozolomide, after radiation therapy, the drug is delivered
only to minimise the impact on neurons but also to maximise the 5 days on, and 23 days off. There have been clinical experiences
effect of the drug on glioma cells. Our aim is to understand how with temozolomide considering pulsed chemotherapy in patients
the neuron population is when c r 10  11 . The therapeutic impli- with glioma (Friedman et al., 2000; Pace et al., 2003). The patients
cation for neurons is shown in Fig. 5, the neuron concentration received radiotherapy before the chemotherapy. It was verified
(colour bar) when c r 10  11 , as a function of the parameters α and that temozolomide chemotherapy was a valid option.
Φ. In this case, we consider a chemotherapy delivered continu- Fig. 7a shows the drug injection pattern for pulsed chemother-
ously. The region of α and Φ values responsible for the reduction apy. We consider 5 days on with Φ ¼ 400 mgðm2 dayÞ  1 , and
of approximately 2% in the neuron concentration (yellow online) 23 days off. Fig. 7b, c, and d exhibit the temporal evolution of the
is 0:96 r n r 0:98, while the region 0:80 r n r 0:84 (dark blue concentration of glial cells, glioma cells, and neurons, respectively.
online) presents approximately a reduction of 17%. This region of There is no relevant decrease in the concentration of glial cells

0.4
10 0.98

0.96
8 0.94
0.3
0.92
6
0.9
α

0.2 0.88
I II III
ζ

4
0.86

0.84
2
0.1
0.82

0 0.8
50 60 70 80 90 100 110 120
0
1 2 3 4 5 Φ
10 10 10 10 10
Φ Fig. 5. Neuron concentration as a function of α versus Φ, where gð0Þ ¼ 0:99,
cð0Þ ¼ 0:01, nð0Þ ¼ 0:99, and Q ð0Þ ¼ 0:0. The colour bar represents the value of the
Fig. 4. Parameter space ζ versus Φ: in the region I the glioma cells kill the glial cells neuron concentration, n, after a successful chemotherapy. (For interpretation of the
and neurons, the region II shows which the equilibrium E1 ðg ; 0; 0; Φζ  1 Þ is locally references to color in this figure caption, the reader is referred to the web version of
stable, and in the region III the equilibrium E0 ð0; 0; 0; Φζ  1 Þ is locally stable. this paper.)
 K.C. Iarosz et al. / Journal of Theoretical Biology 368 (2015) 113–121 119

(Fig. 7b), but the concentration of glioma cells is going to a We studied some aspects of the dynamics of glioma growth, as
suppressed state (Fig. 7c). Whereas, the concentration of neuron well as, we analysed its suppression and elimination varying para-
decreases around only slightly. meters of the system. The main target of treatment is the decrease in
the number of glioma cells. A successful chemotherapy eliminates all
the glioma cells minimising the neurons and glial cells injury.
5. Conclusions Through local stability we found a range of values for the infusion
rate (43:189 o Φ o28 979:255) that allows for the elimination of
We proposed a mathematical model for the evolution of a brain glioma, as well as, the glioma will not return. As a matter of fact the
tumour under the attack of chemotherapeutic agents. Our model temozolomide is a chemotherapeutic drug used for brain tumour,
describes the interactions among glial cells, neurons, and glioma, where the infusion rate is 75 o Φ o 200 ðmgðm2 dayÞ  1 ) (Wick et al.,
with a chemotherapy to suppress the brain tumour. The novelty in 2009). According to our model the rate would kill all the glioma cells,
this model is the glial effect on the neurons. and in addition it would preserve high levels of neural population.
The range of the values for the infusion rate is clinically relevant
10
5 because it reveals the effectiveness of the treatment strategies by
administration of chemotherapeutic drugs. Brock et al. (1998) used an
extended continuous oral schedule of temozolomide against gliomas.
They verified which patients with recurrent glioma were the main
4
group in that tumour responses were seen. Clinical studies conducted
10
by the Cancer Research Campaign (London, United Kingdom) demon-
strated which temozolomide has important efficacy and acceptable
τ

safety profile in the treatment of patients with glioma (Friedman

et al., 2000). New strategies have been developed aiming health
3
10 benefits of patients, that is the elimination of glioma cells (Minniti
et al., 2009). Regardless whether doctors will use the optimal rates
obtained from our model, our work can help doctors to access the
risks of a treatment on an individual basis. For example, Fig. 5 shows
10
2
1 2 3
that depending on the value of α (a parameter that depends on the
10 10 10
individual) the range of vales for the infusion rate can be larger as
Φ smaller for an optimal application.
Fig. 6. τ versus Φ, and the same values as Fig. 5. The slope for Φ r 60 mgðm2 dayÞ  1 We realised numerical simulations and obtained values of the
is about  12.36, and for Φ Z 60 mgðm2 dayÞ  1 the slope is about  1.41. infusion of chemotherapeutic agents in that the glioma growth is

600 0.012

400 0.008
Φ

200 0.004

0 0
0 100 200 300 400 500 0 100 200 300 400 500

1.000 0.990

0.996 0.985

0.992 0.980
g

0.988 0.975

0.984 0.970
0 100 200 300 400 500 0 100 200 300 400 500
t t
Fig. 7. Temporal evolution of the (a) chemotherapy infusion, concentration of (b) glial cells, (c) glioma cells, and (d) neurons.
120 K.C. Iarosz et al. / Journal of Theoretical Biology 368 (2015) 113–121

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Our main result was to show that chemotherapy can be applied Willaime-Morawek, S., 2012. A-disintegrin and metalloprotease (ADAM) 10
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