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arXiv:2211.16328v1 [q-bio.QM] 27 Nov 2022
2 Juan Uriel Legaria-Pe˜na et al.
Noname manuscript No.
(will be inserted by the editor)
Evaluation of Entropy and Fractal Dimension as
Biomarkers for Tumor Growth and Treatment Response
using Cellular Automata
Juan Uriel Legaria-Pe˜na ·F´elix
S´anchez-Morales ·Yuriria Cort´es-Poza(1)
Received: date / Accepted: date
Abstract Cell-based models provide a helpful approach for simulating complex
systems that exhibit adaptive, resilient qualities, such as cancer. Their focus on
individual cell interactions makes them a particularly appropriate strategy to study
the effects of cancer therapies, which often are designed to disrupt single-cell dy-
namics. In this work, we also propose them as viable methods for studying the time
evolution of cancer imaging biomarkers (IBM). We propose a cellular automata
model for tumor growth and three different therapies: chemotherapy, radiotherapy,
and immunotherapy, following well-established modeling procedures documented
in the literature. The model generates a sequence of tumor images, from which
time series of two biomarkers: entropy and fractal dimension, is obtained. Our
model shows that the fractal dimension increased faster at the onset of cancer
cell dissemination, while entropy was more responsive to changes induced in the
tumor by the different therapy modalities. These observations suggest that the
prognostic value of the proposed biomarkers could vary considerably with time.
Thus, it is important to assess their use at different stages of cancer and for dif-
ferent imaging modalities. Another observation derived from the results was that
both biomarkers varied slowly when the applied therapy attacked cancer cells in a
scattered fashion along the automatons’ area, leaving multiple independent clus-
ters of cells at the end of the treatment. Thus, patterns of change of simulated
biomarkers time series could reflect on essential qualities of the spatial action of a
given cancer intervention.
Keywords Avascular tumor modeling ·Complex Systems ·Cellular Automata ·
Imaging Biomarkers ·Shannon Entropy ·Fractal Dimension
(1) Y. Cortes-Poza
IIMAS, Unidad Acad´emica de Yucat´an, Universidad Nacional Aut´onoma de M´exico (UNAM),
Yuc., M´exico. E-mail: yuriria.cortes@iimas.unam.mx
J.U. Legaria-Pe˜na E-mail: walup@ciencias.unam.mx ·
F. S´anchez-Morales E-mail: felixsm@ciencias.unam.mx
Title Suppressed Due to Excessive Length 3
1 Introduction
Cancer is a disease that can develop in multi-cellular organisms, where uncon-
trolled cell proliferation occurs due to genetic mutations. It currently stands as
one of the main threats to human health, accounting for approximately 10 million
yearly deaths according to data provided by the world health organization [1].
As a result, substantial efforts have been made to develop treatments against the
disease, mainly focusing on disrupting the genetic and functional processes that
molecular biology has identified to be related to its development. However, these
treatment efforts have only partially succeeded since cancer tissue’s heterogenous
and adaptive qualities make it resilient and capable of producing various survival
strategies. This difficulty has produced a paradigm shift in recent years. Cancer
is now viewed as a complex adaptive system, and research efforts have been in-
creasingly focused on understanding how underlying interactions between cancer
cells produce the macroscopic structures and versatile dynamics that characterize
the pathology [2]. Multiple mathematical and computational techniques have been
proposed to study how the dynamics of cancer emerge from local microscopic con-
ditions and interactions. One particularly convenient approach is cell-based com-
putational modeling, where the properties of cells are stated as individual discrete
entities that can interact with one another [3]. Models of this kind have been ap-
plied to research multiple aspects of cancer, such as the emergence of calcifications
in breast cancer, vessel morphology in angiogenesis, and adaptive phenomena such
as the go or grow effect observed in Glioblastoma Multiforme [4–6].
A widely applied subclass of cell-based models is cellular automata (CA). CA
methods involve lattices, where each site can be occupied by a single cell. At each
simulation step, rules for how a cell will respond to its environment are applied,
and local actions such as cell division, motility, or death are carried through.
These models have found critical applications in studying cancer therapies’ effect
on tissue since most therapeutical strategies focus on disrupting individual cell
properties or capabilities [7].
Assessment of therapy and disease progression often requires the use of biomark-
ers. A biomarker can be defined as an indicator with a utility for characteriz-
ing normal physiological processes, diseases, or responses to treatments [8]. In
the management of cancer, the introduction of novel imaging techniques such as
positron emission tomography (PET), single-photon emission computed tomogra-
phy (SPECT), and magnetic resonance imaging (MRI) has allowed the proposal
of novel imaging biomarkers (IBM), most of them focused on tumor size charac-
teristics. The use of tumor size as a cancer indicator is mostly prioritized because
the gold standard for tumor response evaluation (RECIST) uses mass longest di-
ameters as a reference to classify therapy effects [9]. Some studies, however, have
highlighted the need to complement such criteria with other biomarkers, particu-
larly for types of tumors such as hepatocellular carcinoma, lung cancer, prostate
cancer, brain glioma, and lymphoma, where other considerations such as necrosis
or cell clustering might become relevant [10].
In this work, we propose a Cellular Automata model for tumor growth and
three cancer interventions: radiotherapy, chemotherapy, and immunotherapy. Syn-
thetic images generated with the model are used to evaluate the time series of
two prospective biomarkers: Shannon’s entropy and fractal dimension, as the tu-
mor develops and when each of the modeled treatments is administered. These
4 Juan Uriel Legaria-Pe˜na et al.
state measures have been shown to change as the complexity and configuration of
systems are altered.
Since exposing a patient to multiple imaging procedures is not clinically viable,
using computational models to study biomarkers and their evolution over time
becomes a handy tool. Our model produces high-resolution images of the evolution
of the tumor during the growing stage and treatment. The proposed biomarkers
of these images are analyzed.
The work presented here has the following structure: Section 2 provides a brief
review; Section 3 details the different models implemented; Section 4 presents our
results; Section 5 discusses the ideas obtained from these results; we conclude our
work in Section 6.
2 Background
The methods used in this study derive from several disciplines, such as mathe-
matics, biology, medicine, physics, and computer science. This section provides a
comprehensive theoretical base that could be useful to researchers from any of the
abovementioned fields.
2.1 Cancer as a pathology
Cancer is a disease where an abnormally high proliferation of cells takes place [11].
This high division rate is often accompanied by other adaptive, resilient traits and
hallmarks, such as immune evasion and a marked glycolytic metabolism [12].
The disease tends to progress in two identifiable stages. In the first one, an
avascular tumor mass grows until reaching a final stationary size. At this point,
the tumor cells struggle to gather nutrients by passive diffusion, and a process
of vessel recruitment known as angiogenesis starts to develop. This phenomenon
constitutes the start of the second cancer stage and has been linked to crucial
signaling chemicals such as vascular endothelial growth factors (VEGF) [13]. In
addition, cancer cells can travel through the bloodstream in their latest stages,
invading distant body parts. This extensive dissemination of the disease receives
the name of metastasis [14].
The model presented in this work will focus solely on avascular tumor growth.
As already mentioned, in this phase, the tumor develops into a stable mass, where
nutrients will be most readily available for those cells lying on the outer surface of
the conglomerate. Differences in nutrient absorption results in a particular arrange-
ment of cells: those at the core of the tumor die, leaving necrotic scar tissue; cells
at intermediate portions of the tumor stay in a quiescent dormant state that is less
metabolically demanding, and actively proliferating cells reside on the outer layers
of the tumor. This morphological organization of tumor cells has been schematized
in Figure 1.
2.2 Cancer therapies
Several known therapies are used to slow the progression of the disease and, in some
cases, to revert it. The ones modeled in this work are radiotherapy, chemother-
Title Suppressed Due to Excessive Length 5
Necrotic
Quiescent
Proliferating
Fig. 1 Layer organization of cells in a tumor. The image was generated with the tumor model
developed in this work.
apy, and immunotherapy. From a systems perspective, these interventions can be
thought of as control methods devised to alter the dynamics of cancer cells and
drive the system out of its diseased state.
Some cancer therapies are designed to act more efficiently at certain stages of
the cell cycle. The cell cycle is a periodic progression of events that a cell goes
through to achieve division. It comprehends four stages named G1, S, G2, and M.
G1 is a phase where the cell grows and increases its number of organelles, the S
phase is where the replication of DNA takes place, G2 constitutes a stage where
cells produce all the material needed for division, and finally, M phase or mitosis
is the process of nuclear division.
It is worth mentioning that selecting the treatment to apply follows established
medical guidelines that consider factors such as cancer stage, location, and other
relevant indicators. These protocols have the purpose of optimizing cancer out-
comes for most patients, and they are constantly revised and updated based on
new findings.
The mechanisms of radiotherapy, chemotherapy, and immunotherapy and the
biological effects they exert on cancer cells are the following:
–Radiotherapy: In radiotherapy, high doses of gamma radiation are applied to the
affected tissue, resulting in a cascade of effects damaging proliferating cells. Its
action proceeds in two stages. In the first one, which takes a short time, ionizing
biochemical material crucial for survival might kill some cells. Furthermore, in
the long term, a phenomenon known as radiolysis takes hold, where radicals
of water molecules generated by irradiation might produce compounds such
as oxygen peroxide, which are toxic to the cell, and may induce its death.
6 Juan Uriel Legaria-Pe˜na et al.
Radiation tends to act more effectively on cells in advanced stages of their cell
cycle [15].
–Chemotherapy: In Chemotherapy, drugs that damage the cancer cell’s genetic
material are administered to the patient. The effects are dependent on local
drug concentration and the cycle stage. Cells in the S-phase are the ones most
likely to die as a result of the treatment [16].
–Immunotherapy: Immunotherapy comprises interventions destined to increase
immune system detection and targeting of cancer cells. The one that we use
in our model consists of training T-cells ex-vivo for cancer antigen recognition.
This therapy is administered gradually to the patient.
2.3 Cellular Automata fundamentals
This work will simulate avascular tumor growth and treatment using Cellular
Automata. This model was first introduced by John von Neumann in his studies
of self-replicating machines and has been widely used for studying how collectively
organized structures can emerge in lattices of individual cells [17,18]. In CA, each
grid cell changes its state at every simulation step based on the state of its close
neighbors and other local conditions.
Proliferating Complex Necrotic
Dead
Cell - Cell interactions
Haptotaxis (ECM)
Chemotaxis (Nutrient)
Immunotherapy
Radiotherapy
Chemotherapy
Fig. 2 Elements considered in the proposed model for cancer growth and therapy.
Title Suppressed Due to Excessive Length 7
The Cellular automaton implemented in this work to simulate tumor growth
and cancer therapies is presented in Figure 2. For tumor growth, a rectangular
grid will be used, and the state of each space will denote the type of biological cell
currently occupying that location. The state values considered are proliferating,
where cancer cells can divide depending on environmental conditions, complexes
formed by entrapment of cancer cells by the immune system; and two types of
death: necrotic, where scar tissue is left, and dead state, where the cell is cleanly
disposed, and can later be replaced by healthy tissue (vacant spaces in the au-
tomaton).
Changes in a cell’s state will depend mainly on cell-cell interactions with its
neighbors. Rules for the possible state transitions will be specified in section 3.1.1.
Aside from communication with adjacent cells, the model considers two crit-
ical factors that regulate the directional proliferation of cancer cells: haptotaxis,
mainly induced by inward constraining contact forces exerted by extracellular
matrix (ECM), and chemotaxis, an effect where nutrient concentration gradients
drive tumor cell propagation outwards. These two effects can oppose each other,
and local differences can determine the tumor’s final morphology and extension
of the tumor [19]. In the automata model, the proliferation and necrosis of cells
at a given grid location will be conditioned on ECM and Nutrient concentrations.
Also, the dynamical degradation of ECM by cancer cells and spatial diffusion of
nutrients will be simulated.
Therapies in the model act mainly by stochastic induction of cancer cell death:
concretely, radiotherapy and chemotherapy will produce either clean death or
necrosis, while immunotherapy will always dispose of cells leaving no scar tissue.
2.4 Biomarkers for complexity: Shannon’s entropy and fractal dimension
Recent advances in medical imaging techniques have opened new possibilities for
the types of analysis that can be conducted to characterize cancer processes accu-
rately. In particular, Shannon’s entropy has been used successfully to segment and
detect tumors in MRI images [20,21]. Some studies have also pointed to entropy as
an appropriate indicator to characterize cancer in CT images, given that a correct
calibration for confounding factors is established [22].
Shannon’s entropy in an image can be computed using its normalized grey-
scale values histogram Pi, where iindexes the bins used to group intensity values.
The equation that defines it is the following:
S=
N
X
i=0
Pilog 1
Pi(1)
In addition to entropy, another complex-systems-related quantity associated
with cancer processes is the fractal dimension. Some studies have found that fractal
dimension in images obtained from histological studies can be a helpful indicator
in detecting cancer cell proliferation [23,24].
The fractal dimension is closely related to the notion of self-similarity, that is,
the presence of structures that repeat themselves at multiple scales in geometrical
patterns or representations of data. Self-similarity is a recurrent property exhibited
8 Juan Uriel Legaria-Pe˜na et al.
by biological systems, being present in neural networks, the immune system and
most relevant to this work in tumor cell arrangements [25–27].
A commonly applied method to compute fractal dimension is box-counting,
where the number Mof boxes of size required to fill the analyzed geometri-
cal structure is obtained. An estimation for fractal dimension can be derived by
applying the following equation:
D=M
1
(2)
In practice, Dis rarely computed for just one value of . Instead, a standard
statistically robust method consists of fitting a linear regression model to the pairs
(, M ), and then the fractal dimension can be found as the slope of the adjusted
line.
3 Proposed model
Details of the proposed cellular automata for tumor growth and treatment will be
given in this section. In addition, our computational model implementation can
be consulted at [28].
3.1 Tumor Growth
As already schematized in Figure 2, the base tumor growth model comprehends
three essential elements to establish: rules for how the state of individual cells
will change (for instance, from healthy vacant spaces to proliferating cancer cells),
haptotaxis qualities of the modeled tissue, namely ECM degradation by the ex-
panding tumor, and chemotactic and necrotic effects due to changes in nutrient
concentration.
3.1.1 Cell - cell interactions and state transitions
The proposed cell-cell automaton interactions are based on the work of Shah-
moradi et al. in [29]. The proposed derivation, however, includes chemotactic-
driven growth and necrosis of cells due to nutrient insufficiency.
A diagram of the possible transitions between cell states is summarized in
Figure 3.
Proliferating cells divide at every step of the automaton’s evolution with a
probability rprolif. Cell division involves randomly placing a new proliferating cell
at any available (healthy) surrounding spaces, and it is also conditioned by the
concentration of extracellular matrix CECM at the location where the system at-
tempts to place a new cell. Namely, if this concentration is more significant than
a threshold TEC M , the placement of the sprung daughter cell will be negated.
The division probability rprolif is limited by the number of proliferating cells
in the tumor. If Nprolif is the current number of proliferating cells and Nprolif,max
is a carrying limit for such cells, then the following equation is applied:
Title Suppressed Due to Excessive Length 9
Proliferating Complex Dead
Necrotic
1−rbinding
rescape
rlysis
rprolif
Place proliferating
cell at a random va-
cant space
rdecay
Dead cell returns to a
healthy (vacant) state
ECM Nutrient
Cn< Tn
CECM < TECM
Fig. 3 State transition diagram of the automaton’s cells
rprolif =rprolif,01−Nprolif
Nprolif,max (3)
where rprolif,0is the initial maximum division probability.
When a proliferating cell does not achieve division, it can form a complex with
immune cells with a probability 1 −rbinding. Antibody immune complexes can
undergo one of two faiths, they could either end up killing the attacked cancer cell
with probability rlysis at every step, or the entrapped cell could escape and return
to its previous proliferating state with probability rescape.
Finally, dead cells will be able to return to healthy vacant spaces with proba-
bility rdecay at every step since they were killed by clean immune catalysis. The
secondary type of death considered, necrosis, can be reached whenever the con-
centration of nutrient Cnat a cell’s position drops below a threshold Tn. Cells in
this state can not be occupied and remain effectively vanished for the rest of the
simulation.
3.1.2 Haptotatic extracellular matrix (ECM)
Haptotaxis involves a simulation of the process where cancer cells progressively
degrade ECM as they come in contact with it. The evolution rule used to update
the concentration of ECM is given by the following equation:
10 Juan Uriel Legaria-Pe˜na et al.
Cnew
ECM(i, j ) = CECM(i, j)−ecI(i, j )CECM(i, j) (4)
where ecis a constant and I(i, j) is the number of proliferating cells adjacent
to the location i, j of the automaton.
3.1.3 Chemotactic nutrient gradient
Nutrient diffusion, which determines chemotaxis and necrosis, was modeled using
the second Fick’s law. Thus, the update rule for the nutrient’s concentration Cn
is given by the following equation:
Cnew
n(i, j) = Cn(i, j ) + Dn∇2Cn(i, j)−Cabs,n (5)
where Dnis a diffusion constant, Cabs,n is an absorption of the nutrient by
the cells, and ∇2Crequires computing the discrete laplacian with concentration
values at neighboring cells, as stated in Equation 6.
∇2Cn(i, j) = Cn(i+ 1, j ) + Cn(i−1, j)
+Cn(i, j + 1) + Cn(i, j −1) −4Cn(i, j ) (6)
For the cell’s nutrient absorption, two values were considered: one Cabs, prolif
which is applied when cells have surrounding vacant spaces to proliferate, and
a lower value Cabs, qui, used for quiescent cells that have no available neighbor
positions where to put new daughter cells.
Figure 4shows images of the automaton’s state taken at six different time
steps and also time series for the counts of the different types of biological cell
states considered in the model. At every grid location, initial concentrations of
ECM are assigned a random value in the interval (0.8,1.2). Cells are initialized
with a nutrient concentration of 1; the ones at the boundary are assigned a fixed
value of 2. Finally, we start with four proliferating cells in a cross shape at the
center of the grid.
Time series for entropy and fractal dimension obtained with the simulated
images as the tumor grew are shown in Figure 5.
3.2 Radiotherapy model
The model for radiotherapy was based on the linear-quadratic cell radiation re-
sponse model used by Sarah C. Br¨uningk et al. in [30].
Radiotherapy delivers a simulated radiation beam at step n0,rad to the automa-
ton’s tissue. The probability that a cell at location i, j is affected by the treatment
is then computed using the following response function:
Prad(i, j ) = 1 −exp −γαdOER (i, j) + βd2
OER(i, j ) (7)
In the previous expression, αand βare constant parameters, weighing the
linear and quadratic responses of the tissue. γis related to the cell cycle stage,
explained in section 2.2, and can be obtained using Equation 8.
Title Suppressed Due to Excessive Length 11
0 50
0
20
40
60
80
Step = 0
0 50
0
20
40
60
80
Step = 120
0 50
0
20
40
60
80
Step = 240
0 50
0
20
40
60
80
Step = 360
0 50
0
20
40
60
80
Step = 480
0 50
0
20
40
60
80
Step = 600
0 100 200 300 400 500 600
Step
0
200
400
600
800
1000
1200
1400
1600
Number of cells
Proliferating
Complex
Necrotic
All Cells
Fig. 4 Automaton images and cell counts obtained in a case where no therapy was adminis-
tered. Evolution took 600 steps and parameters values used were rprolif,0= 0.85, rbinding = 0.1,
rescape = 0.5, rlysis = 0.35, rdecay = 0.35, Nprolif,max = 1000, ec= 0.1, Dn= 0.05,
Cabs,prolif = 0.01, and Cabs,qui = 0.005
0 100 200 300 400 500 600
Step
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Entropy
0 100 200 300 400 500 600
Step
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Fractal Dimension
Fig. 5 Entropy and fractal dimension time series obtained with the automaton images in a
case where no treatment was applied.
γ=γ0×1.5(n−s0)%4 (8)
where γ0is a constant and s0is the step at which the cell in question was first
placed in a vacant space.
The term dOER appearing in Equation 7is called the oxygen status of the
cell, and it is a factor dependent on the nutrient concentration Cn(in this case
12 Juan Uriel Legaria-Pe˜na et al.
interpreted as oxygen saturation) and on the applied gamma beam dose d. Its
calculation can be carried out using Equations 9and 10.
dOER =d
OER (9)
OER = (1Cn(i, j)≥Tn,rad
1−Cn(i,j)
Tn,rad Cn(i, j)< Tn,rad
(10)
In Equation 10,Tn,rad is a low boundary threshold for oxygen concentration,
and those cells where Cndrops below such value will have the greatest probability
of being targeted by radiotherapy.
Cells selected to be targeted by the treatment by the previous response function
will undergo either death or necrosis with probability Prad,0for the first τrad, delay
steps after applying the radiation beam, and with a probability Prad,f for the rest
of the simulation. Here a ratio fnec of deaths that will be of the necrotic type is
fixed at the start of the automaton’s evolution.
Figure 6shows images recorded at six different instants of the automaton’s
evolution and treatment. In this test, the tumor was allowed to grow for the
first 300 steps, and then one single application of radiation was delivered to the
tissue. The right side of the figure also shows the number of cells counted at every
simulation instant.
0 50
0
20
40
60
80
Step = 0
0 50
0
20
40
60
80
Step = 120
0 50
0
20
40
60
80
Step = 240
0 50
0
20
40
60
80
Step = 360
0 50
0
20
40
60
80
Step = 480
0 50
0
20
40
60
80
Step = 600
0 100 200 300 400 500 600
Step
0
500
1000
1500
2000
2500
3000
Number of cells
Proliferating
Complex
Necrotic
All Cells
Fig. 6 Images and cell count time series for the evolved tumor when radiotherapy was ad-
ministered halfway through its growth. Parameters used for the radiotherapy treatment were:
γ0= 0.05, α= 0.1, β= 0.05, d= 1, Tn,rad = 0.35, τrad,delay = 50, Prad,0= 0.02 and
Prad,f = 0.5
Title Suppressed Due to Excessive Length 13
Corresponding time series for entropy and fractal dimension are shown in Fig-
ure 7.
0 100 200 300 400 500 600
Step
0.0
0.1
0.2
0.3
0.4
0.5
Entropy
0 100 200 300 400 500 600
Step
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Fractal Dimension
Fig. 7 Entropy and fractal dimension time series obtained in a case where a beam of radiation
was administered midway through the tumor development (step 300).
3.3 Chemotherapy model
The chemotherapy model applies a cell response function similar to the one pro-
vided for the radiotherapy scenario. In this case, the response model presented by
Fateme Pourhansanzade and S.H. Sabzpoushan in [31] was used.
When chemotherapy starts at step n0,chem, a constant drug concentration C0,d
is set at the borders of the automaton for a period of τchem steps, simulating
the injection of pharmaceuticals. The administered drug will then perfuse the
automaton’s area for the rest of the simulation following the second Fick’s law for
diffusion. An update of the medication concentration at position i, j of the grid
will thus follow the next rule:
Cnew
d(i, j) = Cd(i, j ) + Dd∇2Cd(i, j) (11)
where Ddis the drug’s diffusion constant and ∇2Cdis obtained as the discrete
laplacian (See Equation 6).
Drug concentration Cddetermines a given cell’s response to chemotherapy.
The probability of the cell at location i, j dying at step nof the simulation is
given by Equation 12. Here two types of death are considered: normal death and
necrotic, where the necrotic ratio of dead cells by chemotherapy is established at
the beginning of the simulation.
Pchem =lm(i, j)×P K ×exp (−cm(n−τsim )) (12)
In equation 12,τis the number of steps left in the simulation after starting
chemotherapy, and P K is a pharmacokinetics factor that modulates how quickly
the injected drug is metabolized. Furthermore, cmis an attenuation factor asso-
ciated with live cells in state m(either proliferating or complex in our case), and
lm(i, j) is a linear factor related to the medicine concentration Cd(i, j ). The latter
can be computed using the following equation:
14 Juan Uriel Legaria-Pe˜na et al.
lm(i, j) = km×Cd(i, j)
CR0
m×τsim + 1 (13)
In the previous expression, kmis the killing rate of cells in state m, and CR0
m
is a chemical resistance parameter assigned stochastically with relation 14.
CR0
m=CRm×random(0,1) (14)
where CRmis the maximum possible value for chemical resistances and random(0,1)
denotes a floating point value taken in the interval between 0 and 1.
An essential aspect of the chemotherapy model we considered is that the med-
ication can target only cells at the S-phase of their cell cycle. The cell cycle stage
is computed within the simulation with Equation 15.
cell stage = (n−s0)%4 (15)
Thus, only those cells with cell stage value one will be damaged by radiation.
Also, we consider that a certain fraction Rchemo of cells will be treatment-
resistant and natively unaffected by the administered drugs. These cells have the
quality of asymmetric division; that is, they can either produce a new resistant
cell with probability Pres,chemo, or their offspring can be non-resistant. In contrast,
non-resistant cells will always produce new daughter cells which are non-resistant,
i.e., their division will be symmetric.
Figure 8shows the results of tumor treatment with chemotherapy in a scenario
where drugs are administered in step 300 of the automaton’s evolution. The right
side of this figure shows cell counts obtained for the different cell states considered.
Entropy and fractal dimension time series obtained in the chemotherapy case
are shown in Figure 9.
3.4 Immunotherapy model
Immunotherapy was based on the automaton model devised by Shamoradi et al.
in [29]. It is based on a therapy where T-cells are trained to better detect and
dispose of cancer cells. The therapy is gradually delivered to a patient.
In the automaton, this effect can be imitated by slowly changing the parameters
so that the formation of immune complexes (binding of proliferating cells and the
immune system) increases. These complexes metabolize the attacked cells more
efficiently.
If rfinal and rinitial are the target and initial values for a given tumor growth
parameter, and the effects of chemotherapy are distributed over a number τimmuno
of steps, then the following increment would be added to the parameter value at
every simulation step to modify it:
∆r =rfinal −rinitial
τimmuno
(16)
until the number of steps τimmuno that the therapy takes is reached.
Figure 10 shows the evolution of the tumor automaton, where an immunother-
apy round of treatment that took 250 steps was delivered to the tissue starting at
step 300.
Title Suppressed Due to Excessive Length 15
0 50
0
20
40
60
80
Step = 0
0 50
0
20
40
60
80
Step = 120
0 50
0
20
40
60
80
Step = 240
0 50
0
20
40
60
80
Step = 360
0 50
0
20
40
60
80
Step = 480
0 50
0
20
40
60
80
Step = 600
0 100 200 300 400 500 600
Step
0
500
1000
1500
2000
2500
3000
3500
Number of cells
Proliferating
Complex
Necrotic
All Cells
Fig. 8 Cell counts and images of an evolved tumor automaton, where chemotherapy treatment
is applied at step 300. Value parameters used for this reported test were: Dd= 2, PK =
1, Rchemo = 0.1, τsym = 300, τchem = 2, Cd,0= 1, cProliferating = 0.5, cComplex = 0.5,
CRProliferating = 0.1, C RComplex = 0.05, kProliferating = 0.8, kComplex = 0.01
0 100 200 300 400 500 600
Step
0.0
0.1
0.2
0.3
0.4
0.5
Entropy
0 100 200 300 400 500 600
Step
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Fractal Dimension
Fig. 9 Entropy and fractal dimension time series obtained in a case where chemotherapy was
delivered starting at step 300.
Corresponding time series of entropy and fractal dimension are shown in Figure
11.
4 Results
Figures 4and 5show the results obtained with the avascular tumor growth model
when no therapeutic interventions were administered. As the disease progresses,
16 Juan Uriel Legaria-Pe˜na et al.
0 50
0
20
40
60
80
Step = 0
0 50
0
20
40
60
80
Step = 120
0 50
0
20
40
60
80
Step = 240
0 50
0
20
40
60
80
Step = 360
0 50
0
20
40
60
80
Step = 480
0 50
0
20
40
60
80
Step = 600
0 100 200 300 400 500 600
Step
0
500
1000
1500
2000
2500
3000
Number of cells
Proliferating
Complex
Necrotic
All Cells
Fig. 10 Images of the tumor growth automaton’s state and cell counts obtained for a case
where immunotherapy was delivered at step 300. Parameters used for the treatment were:
rfinal, prolif = 0.65, rfinal, binding = 0.001, rfinal, escape = 0.001, rfinal,lysis = 0.9, rfinal,decay =
0.35 and τimmuno = 250.
0 100 200 300 400 500 600
Step
0.0
0.1
0.2
0.3
0.4
0.5
Entropy
0 100 200 300 400 500 600
Step
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Fractal Dimension
Fig. 11 Entropy and fractal dimension time series computed in a case where immunotherapy
was started at step 300.
the tumor reaches a stationary final state, where cell count, entropy, and fractal
dimension fluctuate around enduring values. This attractor of the cellular automa-
ton’s configuration, which represents the tumor, exhibits resilience and adaptative
qualities toward possible external disturbances [32]. Biologically this is very mean-
ingful and correctly models resistance to treatment and recurrence phenomena,
known traits of cancerous tumors.
Title Suppressed Due to Excessive Length 17
The results obtained when radiotherapy was applied during the tumor evolu-
tion are shown in Figures 6and 7. It is observed that cell count, entropy, and
fractal dimension decrease for some time after the radiation is delivered to the tis-
sue, attempting to escape the tumor attractor. However, the re-incidence of cancer
cell proliferation is observed, manifesting as three distal tumor sprouts, which sug-
gests that the modeled intervention is insufficient to drive the system out of the
tumor’s potential well, so possibly multiple successive irradiation procedures would
be required to eradicate the tumor.
Chemotherapy results, shown in Figures 8and 9, yielded similar patterns to
those obtained with radiotherapy. Namely, a temporary decrease of the analyzed
variables was observed, followed by a recurrence in tumor growth. In addition,
since chemotherapy spatially attacks cells following the drug’s radial perfusion
(in contrast to radiotherapy, where the targeting of the cells is more sparsely
distributed), a more uniform reduction of the proliferating cells in the tumor’s
periphery was achieved.
Finally, results obtained for immunotherapy, presented in Figures 10 and 11,
showed similar patterns of decrease for state variables when the treatment was
applied. In this case, however, the targeting of cancer cells was even more widely
scattered than during radiotherapy, leaving multiple clusters of cells as the immune
system attacked proliferating cells. This simulation resulted in slighter decreases
for both entropy and fractal dimension, and it is an essential spatial aspect to
consider since those cell groups could result in multiple separate tumor formations
later on.
5 Discussion
Comparison of time series for entropy and fractal dimension as the tumor grew
and as each of the modeled treatments was delivered showed that while fractal
dimension grows faster at initial stages of cancer development (greater slope), it
responds slower and on a smaller magnitude than entropy to the applied thera-
peutic interventions. This phenomenon suggests that these biomarkers’ sensitivity
could vary depending on the tumor stage at which they are registered. Namely,
the fractal dimension would be appropriate to detect initial changes in tissue as
the tumor grows, while entropy could provide a useful measure to assess the ef-
fect of therapies on tissue. However, it is important to remark that the previous
observations are limited by assumptions made in the automaton model. In ad-
dition, images generated with the model hold no direct relationship with any of
the available imaging techniques used in a clinical setting. Thus, the sensitivity of
the analyzed biomarkers could depend on the method used to record the tumor
state. Nonetheless, this result highlights the importance of specifying the cancer
stage whenever imaging biomarkers are investigated since their diagnostic and
prognostic utility could be time-dependent.
A comparison of the examined interventions showed that both entropy and frac-
tal dimension decreased more slowly whenever multiple scattered groups of cells
were left due to the treatment (in our results, this occurred with immunother-
apy). This finding suggests that time variations of biomarkers such as entropy and
fractal dimension can reflect on relevant spatial evolution patterns of a complex
system. In the particular case of a tumor, they could be associated with how ef-
18 Juan Uriel Legaria-Pe˜na et al.
ficient treatment is in disrupting communication and interactions of cancer cells
in a tissular area. Both radiotherapy and chemotherapy showed a more homoge-
neous targeting of proliferating cells, obtaining relatively faster response curves
for entropy and fractal dimension.
6 Concluding remarks
A cellular automata model for tumor growth and three treatment modalities (ra-
diotherapy, chemotherapy, and immunotherapy) was developed in this work to
study the evolution of 2 imaging biomarkers: Shannon’s entropy and fractal di-
mension. Results from the model showed that adaptive and resilient properties of
tumors toward treatment emerged naturally from the individual interactions of
cancer cells. Namely, in all three treatments, cancer reoccurrence was observed at
some point after discontinuing the interventions. Time series of entropy and frac-
tal dimension obtained with a sequence of simulated tumor images showed that
fractal dimension increased faster at the onset of cancer cell proliferation, while
entropy exhibited the highest response to effects induced by cancer therapies. The
previous result suggests that these biomarkers’ sensitivity and prospective diag-
nostic utility could vary depending on the cancer stage and treatment conditions.
Also, an imaging technique used to record the state of the tumor could play a crit-
ical role in their application for cancer evaluation purposes. A comparison among
the simulated interventions revealed that entropy and fractal dimension decreased
slower with therapies that left scattered, isolated, proliferating cells as they in-
teracted with the tissue; this suggests that both quantities could reflect essential
characteristics of the spatial targeting of cancer cells in therapies.
Funding and conflict of interest
The research leading to these results received funding from CONACYT-Mexico
under Grant Agreement Fronteras 2019-217367.
The authors have no conflicts of interest to declare that are relevant to the content
of this article.
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