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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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