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"Tumor growth has long been a target of investigation within the context of mathematical and computer modeling. The objective of this study is to propose and analyze a twodimensional stochastic cellular automata model to describe avascular solid

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U N C O R R E C T E DP R O O F
PHYSA: 11782 Model 3G pp. 1–12 (col. fig:
nil
)
ARTICLE IN PRESS
Physica A xx (xxxx) xxx–xxx
Contents lists available atScienceDirect
Physica A
journal homepage:www.elsevier.com/locate/physa
A cellular automata model for avascular solid tumor growth under theeffect of therapy
E.A. Reis, L.B.L. Santos, S.T.R. Pinho
∗
Instituto de Física, Universidade Federal da Bahia, 40210-340, Salvador, Brazil
a r t i c l e i n f o
Article history:
Received 25 September 2008Received in revised form 23 November2008Available online xxxx
PACS:
87.17.Aa05.10.-a87.17.Ee
Keywords:
Tumor growthCellular automataParameter spaceNecrosisTherapy
a b s t r a c t
Tumor growth has long been a target of investigation within the context of mathematicaland computer
∧
modeling. The objective of this study is to propose and analyze a two-dimensional stochastic cellular automata model to describe avascular solid tumor growth,taking into account both the competition between cancer cells and normal cells fornutrients and/or space and a time-dependent proliferation of cancer cells. Gompertziangrowth, characteristic of some tumors, is described and some of the features of the time-spatial pattern of solid tumors, such as compact morphology with irregular borders, arecaptured. The parameter space is studied in order to analyze the occurrence of necrosisand the response to therapy. Our findings suggest that transitions exist between necroticand non-necrotic phases (no-therapy cases), and between the states of cure and non-cure(therapy cases). To analyze cure, the control and order parameters are, respectively, thehighest probability of cancer cell proliferation and the probability of the therapeutic effecton cancer cells. With respect to patterns, it is possible to observe the inner necrotic coreand the effect of the therapy destroying the tumor from its outer borders inwards.
©
2008 Published by Elsevier B.V.
1. Introduction
1
Neoplastic diseases are the cause of 7 million deaths annually or 12% of deaths worldwide [1]. Mathematical and
2
computer
∧
modelingmay lead toagreater understanding of the dynamics of cancer progression in the patient [2,3]. These
3
techniques may also be useful in selecting better therapeutic strategies by subjecting available options to computer testing
4
(
in silico
). Continuous models have been proposed to describe the stages of tumor growth since the middle of the 20th
5
century[4]. Initially, a tumor grows exponentially (linear rate). After this transient stage, the growth rate decreases and a
6
steadystateisattained,duetoseveralfactorsincludingalackofnutrientsandhypoxia.Thisnonlinearbehaviorcharacterizes
7
avascular tumor growth when
∧
neovascularisationhas not yet been triggered. The decelerating avascular growth may be
8
guidedbydifferentrulessuchas,forexample,Gompertzianandlogisticfunctions.Gompertziangrowthhasbeenoneofthe
9
most studied decelerating tumor growth over the past 60 years[5–7]. It is found, for example, in some solid tumors such as
10
breast carcinomas [8,9]. It is also observed in tumors
in vitro
[10,11].
11
Although continuous models are capable of describing the behavior of tumor growth, it would appear more reasonable
12
to adopt a discrete approach when describing the prevascular stage. Due to the fact that the angiogenic process has not
13
triggered early tumor growth, few cancer cells are present and growth depends predominantly on the interactions of these
14
cellswithadjacentcellsandwiththeenvironment[12].Inaddition,inthediscreteapproach,itiseasiertocapturethetime-
15
spatialpatterngeneratedbythemodelinordertocompareitwithactualpatterns[13].Somecellularautomatamodels[14–
16
16] and hybrid cellular automata [12,17,18]have been proposed to study tumor growth.
17
∗
Corresponding author. Tel.: +55 71 32836658; fax: +55 71 32357730.
E-mail addresses:
eareisphysics@gmail.com(E.A. Reis),santosl@ufba.br(L.B.L. Santos),suani@ufba.br(S.T.R. Pinho).
0378-4371/$ – see front matter
©
2008 Published by Elsevier B.V.doi:10.1016/j.physa.2008.11.038Please cite this article in press as: E.A. Reis, et al., A cellular automata model for avascular solid tumor growth under the effect of therapy, Physica A(2008), doi:10.1016/j.physa.2008.11.038
U N C O R R E C T E DP R O O F
PHYSA: 11782
ARTICLE IN PRESS
2
E.A. Reis et al. / Physica A xx (xxxx) xxx–xxx
Anotherimportanttopicthatisanalyzedinmathematicalmodelsistheresponsetotherapy,includinghowtumorgrowth
1
changesundertheeffectofadrug[19].Thefocusisdirectedtowardsidentifyingtheoptimaltherapytomaximizetheeffect
2
on cancer cells and minimize the effect on normal cells [20–22]. There are various continuous chemotherapy models[23–
3
26], with this aim. To the best of our knowledge, the majority of the models cited in the literature, that
∧
analyzeboth the
4
Gompertzian growth and the response to therapy, are continuous models [27,28].
5
In this work, Gompertzian growth, necrosis and the response and therapy effect are
∧
analyzedusing a discrete model for
6
avascular tumor. It is a two-dimensional cellular automata model consisting of 4 states (empty site, normal cell, cancer cell
7
and necrotic tumor cell) to describe avascular tumor growth. It should be emphasized that in this study the term tumor
8
growth is used to refer to the number of cancer cells rather than the volume of the tumor; in other words, it is assumed
9
that the tumor volume is proportional to the number of cancer cells[4]. Assuming that the angiogenic process has not yet
10
beentriggered,thereisnoincreaseinnutrients,whichareuniformlydistributedoverthelattice.Inthissimplemodel,some
11
relevant processes involved in the prevascular phase of tumor growth are assessed: a dynamic proliferation of cancer cells
12
and the competition between normal cells and cancer cells for nutrients and/or space. Since necrosis is often present in the
13
prevascular stage of tumor growth[10,29], the possibility of necrosis in the model must also be taken into consideration.
14
Finally, the effect of therapy is included in order to investigate whether the system evolves to a state of cure.
15
This paper is organized as follows. In Section2,the model is presented, together with its local rules, parameters and the
16
scopeofthealgorithm.Section3describesthesimulatedtimeseriesofcelldensityinthepresenceorabsenceoftreatment,
17
and shows the features of the time-spatial pattern of simulated solid tumors. In Section4,the parameter space related to
18
the process of necrosis and the effect of therapy is analyzed. Finally, in Section5,our results are discussed from the point of
19
view of the phenomenon and some concluding remarks are made.
20
2. The model
21
We propose a stochastic two-dimensional (
L
×
L
) cellular automata model [30] under periodic boundary conditions,
22
usingaMooreneighborhoodwitharadiusof1.Attheinitialcondition(
t
=
0),thereisonlyonecancercell(toensurebetter
23
visualization, this was taken from the center of the lattice). Since the intention is to model a non-viral tumor, normal cells
24
would not be transformed into cancer cells with the exception of the cancer cell that triggers tumor growth at
t
=
0[31].
25
The lattice represents a tissue sample; there is a cell in each site that may be in one of four states: normal cell (NoC), cancer
26
cell (CC), necrotic cell (NeC) or empty site (ES). We assume that the nutrients are uniformly available over the lattice. In this
27
respect, lack of space is identified with lack of nutrients in our model.
28
A local measure, that we call the growth potential, is associated with each normal cell (
P
noc
) or cancer cell (
P
cc
). The
29
growth potential of a cancer cell,
P
cc
, corresponds to the total number of cancer cells, i.e., in addition to the cancer cell on
30
the site of the lattice, the cancer cells generated by these cells.
31
The mitotic probability of cancer cells, that we assume as a global measure, is responsible for the proliferation of cancer
32
cells. Its initial value is given by a parameter
p
0
that measures the available resources at the beginning of the tumor. After
33
that, it decreases by a factor
p
mitot
until reaching the null value:
34
p
mitot
=
exp
−
n
noc
(
t
)
n
cc
(
t
)
2
(1)
35
in which
n
noc
(
t
)
and
n
cc
(
t
)
are the number of normal and cancer cells at time
t
, respectively. As shown,
p
mitot
(
t
)
depends
36
only on the dynamics, setting up a feedback inhibition mechanism [13]: as the tumor grows,
p
mitot
(
t
)
decreases because
37
ofthe combinedeffect ofthe decreaseinthe numberof normalcells andthe increaseinthe numberof cancercells. Inorder
38
to intensify the effect of this mechanism [32], an exponent 2 in Eq.(1)is considered. Since there is no new available source
39
of nutrients and/or space, it decreases as the density of cancer cells increases because the available nutrients and/or space
40
arereduced.Theeffectoftheproliferationofcancercellsisthattheirgrowthpotentialincreasesbyaunitateachtimestep.
41
The local rules are such that:
42
(i) Thecancercellscompetewithnormalcellsfortheemptysites,dependingonthepotentialgrowthofneighboringcells.
43
According to the majority rule, a normal cell is displaced by a cancer cell following local battles occurring between
44
healthy and cancerous cells [14].
45
(ii) Ifthegrowthpotentialofacancercellreachesathresholdvaluethatisafraction
f
ofthelatticesize
L
anditsneighbors
46
are also cancer cells, it becomes necrotic and its growth potential falls to zero.
47
(iii) Dependingontheirgrowthpotentialsandtheirneighborhood,bothnormalandcancercellsmaydie,withprobabilities
48
p
drugn
and
p
drugc
, respectively, due to the continuous infusion of a drug that is applied after
t
ap
time steps; in this case,
49
the site becomes empty.
50
(iv) If there are no cancer cells in the neighborhood of a dead cell (empty site), regeneration of normal cells occurs; if the
51
cancer cells in the neighborhood of a necrotic cell die as a result of the therapy, the necrotic cell is eliminated.
52
The algorithm was computationally implemented in FORTRAN 77 in accordance with the following steps: input data;
53
calculate
p
mitot
(
t
)
;identifythestateofthecell chooseoneofthesubroutines:normalcell(NoC),cancercell(CC),necrotic
Q1
54
cell (NeC), empty site (ES); update the cells of the lattice; after N iterations, output data. A framework of the algorithm is
55
represented inFig. 1.
56
Please cite this article in press as: E.A. Reis, et al., A cellular automata model for avascular solid tumor growth under the effect of therapy, Physica A(2008), doi:10.1016/j.physa.2008.11.038
U N C O R R E C T E DP R O O F
PHYSA: 11782
ARTICLE IN PRESS
E.A. Reis et al. / Physica A xx (xxxx) xxx–xxx
3
Fig. 1.
A block diagram of the mode with the general framework, and the change of states (i-xii) for the subroutines.
The input data are the following cellular automata (CA) parameters:
1
(1) The spatial parameters: lattice size
L
; necrosis threshold fraction
f
of lattice size;
2
(2) The temporal parameters: the length of the time series
t
final
and the initial time of therapy infusion
t
ap
;
3
(3) The probabilities of: the initial proliferation of cancer cells
p
0
; the effect of therapy on normal cells and cancer cells
4
(
p
drugn
and
p
drugc
).
5
The output data are the time series of the density of each type of cell and the final configuration of the lattice at any
6
time step. In addition, the time-spatial configurations, controlled by a package denominated g2 [33] whose commands are
7
inserted into the computer program in FORTRAN, are generated in ‘real time’. This package may be used in C, PYTHON and
8
PERL too.
9
The following is a description of each subroutine based on the local rules. The change of states (i)–(xii) associated
∧
with
10
the subroutines described below, are also indicated inFig. 1.
11
(a)
Normal Cell (NoC)
- a random number
y
is compared
∧
with
p
drugn
. If
y
<
p
drugn
, the growth potential
P
noc
is checked: if
12
P
noc
=
0, the cell dies and the site becomes empty (i); otherwise (
P
noc
=
1), it remains occupied by a normal cell but
13
P
noc
=
0. If
y
≥
p
drugn
,
t
>
t
ap
and if there is at least one neighboring cancer cell, then the normal cell is ’dislocated’,
14
P
noc
=
0 and the site becomes empty (ii); otherwise it remains a normal cell (iii).
15
(b)
Cancer Cell (CC)
- if all of its neighbors are cancer cells and its growth potential reaches a fraction
f
of lattice size
L
, the
16
cancer cell becomes necrotic (iv). Otherwise, a number
y
is randomly chosen. If
y
<
p
drugc
and
t
>
t
ap
, the potential
P
cc
17
is checked: if
P
cc
>
0, it is reduced by a unit and the cell remains a cancer cell (v); otherwise the cell dies and the site
18
becomes empty (vi). Finally, if
y
≥
p
drugc
, the cell remains a cancer cell; however, its growth potential
P
cc
increases by a
19
unit.
20
(c)
NecroticCell(NeC)
-ifatleastoneofitsneighborsisneitheracancercellnoranecroticcell,itiseliminatedandthesite
21
becomes empty (vii); otherwise, it continues necrotic (viii).
22
(d)
Empty Site (ES)
- if there are cancer and normal cells in its neighborhood, the local battle between cancer cells and
23
normal cells is such that if the sum of the potential growth of its neighboring cancer cells is greater or equal to
24
the sum of the potential growth of its normal cell neighbors, then the empty site is occupied by a cancer cell that
25
diffuses from one of the randomly chosen neighbors (ix); otherwise, it is occupied by one of the randomly chosen
26
normal cells that were previously dislocated (x). If there are only normal cells in its neighborhood, it becomes a normal
27
cell through a process of regeneration (xi). Finally, if none of its neighbors
∧
iscancer cells or normal cells, it remains
28
empty (xii).
29
3. Results: Simulated time series and time-spatial patterns
30
Computational simulations of the model were performed in order to analyze two classes of behavior: cases in which no
31
treatment was given and treatment cases. In each subsection, the time series of the simulated tumor as well as time-spatial
32
patterns are shown.
33
Please cite this article in press as: E.A. Reis, et al., A cellular automata model for avascular solid tumor growth under the effect of therapy, Physica A(2008), doi:10.1016/j.physa.2008.11.038
U N C O R R E C T E DP R O O F
PHYSA: 11782
ARTICLE IN PRESS
4
E.A. Reis et al. / Physica A xx (xxxx) xxx–xxx
Fig. 2.
Nonnecrotic tumor: The average of simulated time series (black color) with error bars (grey color) of the density of: (a) normal cells, (b) cancercells. We consider
M
samples
=
100 and the following parameter values:
L
=
225,
p
0
=
0
.
9,
f
=
0
.
4,
p
drugn
=
p
drugc
=
t
ap
=
0
.
0.
Fig.3.
Non-necroticcase: assuming
M
samples
=
100andthe sameparametersvaluesof Fig.2,the averageofsimulatedtime series(blackcolor)with error
bars (grey color) of: (a) the number of cancer cells; (b) the sum of growth potential of cancer cells. The Gompertzian fitting (triangle symbol) is applied to(a) those in which parameters of time series are
α
0
=
4
.
050
±
0
.
043,
β
=
(
1
.
818
±
0
.
011
)
×
10
−
2
and
n
0
=
(
1
.
336
×
0
.
058
)
×
10
2
.
3.1. No treatment case
1
In the case of no treatment, the following parameters are considered:
p
drugn
=
p
drugc
=
t
ap
=
0. The average of the
2
different samples is considered, corresponding to different seeds of random numbers. They do not differ very much
∧
from
3
eachother(theerrorbarsareverysmall).Forfixedvaluesof
L
,
t
final
and
p
0
,butdifferentvaluesof
f
,inFigs.2and4,thetime
4
series of the density of cells for nonnecrotic and necrotic tumors, respectively, are shown. In both cases, the cell densities
5
reach saturated values due to the effects of the competition between normal cells and cancer cells, and the time-dependent
6
mitotic probability. ComparingFigs. 2(b) and4(b), the stationary value of cancer cell density is clearly greater in necrotic
7
tumorsthaninnonnecroticones.Thisisaconsequenceofthefactthat
p
mitot
(
t
)
assumessmallervaluesinnecrotictumors
8
compared to nonnecrotic ones because it does not depend on the density of necrotic cells.
9
Figs. 3(b) and5(b) show that the sum of potential growth of cancer cells, both in the necrotic and non-necrotic cases,
10
reproduces the decelerating process of tumor growth.
11
Figs. 3(a) and5(a) show that the tumor growth obeys the Gompertzian function both in nonnecrotic and necrotic cases.
12
This behavior is observed with respect to the number of cancer cells for a range of values of
p
0
. In Gompertzian growth [4],
13
the specific growth rate of the number of cancer cells decreases logarithmically:
14
1
n
cc
d
n
cc
d
t
=
α
0
−
β
log
n
cc
n
0
(2)
15
where
16
(a)
n
0
is the initial population of cancer cells;
17
(b)
α
0
is the specific growth rate of
n
0
cells at
t
=
0;
18
Please cite this article in press as: E.A. Reis, et al., A cellular automata model for avascular solid tumor growth under the effect of therapy, Physica A(2008), doi:10.1016/j.physa.2008.11.038
U N C O R R E C T E DP R O O F
PHYSA: 11782
ARTICLE IN PRESS
E.A. Reis et al. / Physica A xx (xxxx) xxx–xxx
5
Fig. 4.
Necrotic case: The average of simulated time series (black-color) with error bars (grey color) of density of (a) normal cells, (b) cancer cells. Weconsider
M
samples
=
100 and the following parameter values:
L
=
225,
p
0
=
0
.
9,
f
=
0
.
25,
p
drugn
=
p
drugc
=
t
ap
=
0
.
0.
Fig. 5.
Assuming
M
samples
=
100 and the same parameters values of Fig. 4,the average of simulated time series (black color) with error bars (grey color)
of:(a)thenumberofcancercells;(b)thesumofgrowthpotentialofcancercells.TheGompertzianfitting(trianglesymbol)isappliedon(a)thoseinwhichparameters of time series are
α
0
=
3
.
827
±
0
.
039,
β
=
(
1
.
601
±
0
.
010
)
×
10
−
2
and
n
0
=
(
1
.
870
±
0
.
074
)
×
10
2
.
(c)
β
measureshowrapidlythecurvedepartsfromasingularexponentialandcurvesover,assumingitscharacteristicshape.
1
The solution of (2)is
2
n
cc
(
t
)
=
n
0
exp
α
0
β
[
1
−
exp
(
−
β
t
)
]
.
(3)
3
Itisevidentthatthestationaryvalueof
n
cc
is
n
cc
∞
=
n
0
exp
(α
0
/β)
.AccordingtotheGompertzianfittingrepresentedby
4
Eq.(3),the results of the simulations shown inFigs. 3(a) and5(a) correspond respectively to the following parameters:
5
(I) The nonnecrotic tumor:
α
0
=
4
.
050
±
0
.
043,
β
=
(
1
.
818
±
0
.
011
)
×
10
−
2
and
n
0
=
(
1
.
336
×
0
.
058
)
×
10
2
6
(II) The necrotic tumor:
α
0
=
3
.
827
±
0
.
039,
β
=
(
1
.
601
±
0
.
010
)
×
10
−
2
and
n
0
=
(
1
.
870
±
0
.
074
)
×
10
2
.
7
Comparing the above parameters of (I) and (II), the behavior of nonnecrotic and necrotic tumors was found to be very
8
similar. It was also found that the number of cancer cells obeys the Gompertzian fitting.
9
An important validation of our model is shown by comparing the Gompertzian fitting parameters of simulated tumors
10
withthecorrespondingparametersofactualtumors[6,7].Forinstance,inthecaseofthetesticulartumorsshowninTable3
11
of Ref. [6], the
β
values are in the range of
[
0
.
005
;
0
.
016
]
day
−
1
. If we consider the time step of our simulations to be one
12
day, our simulated
β
value is within the above range for necrotic simulated tumor, and very close to the maximum actual
13
value of
β
for non-necrotic simulated tumor.
14
In relation to the parameters
α
0
and
n
0
, the simulated values are not comparable
∧
withthe actual values, since no
15
information on the initial size of the tumor was included in our model. Both
α
0
and
n
0
are strongly dependent on that
16
information.
17
Please cite this article in press as: E.A. Reis, et al., A cellular automata model for avascular solid tumor growth under the effect of therapy, Physica A(2008), doi:10.1016/j.physa.2008.11.038

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