Authors Accepted Manuscript

What can be learned from a chaotic cancer model?

C. Letellier, F. Denis, L.A. Aguirre

PII: S0022-5193(13)00022-2DOI: http://dx.doi.org/10.1016/j.jtbi.2013.01.003Reference: YJTBI7193

To appear in: Journal of Theoretical Biology

Received date: 17 March 2012Revised date: 20 November 2012Accepted date: 5 January 2013

Cite this article as: C. Letellier, F. Denis and L.A. Aguirre, What can be learned from achaotic cancer model?, Journal of Theoretical Biology, http://dx.doi.org/10.1016/j.jtbi.2013.01.003

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What can be learned from a chaotic cancer model?

Christophe Letellier

CORIA Universite de Rouen, Av. de lUniversite, BP 12, F-76801 Saint-Etienne du

Rouvray cedex, France

Fabrice Denis

Centre Jean Bernard, 9 rue Beauverger, 72000 Le Mans, France

Luis A. Aguirre

Universidade Federeal de Minas Gerais Av. Antonio Carlos 6627, 31270-901 BeloHorizonte MG, Brazil

Abstract

A simple model of three competing cell populations (host, immune and tumorcells) is revisited by using a topological analysis and computing observabilitycoecients. Our aim is to show that a non conventional analysis might sug-gest new trends in understanding the interactions of some tumor cells andtheir environment. The action of some parameter values on the resultingdynamics is investigated. Our results are related to some clinical features,suggesting that this model thus captures relevant phenomena to cell interac-tions.

Keywords: Cancer model, Chaos, Observability, Therapy

1. Introduction

Cancer results from a proliferation of tumor cells which invade nearbyparts of the body. Tumors are microcosms of evolution with a mosaic ofmutant cells that compete for space and resources, evade predation by theimmune system and can even cooperate to disperse and colonize new organs

Corresponding AuthorEmail address: Christophe.Letellier@coria.fr (Christophe Letellier)

Preprint submitted to Journal of Theoretical Biology January 9, 2013

[1]. Unfortunately, interactions between cell populations like host cells, ef-fector immune cells, tumor cells, endothelial cells, or others, are not wellunderstood and the key parameters remain to be identied. Interactionsbetween dierent cell populations in neoplasm not only induce genetic un-stability of tumor cells but also unpredictable tumor growth [2]. Due to this,carcinogenesis population-based models may be used to better understandtumor dynamics and responses to treatment. Competition and predationbetween dierent types of cells in tumor exist in the form of resource con-sumption or in the interaction of neoplastic cells with the immune system,for instance [1]. Neoplastic cells can indeed have direct negative eects oneach other or on immune and stromal cells through soluble factors that maybe targeted by non tumor cell treatment [3, 4]. Carcinogenesis models basedon Lotka-Volterra competition equations dene conditions under which can-cerous cells might be driven up to extinction [5]. By reducing the negativecompetitive eects of cancer cells on normal cells and increasing the negativecompetitive eects of normal cells on cancer cells, the number of cancer cellswhich can be supported in a tissue can be reduced [6]. Such models canhelp to identify the parameters which should be targeted by therapies andto design the most eective strategy for drug treatment regimes [7, 8].

Lotka-Volterra equations are also known to produce chaotic behaviorsin the case of prey-predator model [9, 10] as well as for competing species[11, 12]. Among the models based on Lotka-Volterra equations, a modeldescribing the interactions between three cell populations, one being tumorcells, produces chaotic behaviour [13]. Its parameter values were chosen tomatch with some biological evidences [14]. This model could be thus con-sidered as being qualitatively validated with experimental data. Identifyingaccurately the parameter values of such a model remains an open problemsince there are rather dicult to deduce from biological experiments.

This paper investigates the model proposed in [14], which describes theinteractions among three cell populations: host cells, eector immune cellsand tumor cells. This model was chosen for its chaotic dynamics [13] whichhas numerous analogies to clinical evidences [15, 16]. It will be shown that anextended dynamical analysis can provide insights about cellular interactionsin agreement with some experimental and clinical evidences. The subsequentpart of this paper is organized as follows. Section 2 briey describes the can-cer model and how the parameter values were chosen for the analysis. Fixedpoints of the model are also discussed for these parameter values. The deter-mination of the best variable to measure for investigating the underlying

2

dynamics is discussed in Section 3. A dynamical analysis is performed in Sec-tion 4. Bifurcation diagrams are described and a template which synthetizesthe topological structure of the chaotic attractor is provided. Section 5 givesa discussion and some conclusions.

2. Model equations and xed points

Let us consider a simple model for tumor growth based on Lotka-Volterraequations (see [17, 14] for instance). This model describes the interactionsthat take place in a single tumor-site compartment, between host cells H ,eector immune cells E and tumor cells T . This model is devoted to asite in the neighborhood of a homogeneous tumor. The interactions of theimmune cells with the tumor cells are described as in [17]. Since the natureof the growth law terms does not aect signicantly the tumor growth, thelogistic growth term was retained [14]. This model is therefore a qualitativemodel. In spite of this, this model can describe some important aspects ofthe interactions between the three cell populations: H , E and T . The modelis [14]:

H = 1H

(1 H

1

) 13TH

E =2TE

T + 2 23TE 2E

T = 3T

(1 T

3

) 31TH 32TE ,

(1)

where the rst equation corresponds to the rate of change of host cells growingaccording to a logistic function with a biotic capacity 1. It is here supposedthat tumor cells proliferate more quickly than host cells, a feature obtainedusing 3 > 1. Tumor cells inhibit host cells at the rate 13. The secondequation corresponds to the rate of change of eector immune cells wherethe rst term represents the stimulation of the immunatory system by tumorcells with antigens specic to tumor cells. The rate of recognition of tumorcells by the immune system depends on the antigenicity of tumor cells. Sincethis recognition process is very complex, it is supposed to keep the modelquite simple, that the stimulation of the immune system directly dependson the number of tumor cells with positive constant 2 and 2. Eectorimmune cells are inactivated by tumor cells at the rate 23: they naturallydie at the rate 2. The third equation provides the growth rate of tumor

3

cells. The rst term has the form of a logistic function which governs thetumor cells when they are alone: the growth rate is thus 3 and the bioticcapacity 3. The competition between host cells and tumor cells is describedby a degradation of tumor cells according to the term 31TH . Tumor cellsare killed by eector immune cells at the rate 32. All parameters of thecancer model (1) are supposed to be positive.

The cancer model (1) only diers from the one proposed by de Pillis& Radunskaya [14] by the fact that it does not have a constant inux ofeector immune cells. This was justied by Itik and Banks [13] as follows.It was supposed that eector immune cells are cytotoxic T-cells produced asnaive cells, that is, as not being able to produce any response to tumorcells unless they are activated by antigen presenting cells via some majorhistocompatibility complex MHC-I (found in all nucleated cells, that is, notred blood cells) and HC-II (only found in antigen presenting cells) pathwaysin the presence of tumor specic antigens. Neglecting the constant inux ofeector immune cells was also proposed by Kirschner and Panetta [18, 19].

After renormalization of variables H , E, and T , system (1) can be rewrit-ten in the form

x = 1x(1 x) 13xz

y =2yz

1 + z 23yz 2y

z = z(1 z) xz 32yz,(2)

where we have the correspondence (x, y, z) (H,E, T ), that is, x designatesthe normalized population of host cells, y the population of the eector im-mune cells, and z the tumor cells. In this study, 3 and 31 are set to 1,as in [13]. The set of equations in (2) denes a model for cells in compe-tition as can be seen on the uence graph shown in Fig. 1. Contrary towhat is commonly observed in prey-predator models, there is no particularprey. None is promoting the growth of another one and all the interactionsbetween the dierent populations are repressing. This dynamics thus corre-sponds to highly competitive species. There is no interaction between hostand eector immune cells. Consequently, tumor cells can be viewed as gen-eralist competitors while the two others are specialist competitors, to usea terminology often used in ecology.

4

xhost cells

y z

Tumor cellsEffectorimmune cells

+

++

Figure 1: Fluence graph for the cancer model (2). Solid (dashed) lines represents linear(nonlinear) interactions. The sign describes if the interactions promote (+) or repress (-)the growth of the species.

System (2) will be investigated for the following set of parameters:

1 = 0.6 growth rate of host cells

13 = 1.5 host cell killing rate by tumor cells

2 = 4.5 growth rate of eector immune cells

23 = 0.2 eector immune cell inhibiting rateby tumor cells

2 = 0.5 eector immune cell mortality

32 = 2.5 eector immune cell killing rateby tumor cells ,

(3)

when not specied otherwise. These values were considered by Itik and Banks[13] for producing a chaotic attractor (Fig. 2) which is structured aroundve of the seven real xed points

S0 =

x0 = 0

y0 = 0

z0 = 0 ,

S1 =

x1 = 1

y1 = 0

z1 = 0 ,

S2 =

x2 = 0

y2 = 0

z2 = 1 ,

S3 =

x3 = 0

y3 = 0.347000

z3 = 0.132503 ,

5

S4 =

x4 = 0.668743

y4 = 0.079502

z4 = 0.132503 ,

S5 =

x5 = 0

y5 = 7.147000z5 = 18.867497

and S6 =

x6 = 46.168742y6 = 11.320498

z6 = 18.867497 .

Two of them (S5 and S6) are irrelevant for the resulting dynamics sincethey have negative coordinates (negative populations are not dened and,consequently, the dynamics must take place in the positive octant). The xedpoint at the origin corresponds to a situtation where there is no cell at all.Its eigenvalues are 0 = (0.6,0.5, 1), thus corresponding to a saddle. Thismeans that when x = z = 0, the eector immune would quickly disappearaccording to the negative eigenvalue. With x = z = 0, the cancer model (2)reduces to

x = 0

y = 2yz = 0 .

(4)

As observed in dynamics of competing species, the boundaries of the posi-tive octant cannot be crossed by the trajectory. Moreover, the attractor isconned in a triedron having the origin asapex. For an extended analysisof such constraints in another model for species in competition, see [20].

Fixed point S1 has eigenvalues 1 = (0.6,0.5, 0): this is therefore amarginally stable node. When y = z = 0, that is, when there are onlyhost cells, the dynamics is governed by a logistic equation (as introduced byVerhulst [21]) and the population converges to its maximum value (x = 1).Point S1 is not distinguished from S0 in the y-z plane projection shown inFig. 2.

Fixed point S2 corresponds to a situation where only tumor cells arepresent. The eigenvalues are 2 = (0.9, 1.55,1). The point is thus of asaddle type. By setting x = y = 0 in system (2), the dynamics is reduced to

x = 0

y = 0

z = z(1 z) ,(5)

that is, tumor cells are governed by a logistic function and the populationconverges toward z = 1.

6

0 0,2 0,4 0,6 0,8 1Tumor cells z

0

0,2

0,4

0,6

0,8

1

Effe

ctor

imm

une

cells

y

S2

S3

S0-S1

S4

Figure 2: Chaotic attractor solution to the cancer model (2) plotted with the xed pointsand some of their eigenvectors. Parameter values as in Eq. (3).

The xed point S3 is associated with a situation where only eector im-mune and tumor cells co-exist. This xed point can be considered as beingnot surrounded by the attractor. Its eigenvalues 3 = (0.0660.613i, 0.401)correspond to a saddle-focus. The stable manifold of this xed point belongsto the boundary of the attraction basin of the chaotic attractor. Point S4is associated with the co-existence of the three type of cells; its eigenvalues4 = (0.0403 0.235i,0.614) also dene a saddle-focus.

The xed point S4, which is surrounded by the attractor, has a 2D unsta-ble manifold, in that respect, point S4 diers from point S3 which has a 1Dunstable manifold. It is interesting to notice that the local dinamics aroundS4 display oscillations with natural frequency f = 0.235/2 = 0.037 Hz,which is very close to the pseudo-period of the underlying dynamics associ-ated with a frequency f = 0.039 Hz. The two saddle-foci S3 and S4 structurethe chaotic attractor as the Rossler attractor is structured by two saddle-foci[22]. Such a conguration has a deep impact on the dynamics which will befavourably compared to the Rossler attractor as described below.

3. Observability Coecients

When a single scalar time series s = h(x) from a dynamical systemx = f(x) with x Rm is measured, it is not guaranteed that any statex Rm can be reconstructed (observed) in the space Rm spanned by the

7

reconstructed coordinates X Rn (n m) which are typically delay orderivative coordinates [23]. The determination of whether the system is ob-servable (in that case, any state x Rm can be unambigously reconstructedfrom Rn(X)) is based on the the observability matrix

Os(x) =[

L0fh(x)x

L1fh(x)x

. . .dLn1fds

(x)

]T. (6)

where

Ljfh(x) =Lj1f h(x)

x f(x) , (7)

and where L0fh(x) = h(x). The system is said to be fully observable if OTOis full rank. The observability matrix corresponds to the Jacobian matrixof the coordinate transformation between the original state space Rm(x) andthe reconstructed space Rn(X) when derivative coordinates are used [24].Typically the singular set associated with states x Rm which cannot bereconstructed is dened by Det[OTs Os] = 0. When this determinant is dif-ferent from zero for all x Rm(x), the system is said to be observable andthere is a global dieomorphism between the original state space Rm(x) andthe reconstructed state space Rn(X). To avoid a yes-or-no classication ofobservability, the following coecient can be estimated for monovariable em-beddings [24, 25]

s(x) =| min[OTs Os, x(t)] || max[OTs Os, x(t)] |

, (8)

where max[OTs Os, x(t)] indicates the maximum eigenvalue of matrix OTs Osestimated at point x(t) (likewise for min) and ()T indicates the transpose.Then 0 s(x) 1, and the lower bound is reached when the system isunobservable at point x. The observability coecients can be computed andaveraged along an orbit thus

s =1

T

Tt=0

sx(t), (9)

where T is the nal time considered. The case of multivariate embeddingswas investigated in [26]. Some symbolic observability coecients were alsointroduced [27]. A procedure which only requires data has been put forwardin [28].

8

For instance, when applied to the Rossler system [29]

x = y zy = x+ ay

z = b+ z(x c) ,(10)

the symbolic observability coecients 2x = 0.88, 2y = 1.00, and

2z = 0.44.

These coecients mean that variable-y provides a global dieomorphism be-tween the original R3(x, y, z) and reconstructed R3(X = y, Y = y, Z = y)spaces, that is, the best observability which can be expected. Contraryto this, variable z is a rather poor variable (2z = 0.44) in what concernsobservability and many diculties are encountered when dynamical analy-ses are performed using this variable [30]. Dierential embeddings inducedby these three variables provide intuitive insight for such features (Fig. 3).Thus, the dierential embedding induced by the poorest variable z presentsa small domain near the plane z = 0 where all the dynamics is squeezed andwhere distinguishing dierent states or dierent trajectories is rather di-cult. There is no such a feature in the dierential embedding induced by thetwo other variables. A squeezed region in the reconstructed embedding isthus a signature of poor of observability [28].

-4 -2 0 2 4 6X=x

-4

-2

0

2

4

Y

-6 -4 -2 0 2X=y

-4

-2

0

2

4

Y

0 1 2 3 4 5 6X=z

-6

-4

-2

0

2

4

6

Y

(a) Variable x (b) Variable y (c) Variable z

Figure 3: Dierential embeddings induced by the three variables of the Rossler system.Parameter values: a = 0.398, b = 2, and c = 4.

In the case of the cancer model (2), the dierential embedding inducedby each of the three variables are shown in Fig. 4. The embedding inducedby variable x is the single one which does not show any strongly squeezeddomain. For instance, in the lower left part of the y-induced embedding,

9

all revolutions on the attractor pass through a squeezed region where it isquite dicult to distinguish them from the others. There is a similar domainlocated in the upper left part of the z-induced embedding (Fig. 4c). Fromthese embeddings, it appears that variable x presents the best observable ofthe dynamics.

0 0,2 0,4 0,6 0,8Host cells x

-0,1

-0,05

0

0,05

0,1

dx /

dt

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8Cellules immunitaires effectrices y

-0,2

-0,1

0

0,1

0,2

0,3

0,4

dy /d

t

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7Cellules tumorales z

-0,3

-0,2

-0,1

0

0,1

dz / d

t

(a) Host Cells (b) Eector immune cells (c) Tumor cells

Figure 4: Dierential embedding induced by each of the three variables of the cancermodel (2). Parameter values: 1 = 0.518 and as in Eq. (3) otherwise.

Computing the observability coecients as dened in Eq. (9) wherethe underline has been omitted leads to

x = 0.021 > z = 0.015 > y = 0.001 ,

thus conrming that variable x is the best observable. Variable y is thepoorest. The main message of this analysis is that the dynamics of the cancermodel is observed with a greater reliability from the population of host cells(variable x) rather than from eector immune or tumor cells (variable y andz, respectively). Tumor cells provide an observability of the dynamics that isonly slightly less than that provided by host cells, but eector immune cellsare associated with a very poor observability (y is smaller than x and z byone order of magnitude).

From the therapy point of view, and according to the interplay betweenthe observability coecients and the quality of synchronization [31] or ofcontrol [30], these results suggest to control the dynamics by acting on thepopulation of host cells. Acting on the population of tumor cells bydestroying them for instance is also ecient, but trying an action on theeector immune cells (e.g. cancer vaccine) would be rather inecient asconrmed by [32]. For instance, 82 patients with advanced prostate cancer

10

who received a vaccine did not dier much in terms of time-to-progression(p = 0.052) from 45 patients who received a placebo [33]. A slight benetin survival (25.9 months with vaccine versus 21.4 months with placebo, p =0.010) was observed in that study. But this was not satistically signicant (19months with vaccine versus 15.7 months with placebo) in another study [34].The relative irrelevance of the contribution of immune cells in the dynamicsof this model will be also conrmed in the dynamical study discussed in thenext section.

Let us now assume that we are able to observe simultaneously two pop-ulations among the three considered in this cancer model. We have thusthree dierent possibilities: to measure i) x and y, ii) y and z, and iii) x andz. For each case, there are two possibilities to produce a three dimensionalreconstructed space: adding the rst derivative of the rst i) or of the secondpopulation ii) observed. Using the procedure for multivariate observabilityanalysis discussed in [26], we obtained

R3(x, x, y): x2y = 0.29; R3(x, y, y): y2x = 0.08; R3(x, z, z): z2x = 0.08. R3(y, z, z): z2y = 0.01; R3(x, x, z): x2z = 0.00 (non observable); R3(y, y, z): y2z = 0.00 (non observable).

According to these observability coecients, observing the tumor cells withanother population whose rst derivative is used as a third variable does notprovide a reliable reconstructed space (y2z = x2z = 0), that is, a space whereany state is easily observed. When the evolution of tumor cells is derivatedonce and the population of eective immune cells is observed is added doesnot improve too much the observability (z2y = 0.01). Contrary to this, theobservability is slightly better when the reconstructed space R3(x, z, z) is con-sidered (z2x = 0.08). The best situation is to observe host cells with eectorimmune cells and to derive variable x once. Roughly, such a derivation al-lows to get information about tumor cells and the information about eectorimmune cells is directly obtained by measurements. Derivating variable y isfar less ecient due to the low observability associated with that variable.

11

Working with the reconstructed space R3(x, x, y) would be the best strategywhen two variables are recorded. From that point of view, acting simultane-ously on the host cells and the eector immune cells could be, at least froma dynamical point of view, a possible therapy to reduce the population oftumor cells.

4. A dynamical analysis

4.1. Bifurcation analysis

Since only xed points were investigated in [14] and a single set of pa-rameter values was considered in [13], we choose to start our analysis byvarying one of the parameter of the cancer model. From the observabilitycoecients analysis, acting on the host cells appears to be the best strategyto act on the cancer dynamics. We thus started by varying the growth rate1 of the host cells. The so-obtained bifurcation diagram (Fig. 5a) startswith a period-1 limit cycle. It is followed by a period-doubling cascade. Thechaotic attractor observed beyond the accumulation point (at which the or-bit 2 occurs) must therefore be characterized by a smooth unimodal mapsince a period-doubling cascade is the universal route to chaos in such a map[35, 36]. Many periodic windows can then be observed. The uctuationrange of the population of host cells (Fig. 5a) and tumor cells (Fig. 5b) in-creases when the host cell growth rate is increased up to 1 0.54: betweenthe period-1 limit cycle and this largest attractor, many periodic orbits werecreated. Beyond this value, the pruning of periodic orbits dominates overthe creation of periodic orbits and the uctuation range of the populationsdecreases. This is the signature of an antimonotonicity [37] which requiresthe occurence of at least a third branch in the rst-return map. A simplesequence of reverse bifurcations as observed in bubbling [38] would lead toa completely dierent bifurcation diagram.

To be correctly interpreted from the population point of view, the bifur-cation diagram has to be computed in a slightly dierent way than commonlydone. Indeed, the required information about a population is twofold: wewould like to know its range of variability, that is, its smallest and largestvalues and whether the extrema of the oscillations of the population studiedcan uctuate over this range. Consequently, for computing the bifurcationdiagram, the minimal and maximal values at each oscillations were retained.Note that the dynamical regimes are the same in both original and modiedbifurcation diagrams in spite of dierent ordinates used to compute them: the

12

0,4 0,5 0,6 0,7 0,8 0,9 1Host cell growth rate 1

0,5

0,6

0,7

0,8

0,9

1

Hos

t cel

ls x n

(a) Common bifurcation diagram: host cells versus 1

0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4Host cell growth rate 1

0

0,2

0,4

0,6

0,8

1

Hos

t cel

ls x n

(b) Modied bifurcation diagram: host cells versus 1

0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4Host cell growth rate 1

0

0,2

0,4

0,6

0,8

1

Tum

or c

ells

z n

(c) Modied bifurcation diagram: tumor cells versus 1

Figure 5: Bifurcation diagrams of the cancer model (2) versus the host cell growth rate.Other parameter values as in Eq. (3).

13

modied bifurcation diagram is only easier to interpret from the uctuationsof populations point of view. The modied bifurcation diagrams (Fig. 5b)reveals that when the growth rate of host cells is increased, the population ofhost cells can take larger maximal values (near to 1) but the correspondingminima becomes very low (near to 0). In fact, the cycle-to-cycle variabilityis reduced by increasing the growth rate of host cells in the sense that thepopulation of host cells remains near its maximum value for a quite longperiod but can decrease very quickly to zero (Fig. 6b). Compare to whatis observed for moderate growth rate 1 values (Fig. 6a), the oscillations aremore rare (host cells remain at their maximum for about 1000 units of time,that is, far longer than with a moderate 1-value for which each oscillationtakes about 3 or 4 units of time).

Thus, when the host cell growth rate is large enough, host cells becomesuciently strong to resist to the tumor cells for a certain period of time.Unfortunately, there is still a possibility that tumor cells proliferate: whenthis happens, this is very quickly and almost all host cells are destroyed,thus leading to a deleterious dynamics. A strategy which only stimulates thegrowth of host cells would turn rst to an apparent good status but wouldalso put the patient into conditions for presenting a fast tumoral growth,as biologically observed when tumor and host cells expressed similar growthfactor receptors such as EGFR (epithelial growth factor receptor). Actingpreferentially on host cells rather than on tumor cells could be an interestingstrategy as it can be observed sometimes in radiotherapy [39] and in inter-actions between cancer and host cells [40, 41]. Acting on host cells has alsoa natural interest because they mutate far less often than tumor cells. Nev-ertheless, another action should be applied to better control the growth oftumor cells. This would suggest a combined therapy.

More commonly, the population of host cells increases with the corre-sponding growth rate. For relatively low growth rates, the development ofthe host cells contributes also to develop tumor cells, a fact which is knownfrom biological data [40]. One can remark that tumor cells are developedwith an increase of the amplitude of the oscillations presented by all thepopulations. These uctuations result from the strong competition betweenthe two populations, when one is large, the other is small, and vice versa(Figs. 6). Tumor cells thus destabilize patients health by inducing someuctuations in the population of host cells. When the growth rate is largeenough, host cells hinder the tumor cells from becoming too numerous for aquite long period of time. Fluctuations are thus more rare and the popula-

14

tion of tumor cells decreases (right part of the bifurcation diagrams shownin Figs. 5). Nevetheless, sometimes, tumor cells succeeds in proliferatingand are thus able to kill most of host cells, inducing a rather deleteriousdynamics.

A second bifurcation diagram was computed by varying the rate 13 withwhich tumor cells kill host cells. Decreasing 13 has a rather similar eect asincreasing the growth rate 1 of host cells, that is, to promote the growth ofthe host cells population. For each parameter which inuences the dynamics,the corresponding bifurcation diagram is roughly similar to those shown inFigs. 5. This has to be interpreted with the fact that the parameter spaceof a system can be viewed as made of shells of equivalent behaviours, moreor less as an onion is made.

Another bifurcation diagram was computed by varying the rate 32 withwhich eector immune cells kill tumor cells. No bifurcation was observedover the range of 32-values (Fig. 7). This means that there is no impacton the dynamics played by this nonlinear term, that is, by the interactionbetween eector immune cells and tumor cells. We observed that setting 32to zero leads to an ejection of the trajectory to innity. The chaotic attractortherefore exists only when 32 > 0 but does not depend on the value of thatparameter. The nonlinear term 32yz is thus of a very limited interest fromthe therapeutic point of view, as conrmed by many negative therapeuticprotocols in immunotherapy [42, 43].

4.2. Topological analysis

The structure of the chaotic attractor produced by this cancer model willnow be analyzed for parameter values corresponding to a unimodal attrac-tor, that is, an attractor characterized by a unimodal map. Such a situationoccurs slightly before the 1-value for which the largest uctuations of thepopulations are observed in the bifurcation diagram (Fig. 5). There aremany other parameter values for which an equivalent attractor is observed.In fact, there is a domain of the parameter space associated with such an at-tractor. Dierent sets of parameter values could be associated with dierentpatients or with a patients status at dierent moment of his disease. Theattractor here investigated is selected from topological criteria because it willhelp to identify the general representative structure of most of the attrac-tors produced by this system, as discussed below. The third branch of therst-return map to a Poincare section occurs for 1 = 0.518. Variable x waschosen to perform the topological analysis for the reasons discussed in Sec. 3.

15

0 10 20 30 40 500

0,2

0,4

0,6

0,8

1

Hos

t cel

ls x

0 10 20 30 40 50Adimensional time t

0

0,2

0,4

0,6

0,8

Tum

or c

ells

z

(a) 1 = 0.54

0 2000 4000 6000 8000 10000 12000 140000

0,2

0,4

0,6

0,8

1

Hos

t cel

ls x

0 2000 4000 6000 8000 10000 12000 14000Adimensional time t

0

0,2

0,4

0,6

0,8

1

Tum

or c

ells

z

(b) 1 = 1.40

Figure 6: Time evolution of host and tumor cells for two values of the host cell growthrate 1. Other parameter values as in Eq. (3).

16

0 1 2 3 4 5Tumor cell killing rate 32 by effector immune cells

0,025

0,05

0,075

0,1

0,125

0,15

0,175

0,2

Tum

or c

ells

z n

Figure 7: Bifurcation diagram of the cancer model (2) versus the tumor cell killing rateby the eector immune cells. Other parameter values as in Eq. (3).

Using the space reconstructed from this variable (Fig. 8a) it is possible to geta regular plane projection of couples of periodic orbits (regular meaning thatthere are no more than two segments crossings at the same point). The dif-ferential embedding induced by variable x reveals a rather simple structure,which ressembles to the structure of the spiral Rossler attractor [22].

This is conrmed by a rst-return map (Fig. 8b) to the Poincare sectiondened as

P = {(xn, xn) R2 | xn = 0, xn < 0} . (11)There is a slightly layered structure in the decreasing branch which couldinduce a third branch in the template. Nevertheless these two decreasingbranches will be approximated by a single one in order to yield a fair repre-sentation of the attractor structure using a template. The increasing branchis associated with an even local torsion (preserving order branch) and the de-creasing branch with an odd local torsion (reversing order branch) [22]. Thismeans that the increasing (decreasing) branch presents an even (odd) numberof -twists. These properties are commonly observed after a period-doublingcascade necessarily induced by a smooth unimodal map [22].

From the bifurcation diagram, there is a strong consequence of the lay-ered structure. Indeed, the second decreasing branch would cross the rstincreasing branch around the maximum of the map (Fig. 8b), a feature thatwould break the determinism of the system. This is avoided by a series of

17

0 0,2 0,4 0,6 0,8Host cells x

-0,1

-0,05

0

0,05

0,1

dx /

dt

0,65 0,7 0,75 0,8 0,85 0,9x

n

0,65

0,7

0,75

0,8

0,85

0,9

x n+

1

(a) Chaotic attractor (b) First-return map

Figure 8: Chaotic behavior produced by the cancer model (2) characterized by a smoothunimodal map. Parameter values: 1 = 0.518 and as in Eq. (3) for others.

reverse bifurcations when 1 is increased beyond 0.518. These bifurcationsinduce a pruning of the unstable periodic orbits (similar features are inducedby a homoclinic orbit in the Rossler system [22]). The pruning is associatedwith the existence of a third (increasing) monotonic branch in the rst-returnmap, that is, of a second critical point (see Fig. 11 as discussed later). Con-comitant to this, new periodic orbits are also created. The dynamics is thusmultimodal with an inevitable antimonotonicity [37].

A symbolic dynamics is said to be complete when all symbolic sequencesare realized as periodic orbits within the attractor. Since the crisis whichavoids the intersection between the two decreasing branches occurs quite farfrom the completeness of the symbolic dynamics built on the two symbols0 and 1, many periodic sequences built with these two symbols are notfound in the attractor shown in Fig. 8a. For instance, orbits with period lessthan seven are limited to

(1) (10) (1011) (101110)(101111) (10111) (10110) (101)(100) (100101) .

This can be also viewed in the rst-return map (Fig. 8b) where the increasingbranch remains far from the rst bisecting line (in smooth unimodal mapsas the logistic function or the rst-return map to a Poincare section of the

18

Rossler system, the completeness of the symbolic dynamics is reached whenthe increasing branch touches the rst bisecting line, see for instance [22]).

In order to determine the branched manifold which can be associatedwith the chaotic attractor shown in Fig. 8a, we present (as examples) twoknots made of the two orbits (1) and (10) (Fig. 9a), and of orbits (10)and (10110) (Fig. 9b), respectively. The oriented crossings are signed usingthe third coordinate (variable x) of the dierential embedding induced byvariable x. The linking numbers are thus given by the half-sum of the orientedcrossings [44, 22]. In the present cases, we obtain lk(1, 10) = 1

2(2) = 1,

and lk(10110, 10) = 12(8) = 4.

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9Host cells x

-0,1

-0,05

0

0,05

0,1

dx /d

t

(1)(10)Negative crossings

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9Host cells x

-0,1

-0,05

0

0,05

0,1

dx /d

t

(10)(10110)Negative crossings

(a) Orbits (1) and (10) (b) Orbits (10110) and (10)

Figure 9: Regular plane projections of two knots made of unstable periodic orbits extractedfrom the chaotic attractor solution to the cancer model (2).

These two knots present only negative (anticlockwise rotation) crossings.As we already mentioned, the attractor is indeed similar to the spiral Rosslerattractor which is characterized by the linking matrix [44, 22]

Mij =

[0 11 1

]. (12)

The diagonal elementsMii of such a matrix encodes the number of -twists inthe ith branch of the template and the o-diagonal elements Mij correspondsto the number of permutation between the ith and the jth branches. Thecorresponding template is drawn in Fig. 10 with the three periodic orbits.

19

The oriented crossing can thus be counted from such a picture to check thatthe linking numbers obtained from a template construction are in agreementwith those counted in plane projections of the corresponding orbits (Fig.9). The chaotic attractor solution to the cancer model is thus topologicallyequivalent to the spiral Rossler attractor.

(10110)(10)(1)

Figure 10: Template (or branched manifold) scheming the chaotic attractor solution tothe cancer model (2) with 1 = 0.518. Orbits (1), (10) and (10110) are also shown in orderto count the linking numbers: lk(10, 1) = 1, lk(10110, 1) = 2, and lk(10110, 10) = 4.

When the growth rate of host cells is increased to 1 = 0.73, the rst-return map presents seven monotonic branches (Fig. 11a), that is, it has sixcritical points. The layered structure layered for 1 = 0.518 (Fig. 8b) is nolonger obvious (only a slight double structure can be seen in the rst increas-

20

ing branch). Such a feature justies a posteriori the used approximationaccording to which only two distinct branches for 1 = 0.518 were retained.The new branches are clearly less and less developed as it is similarly ob-served in multimodal chaotic attractors solution to the Rossler system asshown in Fig. 11b (see [22] for details). We observed that the pruning ofperiodic orbits is more important in the case of the cancer model than inthe Rossler system. This was already identied in the unimodal case (Fig.8). This dierence in the population of periodic orbits is the main departurebetween the two systems.

0,75 0,8 0,85 0,9 0,95 1x

n

0,75

0,8

0,85

0,9

0,95

1

x n+

1

-7-6-5-4-3-2-10y

n

-7

-6

-5

-4

-3

-2

-1

0

y n+

1

(a) Cancer model (1) (b) Rossler system

Figure 11: Multimodal rst-return map to a Poincare section for the cancer model (a) andthe Rossler system (b). Parameter values: (a) 1 = 0.73 and others values as in Eq. (3).(b) a = 0.5505, b = 2, and c = 4.

The fact that the dynamics of the cancer model (2) produces chaoticattractors whose structures as topologically equivalent to the attractors so-lution to the Rossler system (even when they are characterized by multimodalmaps) allows to be sure that any attractor produced by this cancer modelhas an equivalent produced by the Rossler system. From the extensive anal-ysis performed in the Rossler system [22], this also certify that the attractorhere investigated is rather representative to what can be observed for otherparameter values. From the cancer point of view, this means that there isnot a too large variety of dynamics which can be produced by this model.

21

5. Discussion and conclusion

A simple model for competing cells published by de Pillis and Radun-skaya, namely host, eector immune and tumor cells, was investigated usingthree lines of attack: observability, bifurcation and topological analysis. Thisapproach might suggest new trends in understanding the growth of somekinds of tumors and, even in designing their treatments. We thus showedthat using bifurcation diagrams, the action of some terms can be identied.In the present paper, we showed that increasing the growth rate of host cellswas equivalent to decreasing the rate with which tumor cells kill host cells.Nevertheless, if promoting host cells helps to reduce the uctuations of thepopulations, and favor long period of time during which host cells are attheir maximal population and tumor cells are very few, there are also stronginversion of populations for which tumor cells become the most importantpopulatoin: such oscillations could correspond to fast growing cancer suchas small lung cancer.

From a topological point of view, a relevant similarity was found betweenthis model for competing populations of cells and the Rossler system, sinceboth produce chaotic attractors which are topologically equivalent. The bi-furcation diagrams are quite similar for both systems. The populations ofunstable periodic orbits is less developed in the cancer model, somethingwhich could be possibly explained by the fact that the attractor must takeplace in the positive octant, a restriction not observed in the Rossler system.The attractors are structured in both systems by two saddle-foci, one (S4)which has a two-dimensional unstable manifold and which is surrounded bythe attractor, and one (S3) which has a one-dimensional unstable manifoldand which is not surrounded by the attractor (although this may be not soobvious on the projection shown in Fig. 2). The latter is associated with theboundary of the attraction basin.

It was thus showed that in order to investigate the underlying dynam-ics, the populations of the host cells, or possibly, the tumor cells should bemeasured. It is rather inecient to measure only the population of eec-tor immune cells, due to the poor observability it provides on the dynamics.This result is interesting because clinicians never use immunological parame-ters to assess tumor behaviour, even during immunotherapy [45]. Moreover,at a larger scale, tumor dynamic is commonly assessed by its impact onweight and health condition (using performance or Karnofsky performancescale [46, 47, 48]), as well as tumor size measurements.

22

Since observability is related to controllability (one is the dual of theother), therapies acting on the host cell population could be eective. Al-though this seems a convenient way to address the problem, further investi-gations using controllability concepts are needed here. A possible approachcould be to increase the growth rate of host cells, and/or to biologically in-uence interactions between these cells and tumor cells. This possible actionresults from the fact that the population of tumor cells decreases when thegrowth rate of host cells increases. This cancer model also indicates that thegrowth rate of tumor cells implies large uctuations of all the populations,leading the patients to present oscillating health conditions (as observed inclinic). Such an approach is thus interesting because models for compet-ing species deal with patients outcome rather than only with the tumoroutcome, that is, with the interactions between the patient and the tumor,which are strongly correlated to patient outcome. The nonlinear behaviorwhich is often observed in patients tumor dynamics contrasts with statisticalor biological models, which are mostly based on single clones of tumor cellgrowth and on linear models, such as linear-quadratic or tumor control prob-ability models of tumor radiosensitivity [50, 49, 51]. Although such modelsare widely used for clinical purposes, they present many known limitationsbecause they do not take into account tumor cells heterogeneity and stromalcells interactions. Ecological models take into account the nonlinear inter-actions between tumoral and other cells and could be useful for therapeuticapproaches such as non-tumoral-cell-targeted treatments.

Indeed, there is no selected therapy that currently increases the growthrate of normal cells without also increasing the tumor cell proliferation. How-ever, there are numerous ways to act on normal cells that lead to tumorshrinkage such as anti-angiogenic therapy, hormonotherapy and early anti-periostine therapy [40]. Moreover, this type of models is useful to betterunderstand the dynamics underlying the growth of tumor cells. Althoughquite simple and possibly incomplete, these models for competing popula-tions of cells capture features which are qualitatively in agreement with thedaily experience of oncologists. Cancer cells are indeed not only responsiblefor tumor response to therapy or metastatic growth [2, 39, 40] but also forsome eects on their environment, that is, on the host cells. Hence, the con-tribution of non-tumor cells in cancer dynamics appears to be very importantin the global behaviour of the system. Cancel cell genomic instability leadsto intra-tumoral cell heterogeneity and to biological advantages which can betaken into account by mathematical and biological models [52], in which the

23

interactions between tumor and non-tumor cells aect in a deterministic waytumor growth, resistance and response to treatment. Otherwise some tumormay be cured without treatment, a feature which is never observed in clinic.Consequently, population dynamical models must be developped to under-stand the population dynamics and the eect of evolutionary parameters ofneoplasms, even if this remains at a qualitative level as is the case nowadays.Moreover, some available measurements provide biomarkers that can be usedfor risk stratication, intermediate endpoints and targets for new drugs [1].

Ecological models of interaction have to be considered for there globaldynamic against or with cancer cells. One of the main advantages of theecological models describing the interactions between tumor cells and theirenvironment is that their qualitative dynamical analysis allows to get newinsights on the resulting global dynamics although some details in the inter-actions between dierent types of tumor cells not only with host cells but alsowith each others are not considered. From that point of view, de Pullis andRadunskayas model is obviously not complete but it has the great interestto be generic enough to reproduce many dierent features which are dailyobserved in clinical practice as well as in biological experiments. Amongothers, this model suggests a chaotic dynamics for tumor growth in a hostenvironment, that is, a nonlinear growth as observed in culture of cancercells. The non conventional analysis we performed beyong investigating thexed points or providing a single set of parameter values for which thereis a chaotic attractor, is hoped to be helpful for designing new diagnosticand control technique to limit the tumor growth. This would redene newstrategy for dynamical therapies.

Acknowledgements

This work has been partially supported by CNPq (Brazil) and CNRS(France).

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Weperformedatopologicalandanobservabilityanalysisofacancermodel.

Increasingthegrowthrateofhostcellsdevelopsthefluctuationsofpopulations.

Increasingthegrowthrateofhostcellsinducesmorerarebutfastgrowingtumors.

Thekillingrateoftumorcellsbytheimmunecellsdoesnotaffectthedynamics.