Dipartimento di Matematica

M. Delitala, T. Lorenzi

Progression and heterogeneity in colorectal cancer dynamics.

A model based on the kinetic theory for active particles

Rapporto interno N. 7, settembre 2009

Politecnico di Torino

Corso Duca degli Abruzzi, 24-10129 Torino-Italia

Progression and Heterogeneity in Colorectal Cancer

Dynamics

A Model Based on The Kinetic Theory for Active

Particles ✩

Marcello Delitala∗,a, Tommaso Lorenzia

aDepartment of Mathematics, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129

Torino, Italy

Abstract

This paper deals with the development of a mathematical model describing col-orectal cancer dynamics at cellular scale within the mathematical frameworksprovided by the kinetic theory for active particles. Compartmental arrange-ment of cells and spatial homogeneity are assumed inside colorectal crypts.Attention is focused on progression and heterogeneity aspects of carcinogenesis.The effects on cancer dynamics of the action exerted by T-lymphocytes againstmutated cells is taken into account. The result of simulations shows how thesimultaneous presence within colorectal crypts of cells at different differentiationstages, characterized by peculiar mutation rates, has an impact on the hetero-geneity of the phenomenon. Moreover, the significant role played by mutationinvolving stem cells in tumour progression is pointed out. A threshold efficiencylevel for the destructive action exerted by T-lymphocytes, if supported by exter-nal drugs, against cancer cells is found, which allows the rejection of malignantcells from the crypts as well as it slows down the mutation process.

Key words:Colorectal Cancer, Kinetic Theory, Active Particles, Stochastic Games,Evolution, Mutations

1. Introduction

Many experimental evidences [1] support the idea that cancer is initiated byunforeseen DNA somatic alterations triggering a multistage genetic alteration

✩Partially supported by FIRB project - RBID08PP3J, Metodi matematici e relativi stru-menti per la modellizzazione e la simulazione della formazione di tumori, competizione con ilsistema immunitario, e conseguenti suggerimenti terapeutici.

∗Corresponding authorEmail address: marcello.delitala@polito.it (Marcello Delitala)

process known as clonal evolution. Cells and their direct descendants accumulatesuccessive mutations leading them from normal phenotypes to highly malignantones and enhancing their ability to escape homeostatic regulation mechanismsas well. The process is often described as a sort of Darwinian microevolutioninvolving normal, cancer and immune cells that could result in neoplasia andmetastasis.

It is to point out that events which take place at molecular scale impactdirectly on the phenomenology of carcinogenesis at cellular scale, which on turnaffect the dynamics at the tissue scale. As a result, cancer generation and pro-gression are both intrinsically multiscale phenomena, and events occurring ata given scale increase the complexity of the scenario at the higher scale. Indetails, DNA somatic alterations cause the simultaneous presence of prolifer-ating heterogeneous subclones at cellular scale, which on turn may lead to theformation of solid aggregates composed of distinct sectors at tissue scale [2].One of the best documented and extensively studied example of such a dis-ease is colon and rectum cancer, which develops through successive malignantmutations involving epithelial cells that are located within colorectal crypts.

The analysis and the comprehension of carcinogenesis represents one of thescientific goals of this century and the Scientific Community is persuaded thatMathematics may play a leading role in attaining it. A consistent mathemat-ical model would be able to provide a framework in which mechanisms thatdetermine experimental results can be better understood, leading to new ther-apies whose validity could be tested through computer simulations, so reducingthe costs of laboratory experiments and the number of animal experiments.Moreover, a mathematical model able to predict behaviours that have not beenobserved yet, the so called emerging phenomena, would allow to study remediesfor new pathologies at an early stage.

Therefore, applied mathematics and biology, theory and experiments, math-ematicians, biologists and clinicians are linked together in a close network ofrelationships and comparisons which may mark a turning point in the researchagainst cancer.

Certainly, from a mathematical point of view, many difficulties have to beovercome, mostly related to the development of new structures and methodsable to deal with living systems. Indeed, they could be different from the onesused in describing inert matter dynamics, and they could result into a newmathematical theory. Additional difficulties descend from the complexity of thebiological phenomenon under consideration, that requires a careful introductoryanalysis in order to select the most relevant events to include in the model.

Bearing all above difficulties in mind, in this paper we propose a model ofcell dynamics within a single colorectal crypt during generation and progressionstages of cancer with the following main purposes:

- describing the interactions between cells at the same or at different stagesof the disease;

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- analyzing the effects of successive mutations occurring within cells at dif-ferent levels of physiological differentiation;

- depicting the competition between the host immune system, or curingagents, and the mutated cells at increasing degrees of malignancy;

- identifying both emerging phenomena and critical parameters related tocarcinogenesis.

This model has been developed at cellular scale as an intermediate scale be-tween the molecular and tissue ones. Spatial homogeneity assumption has beenmade and the mathematical structures provided by the kinetic theory for activeparticles [3], [4], [5] have been employed in order to describe cell expansions, celldeath and immune supervision. This mathematical theory provides a statisticaldescription of the physical system under consideration through a suitable prob-ability distribution function, and it has been implemented by using a stochasticgame theory to describe interactions occurring between cells, that are assumedas organized into interacting modules.

The content of this paper, that consists of five sections apart from thisintroduction, can be summarized as follows:

- Section 2 deals with a qualitative physical description of both the livingsystem and the biological phenomenon under consideration.

- Section 3 provides a brief introduction to the kinetic theory for activeparticles, in its discrete formulation.

- Section 4 is devoted to the presentation of the proposed model of colorectalcancer dynamics.

- Section 5 summarizes the phenomena emerging from some simulationsdeveloped under different parameter settings with an exploratory aim.

- Section 6 contains the conclusions of this work and some perspectivesabout the future researches.

2. Biological Frameworks

This section provides a qualitative biological description of both the physicalsystem and the phenomenon under consideration. Its contents are essentiallyrelated to a paper by Hanahan and Weinberg [1], to a book by Weinberg [2]and to a further paper by Nowak and coworkers [7]. The essential features ofcolorectal crypt structure and the underlying principles of tumourigenesis ingeneral, as well as of colon and rectum carcinogenesis in particular, are hereintroduced. Attention is drawn to mutation phenomena occurring at molecularscale, since they influence strongly the system dynamics at the cellular scale.

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2.1. Genetics of Cancer

Many histopathological evidences support the idea that carcinogenesis inhumans is a multistage process [1] developing through successive DNA somaticalterations, that unable physiological feedback mechanisms and lead to neoplasia.Most of these alterations consist in the modification of DNA sequences, that aremutations which can arise from stochastic replication errors, from exposure tocancerous agents, such as radiations, from defects into DNA repairing processesor, more in general, from our way of life. The dynamics of the phenomenon isformally analogous to a Darwinian microevolution, the so called clonal evolution,in which the single cells represent the competing individuals. At each successivestage the mutated cells acquire a growth advantage. The accumulating of thesemutations implies the progression of cells and their descendants from regularphenotypes to highly malignant ones, that is, the conversion of normal cells intocancer ones.

Genes involved in carcinogenesis are usually grouped into two main cate-gories: caretakers and gatekeepers. Caretakers do not have any direct effect oncellular growth, however they influence the ability of cells in preserving theirgenome integrity. In fact, the deficency of these genes leads to an increase in themutation rates of all genes. Gatekeepers affect in a direct manner the growthof cells and, according to the fact that their action take place in a positive ornegative way, they are further differentiated into proto-oncogenes and tumoursuppressor genes (TSG) respectively. Mutation involving these two classes ofgenes influence cancer progression by keeping under control cellular divisionor apoptosis. Proto-oncogenes in normal cells are strictly controlled, while incancer cells they are subjected to peculiar mutations causing them to evolveinto oncogenes and leading to an increase in the activity of the genetic prod-uct. These mutations take place in a single allele of the proto-oncogene andact in a dominant way, that is, the mutated allele is dominant and overridesthe wild-type one. Tumour suppressor genes are responsible for limiting cellulargrowth.

According to the well-established “two hits” model by Knudson [8] the com-plete loss of function of these genes, that takes place in cancer cells, is a recessivemechanism to the normal allele. In fact, because of the diploid nature of humancells, it requires the inactivation of both the wild-type alleles (Figure 1). Oneof the best documented and most understood example of such a process involv-ing successive DNA somatic alterations is the progression of colon and rectumcancer.

2.2. Colorectal Cancer Progression

The colon and the rectum constitute the last five feet of the digestive system.On an oversimplified point of view, it is possible to look at them as forming atube, whose hollow section is the lumen and the tissue part is the bowel wall,which is composed of four distinct tissue layers. The mucosa, that is the colonepithelium facing the lumen, is lined by about ten millions of eversions thatproject into the lumen, the villi, which are spaced out by as many invaginations,the crypts, in which about two hundred of cells are located.

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Mutation in one copy“first hit”

Deletion of the other copy“second hit”

Tumour progression

Figure 1: Sketch of the “two hits” model by Knudson, [8]

Crypt

Villus

Lumen

Stem cell compartment

Transit-amplifying cell compartment

Highly differentiated cell compartment

Figure 2: Sketch of cellular distribution within the colorectal crypt

Cells within a single crypt can be basically divided into three compartments:stem, transit-amplifying and highly differentiated cells.The stem cell compartment contains an average number of 4 to 6 cells andextends from the bottom to the third portion of the crypt.The transit-amplifying cell (TAC) compartment contains about 150 cells andoccupy the last part of the invagination as far as the top.The highly differentiated cell compartment contains about 3500 cells, but onlya few of them occupy the top of the crypt, while most of these cells line thewalls of the next villi (Figure 2). Actually, cells residing within the crypt canbe assumed as just belonging to stem and transit-amplifying compartments.The stem cell cycle takes about 24 hours; at the end of this time interval thesecells can renew by dividing symmetrically in two identical stem daughter cellsand generate new cells in the transit-amplifying compartment by dividing asym-metrically in a stem daughter and a TAC daughter or can undergo apoptosis.Transit-amplifying cells take about 12 hours to end their life cycle, at the end ofwhich these cells can divide symmetrically in two identical TAC daughter cells,

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Stem cell compartment

Transit-amplifying cell compartment

Highly differentiated cell compartment

Cell death

Figure 3: Sketch of cellular dynamics within the colorectal crypt

die via apoptosis or divide asymmetrically in a TAC daughter and a highly dif-ferentiated daughter. Highly differentiated cells leave the crypt and in about 3days move toward the top of the neighbor villi, where they undergo apoptosisand are shed into the lumen at the end of their life cycle (Figure 3). Transit-amplifying cells divide with a higher frequency than stem cells. In such a way,the twofold goal of guaranteeing a continuous renewing of epithelial tissue liningcolon as well as of partially protecting cells within the crypts from the accu-mulation of successive genetic lesions is achieved. Since TAC cells proliferaterapidly, stem cells have to divide rarely as to maintain a sufficient number ofdifferentiated cells within the epithelial tissue of crypts. Therefore, stem cellsare partially sheltered by stochastic DNA replications errors taking place dur-ing cellular divisions phenomena and transit-amplifying cells are exposed to agreater risk of undergoing malignant mutations. Since sooner or later TACs willenter into a post-mitotic state, leave the crypt and die, the negative effects ofmalignant mutations that could occur during mitotic events are weakened.

Despite this defense arrangement, both stem and transit-amplifying cells canundergo genetic alterations causing the creation of malignant cells and the pro-liferation of their descendants inside crypts (Figure 4). These phenomena breakup homeostatic equilibrium and lead to the formation of solid cellular aggre-gates, the so called adenomas. Constituent cells of adenomas can accumulatefurther genetic lesions which increase their malignancy and make them expressa stronger resistance to immune action. This mutation process drives colorectalcrypts through a series of pathological phases starting from small benign tu-mour, known as adenomatous polyp (ADP), and terminating with carcinoma.

The analysis developed by biologists and clinicians on the DNA of samplesbelonging to colon epithelial tissues at different stages of cancer progressionhave led to the identification of some of the genes usually involved into mu-tations leading to colon and rectum carcinogenesis. These aknowledgmentshave allowed Fearon and Vogelstein [9] to derive a reliable biological model of

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Transit amplifying cells Transit amplifying cellsStem cells

First mutation

Second mutation

Third mutation

Figure 4: Sketch of cellular mutation dynamics within the colorectal crypt

Figure 5: Sketch of a simplified version by Nowak and coworkers [7] of the colorectal cancerprogression model by Fearon and Vogelstein

colorectal cancer progression. A simplified version of this widespread model,which includes the alterations of two tumour suppressor genes and of one proto-oncogene (Figure 5), has been proposed by Nowak and coworkers [7]. The mainsequence of somatic genetic alterations related to biological and clinical colorec-tal cancer progression starts with the inactivation of APC, a gatekeeper genetriggering cells migration process within the crypts. Cells, and their descen-dants, expressing an alteration in this gene, acquire a strong adhesion abilityand they have a higher probability to be retained inside the invaginations. As aconsequence, a clonal expansion of mutated epithelial cells occurs, giving rise toa small adenoma. In the patients suffering from familiar adenomatous polyposis(FAP) this alteration can be inherited; so, these individuals are prone to un-dergo spontaneous growth of small adenomas in their intestine. Subsequentely,carcinogenesis proceeds via alteration from proto-oncogene to oncogene of K-ras ; this oncogene enables mutated cells to grow in an anchorage-independentway [2] supporting the evolution of a small adenoma into a late adenoma. Afterthat, the inactivation of p-53, a tumour suppressor gene responsible for the sup-pression of cell proliferation, causes mutated cells to acquire a growth-promotingpower. As a result, a late adenoma can proceed into a carcinoma. Each of thesesteps can be accelerated by chromosomal instability (CIN) as it contributes to

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an increasing in the rate of heterozygosity loss [7], [10], [11].However, it is relevant to stress out that because of the unforeseeable na-

ture of carcinogenesis, this model should be seen as describing a sort of averagephenomenolegy of colorectal cancer. Therefore, it does not have a general va-lidity. Indeed, both the genes involved in cancer progression and the mutationsequence in which they are implicated could strongly differ from one patient toanother.

3. Mathematical Frameworks

This section is devoted to a brief introduction to the mathematical kinetictheory for active particles [3], [4], [5] in its discrete formulation. Within theframeworks of this theory cellular phenomena are described through a stochas-tic game theory approach, which is coherent with the statistical nature of cancerdynamics. Moreover, since the colorectal crypt can be assumed as a spatial ho-mogeneous system, its dynamics can be efficently modelled by the mathematicaltheory here introduced. This kinetic theory provides a statistical description ofcomplex living systems, through a suitable distribution function, introducingthe dependance of the states of individuals, here cells, on a set of variables,addictional to geometrical and mechanical ones, modelling peculiar organizedbehaviours. Therefore, heterogeneity phenomena, that are related to the widerange of peculiar functions expressed by cells at different progression stages,can be taken into account. The mathematical structures offered by this theoryrepresent the selected mathematical framework in which our model is developed.

According to the interpretation proposed by Bellomo and Forni [4] of thefunctional module theory by Hartwell et alii [6], cells belonging to a complexliving system can be seen as aggregated into some distinct populations composedof cells of different nature which express a collective peculiar and well definedbiological function. This function makes each population distinguishable fromthe other ones, and characterizes it as a functional module.

Therefore, P cell populations have to be chosen, according to the purposesof the following analysis, and to be labelled by an index p = (1, .., P ). Eachcell within a population is an active particle, that is a particle whose stateis identified, in addition to geometrical and mechanical variables, by a scalarvariable u called activity. This variable can be seen as included in the domainDu of the space of states associated to the population p, and it is suitable todescribe the organized behaviour of cells, such as the expression of a peculiarbiological function, i.e. progression and defense ability for cancer and immunecells respectively. If spatial homogeneity is postulated, the microscopic state ofa cell at a given instant of time is identified just by the activity. Moreover, if adiscrete representation of the states is consistent with the biological reality, theactivity u is assumed as a discrete variable and, as a consequence, the domainDu is represented by a close and bounded interval defined as follows:

Du = {u1, ..., uh, ..., uN},

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where uh stands for a general element belonging to Du and the index h ranges inthe set (1; N) ⊆ N. As a result, under this formalism, two cells can be defined asidentical if and only if they are found in the same state of the same population.

The state of the whole system at cellular scale can be described, at timeinstant t, by the total number density of cells n = n(t), that can be defined asfollows:

n(t) =

P∑

p=1

np(t), (3.1)

where np = np(t) is the local number density, that is the number of cells in theunit volume of population p at time instant t.

On turn np can be expressed as the zero order weighted moment of thediscrete probability distribution function f(p,h) = f(p,h)(t) = fp(t, uh), which isrelated to the state of the population p identified by the value uh of the activityvariable. Hence, defining T(p,h) = Tp(uh) as the duration of a life cycle for a cellin the state uh belonging to the population p, np results as follows:

np(t) =

N∑

h=1

f(p,h)(t), (3.2)

where f(p,h) : [0, T(p,h)] × Du → R+ and f(p,h) ≥ 0, ∀t ∈ [0, T(p,h)].

Hence, in order to describe the state of the whole system, an explicit analyti-cal form for the distribution function f(p,h) must be provided. As a consequence,an evolution equation for cells seen as active particles must be derived, whichdescribes the net inlet of cells within the elementary volume of the state space ofpopulation p linked to the state uh. In fact, f(p,h) can be determined by solvingthis equation that acts a paradigm structure for the development of a statisti-cal description, at cellular scale, of the system under consideration. Both theparticular form of this equation and the resultant of the distribution functiondepend on the kind of cellular phenomena and on the interactions taken intoaccount.

Assuming cellular interactions as pair interactions between a nominal testparticle and a nominal field particle, it is possible to include in the model bothconservative and proliferative/destructive interactions [5] as well as loss andproliferation events. Therefore, the evolution equation of interest results asfollows:

df(p,h)

dt(t) + Fp(t) = C(p,h)[f ](t) + P(p,h)[f ](t)

+ S(p,h)[f ](t) + D(p,h)[f ](t) + L(p,h)[f ](t), (3.3)

where the distribution function f(p,h)(t) = fp(t, uh) refers to the test parti-cle, p = (1, ..., P ) and h = (1, ..., N). The effects on population p of externalactions, such as chemotherapeutic or radiotherapeutic agents for cancer cells,are taken into account by Fp(t); therefore, its expression can vary from case

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to case. C(p,h)[f ](t) = Cp[f ](t, uh), P(p,h)[f ](t) = Pp[f ](t, uh), D(p,h)[f ](t) =Dp[f ](t, uh), S(p,h)[f ](t) = Sp[f ](t, uh) and L(p,h)[f ](t) = Lp[f ](t, uh) model theflow of cells, at time instant t, into the elementary volume of the state space ofpopulation p linked to the state uh due to conservative interactions, to prolifer-ative interactions with transition of population, to proliferative and destructiveinteractions without transition of population, to proliferation events and to lossevents respectively. Since proliferation and loss events are not mediated bycellular interactions, both S(p,h)[f ](t) and L(p,h)[f ](t) assume the expression ofsource-sink terms.

The expression of the right-hand side terms of Eq. (3.3) result as follows:

C(p,h)[f ](t) =

P∑

q=1

(

N∑

i=1

N∑

k=1

η(p,i)(q,k)B(p,h)(p,i)(q,k)f(p,i)f(q,k)

− f(p,h)

N∑

k=1

η(p,h)(q,k)f(q,k)

)

(3.4)

P(p,h)[f ](t) =

P∑

l,q=1

N∑

i=1

N∑

k=1

η(l,i)(q,k)µ(p,h)(l,i)(q,k)f(l,i)f(q,k) (3.5)

D(p,h)[f ](t) =

P∑

q=1

N∑

i=1

N∑

k=1

η(p,i)(q,k)µ(p,h)(p,i)(q,k)f(p,i)f(q,k) (3.6)

S(p,h)[f ](t) =P

∑

q=1

N∑

i=1

γ(p,h)(q,i) f(q,i) (3.7)

L(p,h)[f ](t) = τ(p,h)f(p,h). (3.8)

In the expressions above:

- η(p,i)(q,k) = ηpq(ui, uk) is the encounter rate, namely the rate at which atest particle in the state ui of population p encounter a field particle in the stateuk of population q;

- B(p,h)(p,i)(q,k) = Bp

pq(ui, uk; uh) denotes the probability density to find a test

particle belonging to population p with initial state ui into the state uh of thesame population as a consequence of the interaction with a field particle in thestate uk belonging to population q;

- µ(p,h)(l,i)(q,k) = µ

p

lq(ui, uk; uh) models the proliferation rate of particles with

state uh into the population p after the interaction, that is characterized by anencounter rate η(l,i)(q,k), of a test particle with initial state ui of population l

with a field particle in the state uk belonging to population q;

- µ(p,h)(p,i)(q,k) = µpq(ui, uk; uh) describes the net proliferation rate of particles

with state uh within the population p due to interactions that take place atrate η(p,i),(q,k) between a test particle in the state ui of population p and a fieldparticle in the state uk belonging to population q;

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- γ(p,h)(q,i) = γp

q (ui; uh) stands for the rate at which particles in the state ui of

population q create, through division phenomena, particles in the state uh ofpopulation p; that is to say, the proliferation rate of particles in the state uh ofpopulation p due to mitotic events;

- τ(p,h) = τp(uh) stands for the loss rate of particles in the state uh ofpopulation p because of apoptotic events.

Postulating spatial homogeneity and assuming the case of a close system,the analytical expression for f(p,h) can be obtained by solving the followingdifferential equation, which descends from the substitution of Eqs. (3.4), (3.5),(3.6), (3.7) and (3.8) into Eq. (3.3):

df(p,h)

dt=

P∑

q=1

(

N∑

i=1

N∑

k=1

η(p,i)(q,k)B(p,h)(p,i)(q,k)f(p,i)f(q,k)

− f(p,h)

N∑

k=1

η(p,h)(q,k)f(q,k)

)

+

P∑

l=1

P∑

q=1

N∑

i=1

N∑

k=1

η(l,i)(q,k)µ(p,h)(l,i)(q,k)f(l,i)f(q,k)

+

P∑

q=1

N∑

i=1

N∑

k=1

η(p,i)(q,k)µ(p,h)(p,i)(q,k)f(p,i)f(q,k)

+

P∑

q=1

N∑

i=1

γ(p,h)(q,i) f(q,i)

− τ(p,h)f(p,h). (3.9)

Eq.(3.9) acts as a paradigm structure for the development of a statistical de-scription, at cellular scale, of multicellular systems. Its general form can beadapted to a particular case by specifying the analytical expression of the en-counter rate (η), of the transition probability density (B), of the proliferationrates (µ and γ) and of the loss rate (τ).

4. The Model

This section deals with the definition of a model able to describe cell dynam-ics in the colorectal crypts during carcinogenesis, according to the mathematicalframeworks provided by the discrete kinetic theory for active particles. The firstpart is devoted to outline the basic principles and assumptions on which thepresent model relies, while in the second one an evolution equation for cells asactive particles is derived.

4.1. Cellular Dynamics Within The Colonic Crypts

Let us consider, on the basis of the biological considerations drawn in Sec-tion 2 and following a compartmental approach [12], [13], cells within a sin-gle colorectal crypt as active particles divided into three compartments: stem

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Figure 6: Sketch of the mathematical representation of colorectal cancer progression accordingto the proposed model. Connections between cellular, molecular and tissue scale are shownas well

cells, transit-amplifying cells (TACs) and highly differentiated cells. We lookat cell compartments as functional modules and we label them with an indexp = (1, 2, 3) respectively.

As it has been previously pointed out, under physiological conditions stemcells reside at the bottom of the crypt and divide until approximating its thirdportion. Then, they can renew, die or differentiate into transit-amplifying cells,which can renew, die or, approximating the top of the crypt, further differen-tiate into highly differentiated cells. Highly differentiated cells are only able tomigrate out from the crypt and, then, to die. Events taking place within highlydifferentiated cell compartment have a weak impact on crypts dynamics. Thisis a direct consequence of the fact that the number of highly differentiated cellsremaining inside the invagination is rather exiguous and that these cells areprogrammed for dying without undergoing cellular division. For this reason, weconcentrate our attention on phenomena involving stem and transit-amplifyingcells and we look to highly-differentiated cell compartment as a basin storingdifferentiated TACs that are destined to die.

Looking for a description of colorectal cancer dynamics, we assume that theactivity variable u models the progression stage reached by a cell within thecrypt. The domain Du has been defined on the basis of the simplified version ofthe colorectal cancer progression model by Fearon and Vogelstein [9] reported inSection 2, since it guarantees a good compromise between biological consistencyand computational complexity.

We assume u = u1 if a cell in the compartment p = (1, 2) is expressing itshealthy phenotype, u = u2 if the first allele of the APC gene has been neu-tralized by a first stage premalignant mutation and u = u3 if even the secondone has been deactivated. As one of the alleles of the proto-oncogene K-ras hasbeen altered u = u4. Finally, if the first allele of the p-53 gene has been neutral-ized u = u5 and when even the second one has been deactivated u = u6 = uN

(Figure 6). It is worth to notice that Du is defined in the same way for both

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of the cell compartments and that this approach is justified under a biologicalpoint of view. In fact, stem and transit-amplifying cells at a given mutationstage are genetically identical to one another and differ from one another onlybecause they have distinct gene expression programs [2].The lifecycle of a cell can vary according to its population and its state; hence,we describe this quantity through a constant discrete function T(p,i) = Tp(ui)whose values are expressed in day unit.

We assume that cells belonging to the immune system, and able to actagainst cancer cells, are grouped into populations, which are organized as spe-cific functional modules and are labelled by an index p = (1, ..., Z). Each im-mune cell is considered as an active particle whose state is identified by a discretevariable wi, included in the domain Dw of the space of states associated to thepopulation p. The activity wi describes the peculiar defense ability expressedby an immune cell of the population p, with a life cycle requiring T(p,i) = Tp(wi)days.

If a cell belonging to the immune system encounters a normal cell within thecrypt nothing will happen. On the contrary, if the encounter occurs between animmune cell and a cancer cells, an immune competition phenomenon will takeplace. In order to ensure to our model coherence with the biological reality, wecan state that mutated cells in the compartment p = (1, 2) at a different malig-nant stage (u1 < ui ≤ uN) expressing different kinds of antigens are attacked byimmune cells belonging to the population p = (1, ..., Z) and expressing distinctdefense functions. Hence, with reference to colorectal cancer dynamics, wi mustassume at least five values that must be strongly correlated to the ones of thevariable ui. For these reasons, we have decided to identify the domain Dw withthe close and bounded interval

Dw = {w2, ..., wi, ..., wN} = {w2, w3, w4, w5, w6},

where wi represents a general element belonging to Dw, and the index i rangesinside the set (2; N = 6) ⊂ N.

The state of each cell compartment and the one of the whole system, that isthe crypt, can be described, at time instant t, by the local number density np(t)and by the total number density n = n(t) of cells respectively. These functionsare defined by Eq. (3.2) and Eq. (3.1) with N = 6 and P = 2 and depend onthe discrete probability distribution function f(p,h) = f(p,h)(t) = fp(t, uh); thestate of immune cells within the crypt can be characterized in a similar way.

As a result, in order to derive the model equations through the general frame-works introduced in Section 3, we have to define the expressions of functions η,B, µ, γ and τ . Therefore, some preliminary considerations must be made aboutthe cellular phenomena introduced in the proposed model.

We divide cellular phenomena into physiological and pathological phenomena.The former involve normal cells, that are cells in the state u = u1; the latterinvolve both healthy (u = u1) and mutated (u > u1) cells as well as immune

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cells (w). Physiological and pathological phenomena can be further divided intoconservative and non–conservative ones according to the fact that they preserveor not the total number of cells within the colorectal crypt respectively.

Only non–conservative phenomena are taken into account into the presentmodel; they are summarized, together with their mathematical representations,in Table 1 and Table 2 and are described in the following.

Table 1: Cellular physiological phenomena under consideration and their mathematical rep-resentations.

Phenomenon Involved Cells Mathematical

Representation

Differentiation Normal γ(q 6=p,h=i)(p,i=1)

Renewal Normal γ(q=p,h=i)(p,i=1)

Programmed death Normal τ(p,i=1)

Some papers concerning the kinetic theory literature [14] treat prolifera-tive/destructive phenomena as non–conservative interactions by introducing acounter time field particle population whose probability distribution functionmaintains a constant value across time. In this paper, we will describe thesephenomena through a source/sink formalism, since it seems more consistentwith the specific biological reality we are dealing with. The model is based onthe assumptions which follow.

Encounters between cells - A cell within the colorectal crypt can encounter othercells during its life cycle. We assume that a test cell in the compartment p =(1, 2) at a stage of degeneration characterized by (u1 ≤ ui ≤ uN ) can encounterin T(p,i) days a field cell belonging to the compartment q = (1, 2) in the state(u1 ≤ uk ≤ uN ) or a field cell in the state (w1 ≤ wk ≤ wN ) of the populationq = (1, ..., Z). This kind of phenomena, that involve normal, cancer and immunecells, preserve the total number of cells within the single crypt.

We model the encounter rate with the discrete function η(p,i)(q,k) and, inorder to reduce computational complexity, we assume that the value of thisfunction is the same, being the field particle either a normal/cancer cell or animmune cell; so η(p,i)(q,k) = ηpq(ui, uk) = ηpq(ui, wk). To ensure biologicalconsistency to the mathematical way of modelling these phenomena, we assumethat the value of η(p,i)(q,k) defined for i = k is progressively reduced for |i−k| =1 and |i − k| = 2, and that it becomes equal to zero for |i − k| > 2. Infact, malignant cells at a further mutation stage proliferate to the detrimentof cells at the preceding ones, for example taking nutrients away quickly fromthem. Moreover, as cancer progression goes on, cell heterogeneity as well as the

14

Table 2: Cellular pathological phenomena under consideration and their mathematical repre-sentations.

Phenomenon Involved Cells Mathematical

Representation

Differentiation Cancer γ(q 6=p,h=i)(p,i>1)

Unbounded Proliferation Cancer γ(q=p,h=i)(p,i>1)

Stocastich DNA Normal γ(q=p,h=i+1)(p,i≥1)

somatic alteration Cancer

Programmed death Cancer τ(p,i>1)

Destruction due Normal µ(p,h=i)(p,i≥1)(p,k>i)

to cancer progression Cancer

Destruction due Cancer µ(p,h=i)(p,i>1)(q,k=i)

to immune competition Immune

complexity of the scenario in which interactions take place increases; therefore,the encounter rate decreases as much as the values assumed by the activityvariable related to the involved cells differ from each other.Hence, defining a(p,i) as a function assuming a positive constant value for fixedp and ui, η(p,i)(q,k) = ηpq(ui, uk) = ηpq(ui, wk) results as follows:

if p = (1, 2) ∧ (|i − k| = 0) :a(p,i)

T(p,i),

if p = (1, 2) ∧ (|i − k| = 1) :a(p,i)

5T(p,i),

if p = (1, 2) ∧ (|i − k| = 2) :a(p,i)

10T(p,i),

otherwise : 0.

In order to fix the value of a(p,i), a suitable mathematical model describingcellular encounter events must be derived.

Cellular Differentiation and Renewal - When physiological equilibrium is ob-served within the crypt, as well as when pathological conditions occur, stemand transit-amplifying cells can both renew or further differentiate. We as-sume that a normal or a cancer cell at the end of its life cycle divides into two

15

cells; according to the fact that cellular division takes place in a symmetric orasymmetric fashion, we assume that daughter cells are either both identical totheir mother or that one cell is identical to the mother while the other one be-longs to a different compartment. Hence, after T(p,i) days one cell in the state(u1 ≤ ui ≤ uN ) of the compartment p = (1, 2) has a finite probability to un-dergo mitosis and to be replaced by two cells, one identical to it and the otherone belonging to the state uh = ui of the compartment q = (1, 2), if p = 1, orq = (2, 3), if p = 2.

These phenomena do not preserve the total number of cells within the crypt

and they are described by the discrete function γ(q,h=i)(p,i) = γq

p(ui; uh = ui), which

represents both the renewal and differentiation rate of normal cells in the com-partment p = (1, 2), depending on the fact that p is set equal to or differentfrom q respectively.

If the condition ui = u1 holds, γ(q,h=i)(p,i) stands for both physiological differenti-

ation and renewal rate.On the other hand, if ui 6= u1, the same function models the differentiationrate of cancer cells as well as their unbounded proliferation rate. Clearly, dif-ferentiation and renewal rates of normal cells differ from the ones of mutatedcells. Indeed, mutated cells are able to bypass control mechanisms and to renewwith a frequency higher than normal cells. As a result, malignant cells acquirea growth advantage which favours the araising of neoplasia.

At this stage we neglect reverse differentiation events involving normal TACswhich convert to stem cells. Actually, the probability for these phenomena islower than the one related to the phenomena described above.

Stochastic DNA somatic alteration events - As a consequence of the processesmentioned in Section 2, cells within the crypt and their direct descendantscan accumulate unforeseeable mutations moving them through increasing ma-lignancy phenotypes. Division phenomena involving a normal or a cancer cellcan occasionally result into the generation of one or two daughter cells whosestate is different from that of their mother. For this reason, we can state thatafter T(p,i) days a cell in the state (u1 ≤ ui < uN ) of the compartment p = (1, 2)has a finite probability to undergo mitosis and to be replaced by two cells, oneidentical to it and the other one belonging to the state (u1 < uh = u(i+1) ≤ uN )of the compartment q = p.

We model the rate of these non–conservative phenomena through the discrete

function γ(q=p,h=i+1)(p,i≥1) = γq

p(ui ≥ u1; uh = u(i+1)). We describe in the same way

all mutation events, independently from the phase of the cell life cycle at whichthey take place. In fact, we assume a mutation as becoming effective when amutated cell divides and creates mutated descendants.

The non–conservative phenomena modelled by γ(q,h)(p,i) are visualized in Figure

7. Defining b(p,i), c(p,i) and d(p,i) as functions assuming a positive constantvalue for fixed p and ui, the analytical expressions for the not null terms of

16

Figure 7: Sketch of physiological and pathological non–conservative phenomena related to the

function γ(q,h)(p,i)

γ(q,h)(p,i) = γq

p(ui; uh) are:

if p, q = (1, 2) ∧ q = p ∧ (1 ≤ i = h ≤ N) :b(p,i)

T(p,i),

if p, q = (1, 2) ∧ q = p ∧ (1 ≤ i < h = i + 1 ≤ N) :c(p,i)

T(p,i),

if p = (1, 2) ∧ q = (2, 3) ∧ q 6= p ∧ (1 ≤ i = h ≤ N) :d(p,i)

T(p,i).

The domain of b(p,i), c(p,i) and d(p,i) can be defined on the basis of suitablebiological considerations.

As cancer progression develops, both genomic instability and proliferationability expressed by mutated cells rise; therefore, functions b(p,i) and c(p,i) as-sume greater values for increasing values of i. Moreover, if stable physiologicalequilibrium is observed within the crypt condition c(p,i) = 0 holds.

If d(p,i) and c(p,i) are defined as suitable stochastic functions of time, thatcan alternatively double or assume their fixed initial values again, also divisionphenomena resulting into the creation of two differentiated daughter cells or twomutated daughter cells, that sometime take place in the biological reality, can bemimicked by the proposed model respectively.

Programmed cell death - In order to maintain equilibrium within the crypt,programmed death of both stem and transit-amplifying cells occurs, every T(p,i)

days, at the end of cellular life cycle (Figure 8). The number of mutated cellsundergoing apoptosis decreases across cancer progression and it becomes equalto zero for cells at the two last mutation stages (p-53+ - and p-53- -). In fact,as it has been remarked in Section 2, mutated cells acquire the ability to eludemitotic control mechanism, such as programmed cell death. We associate toapoptotic phenomena a discrete function τ(p,i) = τp(ui), standing for the death

17

Figure 8: Sketch of physiological and pathological non–conservative phenomena related to thefunction τ(p,i)

rate of cells in the state (u1 ≤ ui < uN−1) of the compartment p = (1, 2), heredefined:

if p = (1, 2) ∧ (u1 ≤ ui < uN−1) :r(p,i)

T(p,i),

otherwise : 0.

r(p,i) is a function assuming a positive constant value for fixed p and ui. Theanalytical expression of this function for p = (1, 2) and ui = u1 descends fromimposing preservation of the numbers of stem and transit-amplifying cells withinthe crypt. The value assumed by r(p,i) decreases for increasing values of ui,becoming equal to zero for ui = uN−1 and ui = uN . Condition r(p,i) = 0 impliesthat cells in the state ui of compartment p are not involved into apoptosis events.

The following restriction applies to b(p,i), c(p,i), d(p,i) and r(p,i) for eachadmitted value of p and ui:

b(p,i) + c(p,i) + d(p,i) + r(p,i) = 1 ∀(p, i). (4.1)

Destruction events due to cancer progression - Cancer cells proliferate to thedetriment of both normal cells and other cancer cells at less advanced progres-sion stages. We take into account these non–conservative phenomena by statingthat binary interactions occurring during T(p,i) days between a test particle inthe state (u1 ≤ ui < uN) of the compartment p = (1, 2) and a field particlebelonging to the same compartment in the state (u1 ≤ ui < uk ≤ uN ) couldcause the destruction of the test particle leaving the state of the field one un-changed (Figure 9). If the test cell survives in the competition, it remains inthe unchanged state uh = ui of the same compartment p. The related discrete

function µ(p,h=i)(p,i≥1)(p,k>i) = µpp(ui ≥ u1, uk > ui; uh = ui) can be written in the

following form:

if p = (1, 2) ∧ (1 ≤ i = h < k ≤ N) :m(p,i)

T(p,i),

otherwise : 0,

18

Figure 9: Sketch of pathological non–conservative phenomena related to the function

µ(p,h=p)(p,i≥1)(p,k>i)

Figure 10: Sketch of pathological non–conservative phenomena related to the function

µ(q,h=i)(p,i>1)(q,k=i)

where m(p,i) is a function assuming a positive constant value for fixed p andui, whose domain can be defined through consistent biological assumptions.However, taking into account interactions between cells whose states ui and uk

are such that |i−k| > 2 would not be reasonable from a biological point of view.

Destruction events due to immune competition - As cancer progression devel-ops, competitive interactions between cancer cells and cells belonging to the hostimmune system occur. In terms of binary interactions between active parti-cles, we can say that a test particle in the state (u1 < ui = uh ≤ uN) ofthe compartment p = (1, 2) interacts with a field particle belonging to the im-mune population q = (1, ...Z) and expressing the defense ability described by(w2 ≤ wk ≤ wN ). Since immune cells exert their action selectively on cancercells expressing a specific antigen, immune competition events take place if andonly if ui = wk.

We assume that this kind of non–conservative interactions occurring duringT(p,i) days can lead to the death of the cancer cell at a rate represented by the

discrete function µ(p,h=i)(p,i>1)(q,k=i) = µp

pq(ui > u1, wk = ui; uh = ui).

The state of the surviving cell remains unchanged (Figure 10). The analytical

19

expression for µ(p,h=i)(p,i>1)(q,k=i) results as follows:

if p = (1, 2) ∧ q = (1, ..., Z) ∧ (1 < i = k = h ≤ N) :z(p,i)

T(p,i),

otherwise : 0,

where z(p,i) is a function assuming a positive constant value for fixed p and ui,whose domain definition depends on the properties of the biological system. Ascancer progression develops, immune action efficiency weakens, and thereforez(p,i) assumes lower values as ui increases and it can be equal to zero if thedefense action expressed by an immune cell in the state wk of the population q

has no effect on the cancer cell in the state ui of the compartment p, that is, ifthe condition ui = wk is not verified. On the other hand, as previously noticed,the condition z(p,i) = 0 also holds if the test particle under consideration is anormal cell, that is if ui = u1.

4.2. The Mathematical Model

All elements have been introduced to model the flow of cells under bothphysiological and pathological conditions, at time instant t, into the elemen-tary volume of the state space of compartment p = (1, 2) linked to the state(u1 ≤ uh ≤ uN). We take into account only non–conservative events, neglectinginteractions between both normal and cancer cells belonging to different com-partments and, at this stage, we leave out interactions between cancer cells andexternal agents; as a consequence, terms of Eq. (3.3) can be defined as follows:

F(p,h)[f ](t) = 0 (4.2)

C(p,h)[f ](t) = 0 (4.3)

P(p,h)[f ](t) = 0 (4.4)

D(p,h)[f ](t) = −

Z∑

q=1

η(p,i)(q,k)µ(p,h=i)(p,i>1)(q,k=i)f(p,i)f(q,k)

−

N∑

k=2

η(p,i)(p,k)µ(p,h=i)(p,i≥1)(p,k>i)f(p,i)f(p,k) (4.5)

S(p,h)[f ](t) =

2∑

q=1

N∑

i=1

γ(p,h)(q,i) f(q,i) − γ

(q 6=p,i)(p,h) f(p,h) − γ

(q,i6=h)(p,h) f(p,h) (4.6)

L(p,h)[f ](t) = −τ(p,h)f(p,h). (4.7)

According to Eq. (3.1) and Eq. (3.2) the total number density of cellsn = n(t) can be calculated after that a numerical expression for the probabilitydistribution function f(p,h) for p = (1, 2) and (u1 ≤ uh ≤ uN ) has been deliveredby the solution of the initial value problems generated by linking the followingset of differential equations, descending from substituting Eqs. (4.2), (4.3),

20

(4.4), (4.5), (4.6) and (4.7) into Eq. (3.3), to the initial conditions defined in(4.9):

df(p,h)

dt=

2∑

q=1

N∑

i=1

γ(p,h)(q,i) f(q,i) − γ

(q 6=p,i)(p,h) f(p,h) − γ

(q,i6=h)(p,h) f(p,h)

− τ(p,h)f(p,h)

−

Z∑

q=1

η(p,i)(q,k)µ(p,h=i)(p,i>1)(q,k=i)f(p,i)f(q,k)

−N

∑

k=2

η(p,i)(p,k)µ(p,h=i)(p,i≥1)(p,k>i)f(p,i)f(q,k), (4.8)

where f(p,h) = fp(t, uh),

if p = 1 ∧ h = 1 : f(p,h)(t = 0) = f0,

if p = 2 ∧ h = 1 : f(p,h)(t = 0) = g0,

otherwise : f(p,h)(t = 0) = 0,

(4.9)

where f0 and g0 are biologically coherent positive constant values.We can postulate that these values for f(p,h) are stable solutions of Eq. (4.8)when physiological equilibrium is observed within the crypt; therefore, the valueassumed by the constant function r(p,h) for p = (1, 2) and uh = u1, whichprovides an analytical expression for τ(p,h=1), can be determined by equatingthe right-hand side of Eq. (4.8) to zero and solving it with respect to τ(p,h=1).The resulting analytical expression for r(p,h) is here reported:

if p = 1 ∧ uh = u1 :b(p,h)

T(p,h)−

d(p,h)

T(p,h),

if p = 2, q = 1 ∧ uh = u1 :b(p,h)

T(p,h)−

d(p,h)

T(p,h)+ d(q,h)

f0

g0

T(p,h)

T(q,h),

where r(p,h) must also meet the requirements expressed by (4.1).The function f(q,k) = fq(t, wk) which appears in the third right-hand side

term of Eq. (4.8) stands for the discrete probability distribution function relatedto immune cells in the state (w2 ≤ wk ≤ wN ) of population q = (1, ..., Z).

Since we assume that for each new cancer cell at the malignant stage (u1 <

uk ≤ uN ) in the compartment q = (1, 2) a new immune cell expressing thedefense ability (w2 ≤ wh ≤ wN ) with wh = uk is created in the populationp = (1, ..., Z), and that an immune competition event can lead the cancer cellto die and the immune cell to survive, or vice versa, what results for immunecells is:

D(p,h)[f ](t) = −η(p,i)(q,k)

(

1 − µ(p,h=i)(p,i=k)(q,k>1)

)

f(p,i)f(q,k) (4.10)

S(p,h)[f ](t) =

2∑

l=1

2∑

q=1

γ(q,i=h)(l,i=h) f(l,i=h). (4.11)

21

We set terms Fp(t), C(p,h)[f ](t) and P(p,h)[f ](t) of Eq. (3.3) equal to zero forthe same reasons above mentioned for normal/cancer cells. Furthermore, sinceat this stage we are interested in cancer cell dynamics and the life cycle durationof immune cells is much longer than the one of normal/cancer cells, we do nottake into account programmed death events involving immune cells. As a result,term L(p,h)[f ](t) is equated to zero.

Hence, the analytical expression of f(p,h) = fp(t, wh) can be determined bysolving the initial value problem generated by linking the following differentialequation to the initial conditions defined in (4.9):

df(p,h)

dt=

2∑

l=1

2∑

q=1

γ(q,i=h)(l,i=h) f(l,i=h)

− η(p,i)(q,k)

(

1 − µ(p,h=i)(p,i=k)(q,k>1)

)

f(p,i)f(q,k). (4.12)

The action of just one immune cell population is included in the model, that isT-lymphocytes. As a result, we state Z = 1. Indeed, competition events betweenthe host immune system and cancer cells represent a quite delicate biologicalmatter and the only kind of well documented immune action is the one exertedby T-lymphocytes [2].

5. Parameters Choice and Simulations

The first part of this section is devoted to organize the biological reasoningleading to define the domain of the constant functions modelling η, γ, τ andµ. The second part, where the attention is focused on emerging phenomena,resumes the results of simulations obtained by solving numerically the initialvalue problems generated by linking Eq. (4.8) to the initial conditions definedby (4.9).

5.1. Assessment of the parameters of the Model

Before starting simulations, we have made some biological considerations andmathematical assumptions in order to define in a consistent way the domain ofthe functions representing the parameters of the model and to reduce compu-tational complexity too. The values assigned to these functions, depending oncompartment p and state uh, are summarized in Table 3.

Cellular cycle duration - According to the biological reality, we can state that theaverage time required by a stem cell to terminate a life cycle is 1 day, the doubleof that of a TAC, and we keep these values fixed for both of the compartments,independently from the mutation stage of cells, that is T(p,h) = T(p,h+1) forp = (1, 2) and (u1 ≤ uh < uN).

Initial values of cell local number density - The physiological average numbersof stem and TAC cells are 5 and 150 respectively. Therefore, we can state that

22

Table 3: Assumed rate per life cycle of phenomena described by the parameters of the proposedmodel.

p h T(p,h) a(p,h) b(p,h) c(p,h) d(p,h) r(p,h) m(p,h) z(p,h)

(%) (%) (%) (%) (%) (%) (%) (%)

1 1 1 100 49.99 0.02 30 19.99 10 0

1 2 1 10 69.88 0.02 30 0.1 0.1 20

1 3 1 10 69.88 0.02 30 0.1 0.1 20

1 4 1 10 69.97 0.02 30 0.01 0.01 0.2

1 5 1 10 69.98 0.02 30 0 0 0.02

1 6 1 10 70 0 30 0 0 0.02

2 1 0.5 8 49.74 0.02 40 10.24 10 0

2 2 0.5 0.8 59.88 0.02 40 0.1 0.1 20

2 3 0.5 0.8 59.88 0.02 40 0.1 0.1 20

2 4 0.5 0.8 59.97 0.02 40 0.01 0.01 0.2

2 5 0.5 0.8 59.98 0.02 40 0 0 0.02

2 6 0.5 0.8 60 0 40 0 0 0.02

initial conditions defined by (4.9) result as follows:

if p = 1 ∧ h = 1 : f(p,h)(t = 0) = f0 = 5,

if p = 2 ∧ h = 1 : f(p,h)(t = 0) = g0 = 150,

otherwise : f(p,h)(t = 0) = 0.

(5.1)

Encounter rate - At this stage we look at cells as spheres, and approach to thedefinition of the encounter rate values in physiological conditions as a spherepacking problem. Hence, assuming the initial numbers of stem cells and TACsas defined by (4.9), we fix the values of a(p,h=1) for p = (1, 2). Moreover, ascancer progression develops, the increasing of both cell number and cellularheterogeneity leads to the condition a(p,h+1) ≤ a(p,h) for p = (1, 2) and (u1 ≤uh < uN).

Renewal, differentiation, mutation and programmed death rates - After defin-ing the values of renewal and differentiation rate for normal cells according tobiological reality, the value of the programmed death rate is fixed through the

23

expression of r(p,h) for p = (1, 2) and uh = u1 introduced in Section 4. Underpathological conditions cells belonging to both compartments acquire a finiteprobability to undergo mutations and mutated cells acquire the ability to re-new with a higher frequency and to elude programmed cell death; as a result,biological coherence in conjunction with condition expressed by (4.1) requiresthat b(p,h) increases, while r(p,h) decreases across cancer progression, that is,b(p,h+1) ≥ b(p,h) and r(p,h+1) ≤ r(p,h) for p = (1, 2) and (u1 ≤ uh < uN).

We state that the renewal rate of cells at a given progression stage is muchhigher than their mutation rate for both compartments, that is b(p,h) � c(p,h)

for p = (1, 2) and (u1 ≤ uh ≤ uN).In order to reduce computational complexity we assume that the value of

d(p,h) defined under physiological conditions is not altered under pathologicalconditions (d(p,h+1) = d(p,h) for p = (1, 2) and (u1 ≤ uh < uN )). Moreover, weassign to c(p,h) a constant value greater than zero for uh > u1, which is equalfor both p = 1 and p = 2.

Destruction rates - We fix the values of m(p,h) equal for both of the cell com-partments (m(p,h) = m(q,h), for p, q = (1, 2) and (u1 ≤ uh ≤ uN)). Moreover,according to the biological reality, we can state that for normal cells as well asfor cells at the two first malignant stages the destruction rate related to theinteractions with cells at a further level of malignancy is higher than the mu-tation rate, even if it is lower than the renewal one (c(p,h) < m(p,h) < b(p,h) forp = (1, 2) and (u1 ≤ uh ≤ u3)).

The destruction rates related to immune competition events for cells at thetwo first malignant stages are set on the same order of magnitude of the prolif-eration rate (z(p,h)

∼= b(p,h) for p = (1, 2) and (u2 ≤ uh ≤ u3)).Furthermore, destruction rates are assumed to decrease across cancer pro-

gression(m(p,h+1) ≤ m(p,h) for p = (1, 2) and (u1 ≤ uh < uN), z(p,h+1) ≤ z(p,h) forp = (1, 2) and (u2 ≤ uh < uN)).

Finally, we want to stress out that, bearing in mind the above outlinedconsiderations, the values of the model parameters have been selected in orderto allow a new mutation to start when the number of cells at given mutationstage has reached a value near to or greater than 106 [2]. On the other hand,assuming that the average time required by the genesis of colorectal cancer toreach its final stage is about 40 years [2], in order to reduce the time required bythe development of simulations, values reported in Table 3 represent a suitablenormalization of the original ones so that the mutation process ends by about100 days.

5.2. Simulations

We are convinced that the usefulness of a mathematical-biological modeldepends on its ability to show emerging phenomena. With this aim, we havenumerically solved the mathematical problem generated by linking Eq. (4.8) tothe initial conditions fixed by (5.1) varying the values of the model parameters

24

10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

x 108

Time (days)

Num

ber

of C

ells

f1

f2

f3

f4

f5

f6

Figure 11: Dynamics of the whole crypt with z(p,h) = 0% for p = (1, 2) and (u1 < uh ≤ uN )

summarized in Table 3. The most relevant results are summarized in Table 4,Figure 11 and Figure 12, and they are here briefly analyzed in a critical way.

We have first solved the mathematical problem defined by Eq. (4.8) linked toinitial conditions defined by (5.1) setting the parameters of the model accordingto Table 3, but assuming the immune system action as not effective (z(p,h) = 0for p = (1, 2) and (u1 ≤ uh ≤ uN)). We have found out that heterogenity incarcinogenesis at colon and rectum descends from the simultaneous presenceof cells at different differentiation stages, if the same probability to undergomalignant mutations is accorded to them (c(p,h) = c(q 6=p,h), for p, q = (1, 2) and(u1 ≤ uh ≤ uN)).

In fact, even if cells at different mutation stages coexist within the samecompartment during the transition phase between the occurrence of two subse-quent mutations, the presence of different compartments causes to find in thecrypt cells at additional mutation stages.

In other words, the dynamics of the two compartments superimpose overeach other and the heterogeneity of the scenario of cellular interactions increases(Figure 11). Subsequently, we have done some simulations varying the mutationrate of stem cells and TACs. The obtained results show that the situation abovedescribed can be altered by assigning to one cell compartment a higher mutationrate (about two order of magnitude) than that of the other one (c(p,h) � c(q 6=p,h),for p, q = (1, 2) and (u1 ≤ uh ≤ uN)). Indeed, in such a way the dynamics ofcarcinogenesis within the crypt is led by the cell compartment characterizedby the higher mutation rate and the effect of the other one becomes negligible

25

(Table 4).

Table 4: Effects on system dynamics of mutation rate.

c(1,h) � c(2,h) System dynamics is dominated by the one of stem cells

c(1,h) � c(2,h)System dynamics is dominated by the one of

transit-amplifying cells

c(1,h) = c(2,h)Stem and transit-amplifying cell dynamics superimpose

each other

Since the number of transit amplifying cells is much greater than the oneof stem cells, it seems natural that the system dynamics will be dominatedby the one of TAC compartment if a mutation rate higher than that of stemcell compartment is assigned to TACs (c(1,1) � c(2,1)). On the other hand,in spite of the small number of cells in the stem compartment, defining forthem a higher mutation rate than the one defined for transit-amplifying cells(c(1,1) � c(2,1)), simulation results stress out how mutations involving stem cellsplay a more important role in tumour progression than the ones undergone bytransit-amplifying cells. This is because mutated stem cells express a higherproliferative potential than that of the other cells within the crypt.

Then, we have developed further simulations setting z(p,h) for p = (1, 2) and(u1 ≤ uh ≤ uN) according to Table 3. The obtained results show that the trendsof curves describing the evolution across time of cell number within the cryptare kept unchanged, even if both the unbounded proliferation and the mutationprocess slow down.

Finally, we have varied the immune destruction rate looking for a thresholdvalue allowing the rejection of cancer cells from the crypt. We have foundout that such a value exists and that it is z(p,h) = 70% for p = (1, 2) and(u1 ≤ uh ≤ uN ). Under this condition the number of cancer cells tends to zeroacross time and even if the mutation and destruction processes involving normalcells are not stopped, actually c(p,h) and m(p,h) remain different from zero forp = (1, 2) and (u1 ≤ uh < uN ), but at least they slow down (Figure 12). Aswe could expect, the number of T-lymphocytes in the state (w2 < wh ≤ wN )increases until it reaches the stable equilibrium value that allows, under thefixed immune efficiency level, the complete destruction of cancer cells at therelated progression stages (u2 < uh ≤ uN ).

On the other hand, since the mutation rate of normal cells (uh = u1) remainsdifferent from zero, the number of cells at the first mutation stage (uh = u2)would increase linearly. As a result, in order to reach and to maintain a nullnumber of cells in the state uh = u2, the number of immune cells in the statewh = w2 must increase linearly across time.

The required threshold level for the destruction rate of cancer cells can bereached by joining the action exerted by T-lymphocytes and the one expressedby drugs able to act in a selective way against mutated cells.

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10 20 30 40 50 60 70 80 90 1000

50

100

150

Time (days)

Num

ber

of C

ells

f1

f2

f3

f4

f5

f6

Figure 12: Dynamics of the whole crypt with z(p,h) = 70% for p = (1, 2) and (u1 < uh ≤ uN )

6. Closure

A model of cell dynamics within a single crypt during carcinogenesis un-der spatial homogeneity assumption has been developed. The mathematicalframeworks offered by the mathematical kinetic theory for active particles havebeen employed, focusing on progression and heterogeneity aspects; phenomenain which cells are involved have been modelled by a stochastic game approach.

Flexibility is one of the distinctivenesses of the present model. It can beapplied to tumours occurring within different human body areas. Indeed, itwill be enough to vary the reference biological frameworks, and so to consider adifferent cell arrangement and to define different activity domains Du and Dw.

Some additional efforts to make the description of colorectal carcinogenesisprovided by the proposed model more consistent with the biological reality canbe proposed.The first one can be represented by the removal of spatial homogeneity assump-tion and by the introduction of a more advanced model for the definition of thefunction describing the encounter rate. Furthermore, immune system dynamicscan be made more complex, for example by involving immune cells in a peculiarlife cycle and including ageing effects.

The results of simulations suggest the fulfillment of our main purposes.The crucial role played by the mutation rate in regulating the heterogeneity ofcolorectal cancer dynamics has been stressed out. The strict relation betweenheterogeneity and simultaneous presence of cells at different stage of differenti-

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ation within the crypt has been shown. Besides, the important role carried bymutation involving stem cells in tumour progression has been pointed out.

The proposed model leads to a satisfactory qualitative description. In par-ticular, it has been shown that if the destruction rate of cancer cells is increasedto reach a given threshold value within the patient suffering from colorectalcancer, for example through the inoculation of suitable drugs acting selectivelyagainst malignant cells and joining the host immune system action expressedby T-lymphocytes, mutated cells will be rejected from the crypts and the pro-gressive reduction of the normal cell number will be slowed down even if the cellmutation rate still remains different from zero.

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