Mathematical models of therapeutical actions related to tumour and immune system competition

MATHEMATICAL METHODS IN THE APPLIED SCIENCESMath. Meth. Appl. Sci. 2005; 28:2061–2083Published online 18 July 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/mma.656MOS subject classi�cation: 92D 25; 82C 40; 42K 05

Mathematical models of therapeutical actions relatedto tumour and immune system competition

Elena De Angelis1;∗;† and Pierre-Emmanuel Jabin2;‡

1Dipartimento di Matematica; Politecnico di Torino; Corso Duca degli Abruzzi 24;10129 Torino; Italy

2 �Ecole Normale Sup�erieure, D�epartement de Math�ematiques et Applications; CNRS UMR 8553;45 rue d’Ulm; 75230 Paris Cedex 05; France

Communicated by J. Banasiak

SUMMARY

This paper deals with the qualitative analysis of a model related to the description of two medicaltherapies which have been intensively developed in recent years. In particular, we refer to the modellingof the actions applied by proteins, to activate the immune defense, and to the control of angiogenesis,to contrast the growth of tumour cells by preventing the feeding actions of endothelial cells. Thetherapeutical actions which are object of the modelling process developed in this paper have to beregarded as applied within the framework of the competition between the immune system and tumourcells. We prove the existence of solutions to the Cauchy problem related to the model. The e�ciencyof the therapies and the asymptotic behaviour in time of our solutions is also investigated. Copyright? 2005 John Wiley & Sons, Ltd.

KEY WORDS: Vlasov kinetic theory; Cauchy problem; cell population; tumour–immune competition

1. INTRODUCTION

Methods of the mathematical kinetic theory have been applied in the last few years tomodel the competition between tumour and immune cells. The literature in the �eld anda critical analysis on the existing results and open problems can be found in the reviewpapers [1,2].Mathematical models are expected to describe the interactions and competition between

tumours and the immune system. The evolution of the system may end up either with theblow-up of the host (with inhibition of the immune system), or with the suppression of the

∗Correspondence to: Elena De Angelis, Dipartimento di Matematica, Politecnico di Torino, Corso Duca degliAbruzzi 24, 10129 Torino, Italy.

†E-mail: [email protected]‡E-mail: [email protected]

Copyright ? 2005 John Wiley & Sons, Ltd. Received 21 October 2004

2062 E. DE ANGELIS AND P.-E. JABIN

host due to the action of the immune system. The mathematical structure of the equationssuitable to deal with the modelling of the above system have been developed and criticallyanalysed in Reference [3], while motivations from scientists operating in the sciences ofimmunology in favour of development of methods of non-equilibrium statistical mechanicscan be recovered, among others, in References [4,5].The mathematical methods are those typical in non-equilibrium statistical mechanics and

generalized kinetic theory. The general idea, as documented in Reference [6], consists inderiving an evolution equation for the �rst distribution function over the variable describingthe microscopic internal state of the individuals. Generally, this variable may include positionand velocity, but it can also refer to some additional speci�c microscopic features. Interactionsbetween pairs have to be modelled taking into account not only mechanical rules but alsomodi�cations of the non-mechanical physical (internal) state.Speci�cally we are interested in a development of the model proposed in Reference [6],

whose analytical properties have been studied in Reference [7], toward the description ofmedical therapies which have been intensively developed in recent years. More preciselya model of competition between the immune system and tumour cells was proposed inReference [6], this model did not include the e�ect of any therapy. In Reference [7], it wasproved that there exists global solutions to this model and the possible asymptotic behavioursin time were detailed.The aim of this paper is �rst to present a general framework (in which the model of

Reference [6] �ts) which is then used to build a new model. The di�erence with the previousmodel of Reference [6] is that the e�ect of two di�erent therapies is now taken into account.The last part of the paper studies the properties of the new model; Existence of global solutionsis a consequence of the proof given in Reference [7] and consequently we only state the result.However, the analysis of the asymptotic behaviour in time is more complicated for the newmodel than what it was for the one of Reference [6], thus requiring new ideas and new proofswhich are presented.Concerning the therapies, we refer to the modelling of the actions applied by proteins

to activate the immune defense [8] and to the control of angiogenesis [9,10], to contrastthe growth of tumour cells by preventing the feeding actions of endothelial cells. All abovetherapeutical actions which are object of the modelling process developed in this paper have tobe regarded as applied within the framework of the competition between the immune systemand tumour cells. Referring to the literature on the immune competition, developed withinmedical sciences, the interested reader is addressed to the survey [11].Another development of the model introduced in Reference [6] is presented in

Reference [12], where the capacity of the body to repair cells damaged is taken into ac-count. The idea is that for a long time range the body produces new cells to replace the deadones in order to try to reach its normal healthy state.After this introduction, the contents of this paper are organized in �ve more Sections:

◦ Section 2 deals with the derivation of a mathematical framework for the design of speci�cmodels suitable to describe the therapeutical actions indicated above. This means derivinga class of integro-di�erential equations to describe, by methods of the mathematicalkinetic theory, the evolution over the biological state of the cells;

◦ Section 3 deals with a short description of the two therapies we will consider: angiogen-esis control and activation of the immune defense;

Copyright ? 2005 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2005; 28:2061–2083

MATHEMATICAL MODELS OF THERAPEUTICAL ACTIONS 2063

◦ Section 4 deals with some phenomenological assumptions about microscopic interactions,by which we obtain, within the general framework, the speci�c models object of ourinterest;

◦ Section 5 deals with the presentation of the analytical results related to the qualitativeanalysis of the solution to the initial value problem introduced in Section 4. This Sectionis also dedicated to the biological interpretation of the analytical results;

◦ Section 6 deals with the technical proofs of the results listed in Section 5.

2. MATHEMATICAL FRAMEWORK TOWARDS MODELLING

This section deals with the design of the general framework within which the speci�cmodels proposed in the next section will be developed. The objective is presented throughthree sequential steps. First we deal with the characterization of cell or particles population,then with the modelling of microscopic interaction, and �nally with the derivation of a classof evolution equation. Speci�c models can be derived, as we shall see, by specializing theabove-mentioned microscopic interactions.

2.1. Cell populations and statistical representation

Consider a large system of cells or particles homogeneously distributed in space. Modelling theimmune competition between tumour and immune cells under the medical action developedby particles which are arti�cially inserted into a vertebrate, needs the de�nition of the variouspopulations which play the game.In particular, we adopt the following assumptions:

Assumption 2.1Cells and particles are homogeneously distributed in space. The system is constitutedby the following populations: cells or particles of the aggressive host, immune cells,environmental cells, and particles of the therapeutical host (with possibly several di�erentpopulations in this last class). Each population is labelled, respectively, by the indicesi=1; : : : ; n.

Assumption 2.2Each cell is characterized by a certain state, a real variable u∈ I =[0;+∞) describingits main properties: progression for the host cells, activation for the immune cells, feedingability for the environmental cells, and therapeutical ability for the particles of thetherapeutical host.

Remark 2.1The therapeutical ability needs to be specialized according to the speci�c medical actionwhich is being modelled. In the cases which will be studied in the next section, we willconsider two populations of particles with therapeutical e�ect and the therapeutical abilitycan hence be either the control of the activation ability of the immune system (for the �rstpopulation), or the control of the feeding ability of the environmental cells (for the secondpopulation).



Assumption 2.3The statistical description of the system is described by the number density functions

Ni=Ni(t; u)

which are such that Ni(t; u) du denotes the number of cells per unit volume whose stateis, at time t, in the interval [u; u + du]. Moreover, if n0 is the number per unit volume ofenvironmental cells at t=0

n0 =∫IN3(0; u) du

the description of the system can be given by the following distributions, normalized such asto have total density 1 at t=0 of environmental cells

fi=fi(t; u)=1n0

Ni(t; u)

Other normalizations are of course possible. We opted for this one as the density of environ-mental cells represents in some sense the normal healthy state of the body.

Remark 2.2If the distribution function fi is given, it is possible to compute, under suitable integrabilityproperties, the size of the population still referred to n0

ni(t)=∫Ifi(t; u) du (1)

and �rst-order moments such as the activation

Ai(t)=Ai[fi](t)=∫Iufi(t; u) du (2)

of each population.

2.2. Modelling microscopic interactions

This section deals with the modelling of microscopic interactions. Speci�cally we refer tothe framework proposed in Reference [6], which may be classi�ed as mean �eld modelling,according to the fact that a test cell feels the presence and interacts with the surrounding�eld cells localized in a suitable volume around the test cell.

Assumption 2.4Interactions are homogeneously distributed in space and can be divided into three types ofencounters: mass conservative interactions, which modify the state of the pair, but not thesize of the population, proliferative and destructive interactions, which produce proliferationor destruction of the interacting subjects, and population shifting interactions which gener-ate individuals into a third population out of interactions within individuals of two di�erentpopulations.



Assumption 2.5Conservative encounters are interactions modelled by the term Pik =Pik(u; u∗) which de�nesthe action on the cell of the ith population with microscopic state u due to the cell, withstate u∗, of the kth population, so that the resultant action is

Fik[f](t; u)=∫IPik(u; u∗)fk(t; u∗) du∗ (3)

where f = {f1; : : : ; fn}.Additional conditions should be considered on the term Pik for u=0 to guarantee the

conservation property of term (3) on the interval [0;∞[. We will see, in Section 4, that underthe phenomenological assumptions we consider, such conditions are automatically satis�ed.

Assumption 2.6The term describing proliferation and=or destruction phenomena, within the same population,in the state u due to pair interactions between cells of the ith population with microscopicstate u∗ with cells of the kth population, with microscopic state u∗∗, is modelled by thesource=sink term

Sik[f](t; u)=∫I

∫I�ik(u∗; u∗∗; u)fi(t; u∗)fk(t; u∗∗) du∗ du∗∗ (4)

where �ik is a suitable proliferation–destruction function.

Assumption 2.7The term describing proliferation and=or destruction phenomena, within the ith population, inthe state u related to pair interactions between cells of the jth population with microscopicstate u∗ due to the subject of the kth population, with microscopic state u∗∗ is given by

Q(i)jk [f](t; u)=

∫I

∫I (i)jk (u

∗; u∗∗; u)fj(t; u∗)fk(t; u∗∗) du∗ du∗∗ (5)

where (i)jk is a suitable proliferation–destruction function.

Assumption 2.8The above framework refers to the evolution in absence of source=sink terms. This meansthat cells are contained in a vessel and the system is closed. Tumour cells can then replicateexploiting the existing environmental cells. On the other hand, the above-mentioned supplyor consumption can be modelled for the ith population by suitable source=sink terms Ii(t; u).

2.3. Evolution equations

The mathematical model consists in an evolution equation for the distribution functions fi

corresponding to the above-mentioned cell populations. The mathematical structure of themodel proposed in Section 5 of Reference [6] is as follows:

@@t

fi(t; u) +Fi[f](t; u)=Si[f](t; u) + Qi[f](t; u) +Ii(t; u) (6)



where, according to the framework for microscopic modelling described in Equations (3)–(5),we have

Fi[f](t; u) =@@u

[fi(t; u)

n∑k=1

Fik[f](t; u)]

Si[f](t; u) =n∑

k=1Sik[f](t; u)

and

Qi[f](t; u)=n∑

j=1

n∑k=1

Q(i)jk [f](t; u)

This general framework can generate speci�c models after a detailed modelling of microscopiccell interactions, as it will be shown in Section 4.

3. ON THE CONTROL OF ANGIOGENESIS AND IMMUNE ACTIVATION

The framework described in Section 2 can be exploited to derive speci�c models after havingproperly modelled all microscopic interactions among the various subjects playing the game.Of course the speci�c therapeutical actions which are applied has to be properly speci�ed.A large variety of therapeutical actions are known in the �eld of medicine: a brief account

is given in this section with reference to the following two speci�c actions (we refer theinterested reader to Reference [2] for more details)

• Modelling of the actions applied by proteins to activate the immune defense, e.g.Reference [13] or [8], thus preventing the ability of tumour cells to inhibit immunecells.

• Control of angiogenesis phenomena, that is the formation of new blood vessel from pre-existing vasculature, e.g. References [9,10], thus preventing tumour growth by limitingthe feeding ability from blood vessels.

There is an experimental evidence that immunotherapy have the potential to treat manytumour types, see Reference [14]. The immunotherapy approach consists in the activationof speci�c tumour antigen combined with incorporation of an immunological adjuvant into avaccine regime. In detail, cancer vaccines involve the induction of an active immune responsethat may lead to the subsequent destruction of tumour tissue. On the other hand, as reportedin Reference [15], an adoptive cancer immunotherapy involves the use of tumour-killing lym-phocytes and lymphokines engaging in a search and destroy anti-cancer activity. FollowingReference [13], tumour vaccine and cytokine therapy are two methods of promoting an anti-cancer immune response and these techniques are highly e�ective when combined. In cancerprevention using cancer vaccines the target is not the tumour mass but the potential risk ofcancer (the so-called primary prevention), a preneoplastic lesion (the so-called secondary pre-vention) or a small number of isolated neoplastic cells remaining after a temporarily successfultherapeutical treatment (the tertiary prevention). Moreover, vaccination after the removal of atumour mass can stop the formation of minimal residual disease or metastatic di�usion.



Referring to the second therapy, e.g. the control of angiogenesis phenomena, tumour pro-gression and growth cannot occur without angiogenesis, which supplies the necessary oxygenand nutrients to the growing tumour. Various angiogenesis inhibitors have been developedto target endothelial cells and stop the process. Referring to Reference [10], a new class ofdrugs is represented by di�erent type of angiogenic inhibitors and they are extremely importantin cases for which the general rules involving conventional chemotherapy might not apply.Inhibitors like angiostatin prevent vascular endothelial cells from proliferating and migrating,while indirect angiogenesis inhibitors can prevent the expression of the activity of one ofthe tumour proteins which drive the angiogenic switch. Another feature of the angiogenesisprocess is the evident abnormal vasculature as a hallmark of solid tumour, see Reference [16],and the normalization of this abnormal vasculature can facilitate drug delivery to tumours andit represents another important goal in the antiangiogenic therapy.

4. MODELLING THE IMMUNE COMPETITION AND THETHERAPEUTICAL ACTIONS

The general principles followed toward modelling are precisely the same we have seen inSection 2. Interactions with cells of the other populations modify the biological state andmay generate proliferation and=or destruction phenomena. In detail, referring to interactionsbetween host, immune and endothelial cells, we shall essentially extend the model proposed inReferences [6,7]. For i=1; 2; 3 we will indicate, respectively, the aggressive host, the immunesystem and the environmental cells. Concerning the angiogenesis control and the activationof the immune system, let the related population of particles be denoted, respectively, by thesubscript i=4 and i=5. The assumptions which de�ne the microscopic interactions can bestated as follows.Consider �rst conservative encounters described by Equation (3).

Assumption 4.1The progression of neoplastic cells is not modi�ed by interactions with other cells of thesame type. On the other hand, it is weakened by interaction with immune cells (linearlydepending on their activation state) and it is increased by interactions with environmental cells(linearly depending on their feeding ability). The e�ect increases with increasing values of theprogression. Moreover, the progression of the aggressive host is not modi�ed by interactionswith both particles of the anti-angiogenesis and of the immune activation therapeutical actions

P11 = 0; P12(u; u∗)=−�12uu∗; P13(u; u∗)= �13uu∗; P14 =P15 = 0

Assumption 4.2The defense ability of immune cells is weakened by interactions with tumour cells (linearlydepending on their activation state) due to their ability to inhibit the immune system. Onthe other hand, it is not modi�ed by interactions with other cells of the same type, withenvironmental cells and with the angiogenic therapeutical host. Moreover, it is increased byinteraction with the immune activation therapeutical host and the e�ect is linearly dependingon their activation state

P21(u; u∗)=−�21uu∗; P22 =P23 =P24 = 0; P25(u; u∗)= �25uu∗



Assumption 4.3The feeding ability of the endothelial cells is weakened by interaction with tumour cellslinearly depending on their activation state. On the other hand, it is not modi�ed by interactionswith immune cells and with other cells of the same type. Moreover, it is weakened byinteraction with the therapeutical host and we assume that this action depends linearly ontheir activation state. Finally it is not modi�ed by interactions with the immune activationtherapeutical host

P31(u; u∗)=−�31uu∗; P32P33 = 0; P34(u; u∗)=−�34uu∗; P35 = 0

Assumption 4.4The therapeutical ability of the angiogenic therapeutical host is not modi�ed by interactionswith all the other populations and with cells of the same type

P41 =P42 =P43 =P44 =P45 = 0

Assumption 4.5The therapeutical ability of the immune activation host is not modi�ed by interactions withall the other populations and with cells of the same type

P51 =P52 =P53 =P54P55 = 0

Consider now the non-conservative encounters described by Equation (4). A simple modellingcan be based on the assumption that the terms �ij are delta functions over the state u∗ of theinteracting test cell

�ij(u∗; u∗∗; u)= sij(u∗; u∗∗)�(u − u∗)

for all i; j=1; : : : ; 5.

Assumption 4.6No proliferation of neoplastic cells occurs due to interactions with other cells of the same type.On the other hand, interactions with immune cells generate a destruction linearly dependingon their activation state, while interactions with environmental cells generate a proliferationdepending on their feeding ability and the progression of tumour cells. Moreover, no prolif-eration arises due to the interactions with the therapeutical hosts

s11 = 0; s12(u∗; u∗∗)=−�12u∗∗; s13(u∗u∗∗)=�13u∗u∗∗; s14 = s15 = 0

Assumption 4.7Proliferation of immune cells occurs due to interactions with tumour cells, linearly dependingon their defense ability and on the activation state of tumour cells. On the other hand, noproliferation of immune cells occurs due to interactions with other cells of the same type,with environmental cells and with cells of the therapeutical hosts

s21(u∗; u∗∗)=�21u∗u∗∗; s22 = s23 = s24 = s25 = 0

Assumption 4.8A destruction of the environmental cells occurs due to interactions with tumour cells, linearlydepending on the activation state of tumour cells. On the other hand, no proliferation of



environmental cells occurs due to interactions with immune cells, with other cells of the sametype and with cells of the therapeutical hosts

s31(u∗; u∗∗)=−�31u∗∗; s32 = s33 = s34 = s35 = 0

Assumption 4.9No proliferation of cells of the angiogenic therapeutical host occurs due to interactions withtumour cells and with immune cells. On the other hand, a destruction occurs due to interactionswith environmental cells, linearly depending on the activation state of environmental cells.No proliferation occurs due to interactions with cells of the same type and with cells of theimmune activation therapeutical host

s41 = s42 = 0; s43(u∗; u∗∗)=−�43u∗∗; s44 = s45 = 0

Assumption 4.10No proliferation of cells of the immune activation therapeutical host occurs due to interac-tions with tumour cells. On the other hand, a destruction occurs due to interactions withimmune cells, linearly depending on the activation state of the cells of the host. Moreover,no proliferation occurs due to interactions with the immune cells, with cells of the angiogenictherapeutical host and with other cells of the same type

s51 = 0; s52 =−�52u∗∗; s53 = s54 = s55 = 0

Assumption 4.11For all the actions we consider, the terms Q(i)jk in Equation (5) are equal to zero. This meansthat the possibility of shift between populations is not relevant.

Assumption 4.12No source terms are considered in the evolution equations for all the distribution functions fi,expressing the absence of source=sink terms.

Based on the above modelling of cell interactions, we are now able to derive the evolutionequations (6) for each fi

@f1@t(t; u) =

@@u[uf1(t; u)(�12A2[f2](t)− �13A3[f3](t))]

+f1(t; u)(−�12A2[f2](t) + �13uA3[f3](t)) (7)

@f2@t(t; u) =

@@u[uf2(t; u)(�21A1[f1](t)− �25A5[f5](t))]

+�21uf2(t; u)A1[f1](t) (8)

@f3@t(t; u) =

@@u[uf3(t; u)(�31A1[f1](t) + �34A4[f4](t))]

−�31f3(t; u)A1[f1](t) (9)



@f4@t(t; u) =−�43f4(t; u)A3[f3](t) (10)

@f5@t(t; u) =−�52f5(t; u)A2[f2](t) (11)

for all t; u∈R+, where the activations Ai[fi] are de�ned in Equation (2).The Cauchy problem for system (7)–(11) is de�ned given the initial conditions

fi(t=0; u)=f0i (u); ∀ i=1; : : : ; 5 (12)

for all u∈R+.All the parameters � and �, which appear in the above assumptions, have to be regarded

as positive, small with respect to one, constants to be identi�ed by suitable experiments.

5. ANALYTIC RESULTS AND BIOLOGICAL INTERPRETATION

In the �rst part of this section we present the analytical results related to the qualitativeanalysis of the solution to the Cauchy problem (7)–(11) and we refer to the next Section 6for the technical proofs.The following theorem states the existence of the solutions of the Cauchy problem

(7)–(12).

Theorem 5.1Assume that the initial conditions f0i , for i=1; : : : ; 5, satisfy, respectively, the followingassumptions:

∫ ∞

0e�uf01 (u) du¡ ∞; ∀�¿ 0 (13)

∫ ∞

0(1 + u) e�u f02 (u) du¡+∞; ∀�¿ 0 (14)

∫ ∞

0(1 + u)f0i (u) du¡ ∞; ∀i=3; 4; 5 (15)

Then there exists at least one solution (f1; f2; f3; f4; f5)∈C([0;∞); L1((1+u) du)) to the initialvalue problem (7)–(12) (in the distributional sense) which satisfy

∫ ∞

0e�uf1(t; u) du ∈ L∞([0; T ]); ∀�; T ¿ 0

∫ ∞

0(1 + u) e�uf2(t; u) du ∈ L∞([0; T ]); ∀�; T ¿ 0 (16)

∫ ∞

0(1 + u)fi(t; u) du6

∫ ∞

0(1 + u)f0i (u) du; ∀i=3; 4; 5



Notice that to obtain the existence of solutions, we would only require to ask that∫e�uf02 (u) du

be �nite for all �¿ 0. The stronger hypothesis (14) is however useful in what follows, whichconcerns the asymptotic behaviour in time.We also point out that the equation for f4 and f5 are not really necessary. Indeed the only

interesting quantities are A4 and A5 and the relative equations, together with the equationfor A3, are listed in the following lemma.

Lemma 5.1For the solutions to the initial value problem (7)–(12) given by Theorem 5.1 we have

ddt

A3(t) =−[(�31 + �31)A1(t) + �34A4(t)]A3(t) (17)

ddt

A4(t) =−�43A3(t)A4(t) (18)

ddt

A5(t) =−�52A2(t)A5(t) (19)

Remark 5.1From now on we will use the notation

A0i :=Ai(0)=∫ +∞

0uf0i (u) du and n0i := ni(0)=

∫ +∞

0f0i (u) du

∀ i=1; : : : ; 5.

In general we are not able to predict exactly the asymptotic behaviour given a set of initialdata, except in the case where only the host and the immune system are present where wehave the following result.

Proposition 5.1For f03 =f04 =f05 = 0, we de�ne n∗

2 =∫e�21 u=�21f02 (u) du and we have that

(i) If (�12 + �12) n02 + �21 A01¡ (�12 + �12) n∗2 , then A1 → 0 as t → ∞ and A2 is uniformly

bounded from below.(ii) If (�12 + �12) n02 + �21 A01¿ (�12 + �12) n∗

2 , then A2 → 0 as t → ∞.Before presenting the partial results which we are able to obtain in the general case, we

state the next lemma concerned with an estimate for the distribution f2.

Lemma 5.2Assume f0i satisfy (13)–(15), i=1; : : : ; 5. Then we have for the activation A2(t) the followingestimate:

A02e∫ t0 A(s) ds6A2(t)6

( ∫u eCuf02 (u) du

)× e

∫ t0 A(s) ds (20)

where C= �21�21+ �21�25A05

�21�52A02and A(s)= �25A5(s)− �21A1(s).



The next two propositions will show that it is always possible to have a therapeutical actionsu�ciently high in order to have, asymptotically, the destruction of the tumour cells.

Proposition 5.2It is always possible to choose A05 su�ciently high so that the activity of the immune system A2together with A3 are bounded from below by a positive constant and A1 converges toward 0as time tends to in�nity.

Concerning the �rst therapy, the result is a bit weaker as it reads:

Proposition 5.3If A05 = 0, and (�12 + �12) n02 + �21 A01¿ (�12 + �12) n∗

2 , it is possible to choose A04 such that A2is bounded from below and both A1 and A3 converge toward zero as time tends to in�nity.

Notice that this result is optimal in the sense that if A05 = 0, and 2n02 + A01¡ 2n∗

2 , then forany A04, it is A2 which converges to 0 since we have convergence toward the asymptoticbehaviour prescribed by Proposition 5.1. We do not know the precise condition if A05 isnon-zero.We stress the point that the previous two propositions are independent from each other.Let us now investigate the asymptotic behaviour in time of the solution.We will use the following lemma.

Lemma 5.3For the solutions to the initial value problem (7)–(12) given by Theorem 5.1 we have

∫ +∞

0A2(t)A5(t) dt ¡+∞ (21)

∫ +∞

0A3(t)A4(t) dt ¡+∞ (22)

∫ +∞

0A1(t)A3(t) dt ¡+∞ (23)

From Equation (17), A3(t) is decreasing, and hence

either A3 → 0 or ∃c¿ 0 s:t: A3(t)¿ c; ∀t

First, we consider the asymptotic behaviour in the case A04 = 0.

Theorem 5.2Assume that A05 �=0 and A04 = 0. Then as t →+∞, there are only the following two possibilities:(i) If A3(t) is bounded from below by a positive constant, then also A2(t) is bounded from

below by a positive constant,∫ +∞0 A1(t) dt ¡+∞ and

∫ +∞0 A5(t) dt ¡+∞;

(ii) If A3 → 0, then also A2(t)→ 0 and∫ +∞0 A1(t) dt=+∞.



We are not able to indicate all possible asymptotical features of the system, if A05 �=0 andA04 �=0, but we can still give some partial answers. In particular, property (i) of Theorem 5.2remains true.

Proposition 5.4If A3(t) is bounded from below by a positive constant, then also A2(t) is bounded from belowby a positive constant,

∫ +∞0 A1(t) dt ¡+∞, ∫ +∞

0 A4(t) dt ¡ +∞ and∫ +∞0 A5(t) dt ¡+∞.

An interesting biological interpretation can be given to the main results described in thissection. First we observe that conditions (13)–(15) needed to prove the existence theoremare consistent with the biological requirements of cell populations. Speci�cally the decayto zero at in�nity implied by conditions (13)–(15) can be regarded as natural, since thedistribution functions fi must e�ectively decay to zero at in�nity to have a bound to theinitial density and activation as given by Equations (1) and (2). Exponential might neverthelessseem too demanding but we recall the result obtained in Reference [7], which also applieshere and which tells that, without some exponential decay, no solution exists even for ashort time.The theorem states in (16) the L∞ boundedness of the weighted distributions f1 and f2

and a bound of f3–f5 with respect to the corresponding quantities at t=0. The theoremprovides a useful background for simulations, while the results which are useful for a biolog-ical interpretation are stated in Propositions 5.1–5.4 and Theorem 5.2. In detail, Proposition5.1 concerns the simplest case where only the tumour cells and the immune system arepresent and it indicates how the system chooses between the two asymptotic behaviours,already described in Reference [7], in terms of the initial data and of the parameters ofthe model. We recall that in the general case studied in Reference [7] we did not havesuch a result.Propositions 5.2 and 5.3 show the e�ciency of the treatment in both the cases of the two

therapies considered, even if the analytical results obtained in the case of the angiogenesiscontrol is a bit weaker. The two propositions state how in principle it is always possibleto reach the situation where the immune system wins and completely eliminates the tumourcells of the organism. As a consequence of the activation of the immune system or of theweakening of the feeding ability of the environmental cells, respectively, the activation of thetumour system evolves toward lower degrees of malignity. Unfortunately, these mathematicalresults do not always correspond to the reality, in the sense that the amount of the initialconditions of the treatments cannot be realistic (if a high dose of treatment is dangerous forother reasons).Theorem 5.2 shows the complete asymptotic analysis in the presence of the second therapy.

Recalling that this is the therapy referring to the actions applied by proteins to activate theimmune defense, the analysis corresponds to a well de�ned medical motivation related to theaction of cytokine signals [8,13]. The competition between the tumour cells and the immunesystem may end up with the regression of progressed cells, due to the action of the immunesystem, or with the blow up of progressed cells and inhibition of the immune system.Finally, Proposition 5.4 gives only a partial answer concerning the asymptotic behaviour in

the general case of the two therapies, showing a case where the destruction of the host is stillpossible. In that case, some more complicated asymptotic behaviours are probably possible,corresponding to chronical diseases for instance (with oscillations for the numbers of immunecells and aggressive hosts).



6. ESTIMATES AND PROOFS

This section is devoted to the technical proofs of the qualitative properties of the solution tothe Cauchy problem (7)–(12).

6.1. Existence Theorem 5.1

The proof of the existence Theorem 5.1 is the same as that of Theorem 2.1 in Reference [7].We just list here the main steps of the proof:

• A priori estimates: Any solution fi, for i=1; : : : ; 5, to the system which is the weaklimit of compactly supported solutions in L∞

t L1u, satis�es estimates (16).• Continuity in time: Given Ai ∈ L∞([0; T ]), for i=1; : : : ; 5, the weak solutions fi, for

i=1; : : : ; 5, to the system, satisfying conditions (16), also belong to

C([0; T ]; L1((1 + u) du)); ∀T ¿ 0

• Stability result: Let fni in C([0;∞); L1((1 + u) du)), for i=1; : : : ; 5, be a sequence of

solutions in the distributional sense. Assume that they satisfy the a priori estimates (16)uniformly in n. Then any weak limit fi, for i=1; : : : ; 5, of converging subsequences alsobelongs to C([0;∞); L1((1 + u) du)) and solves the system in distributional sense withinitial data the weak limits of the initial data.

• Existence: We construct sequences of approximate solutions of our system, satisfyingevery a priori estimate. Taking weak limits, we obtain solutions for any initial datasatisfying the hypotheses (13)–(15) of the theorem, on the interval [0; T ]. Since thesolutions satisfy the estimates (16) uniformly in t, we may extend inde�nitely the timeof existence.

6.2. Proof of Proposition 5.1

If f03 =f04 =f05 = 0, they remain so at any latter time, which obviously simpli�es the equationson f1 and f2. Multiplying Equation (7) by u and integrating we formally obtain in this case

ddt

A1(t)=−(�12 + �12)A1A2

This formal computation can easily be made rigorous (see Reference [7] for more details).Integrating Equation (8) we get

ddt

n2(t)=�21 A1 A2

Then with these two relations, we conclude that

ddt((�12 + �12)n2 + �21A1)=0 (24)

We now recall the main result from Reference [7] which states that either A1 convergestoward 0, A2 is bounded from below and n2 is bounded away from n∗

2 , or A2 convergestoward 0 and n2 converges toward n∗

2 .



The �rst case means that

limt→∞ ((�12 + �12)n2(t) + �21A1(t))¡ (�12 + �12)n∗

2

and the second that

limt→∞ ((�12 + �12)n2(t) + �21A1(t))¿ (�12 + �12)n∗

2

These are exactly the two conditions given in Proposition 5.1.

6.3. Proof of Lemma 5.1

It is easy to see that the estimates given in Lemma 5.1 are formally true. For a rigorous proofwe refer to Lemma 5.5 in Reference [7].


Let us consider Equation (8) for the distribution f2 and the characteristics

U (t; u)= u × e∫ t0 A(s) ds

where A(s)= �25A5(s)−�21A1(s). Then the equation for f2 along the characteristics is given by

ddt[f2(t; U (t; u))]@tf2(t; U (t; u)) + @tU (t; u)@uf2(t; U (t; u))

= @tf2(t; U (t; u)) + A(t)U (t; u)@uf2(t; U (t; u))

= [�21U (t; u)A1(t) + A(t)]f2(t; U (t; u))

so that

f2(t; U (t; u))=f02 (u)× e∫ t0 [�21U (s;u)A1(s)+A(s)] ds

and hence∫

Uf2(t; U ) dU

=∫

u × e∫ t0 A(s) ds × f02 (u)× e

∫ t0 [�21U (s;u)A1(s)+A(s)] ds × e

∫ t0 A(s) ds du

Consequently, we get for A2 the expression

A2(t)=( ∫

uf02 (u)× e∫ t0 [�21U (s;u)A1(s)] ds du

)× e

∫ t0 A(s) ds (25)

Let us now consider

06∫ t

0�21U (s; u)A1(s) ds�21u ×

∫ t

0e∫ s0 A(�) d� ds=:�21uI(t)



and

�21I(t)=−∫ t

0e∫ s0 A(�) d� × A(s) ds+ �25

∫ t

0e∫ s0 A(�) d� × A5(s) ds=: I1(t) + I2(t)

For the �rst integral we have

I1(t)=−∫ t

0@s

[e∫ s0 A(�) d�

]ds1− e

∫ s0 A(�) d�6 1

while from (25) we deduce that

A2(t)¿( ∫

uf02 (u) du)

× e∫ t0 A(�) d�A02 e

∫ t0 A(�) d�

Therefore,

06 I2(t)6�25A02

×∫ t

0A2(s)A5(s) ds

�25�52A02

(A05 − A5(t))

where the last equality is obtained from Equation (19). Finally,

06 I2(t)6�25A05�52A02

and

�21I(t)6 1 +�25A05�52A02

so that we obtain

A02e∫ t0 A(s) ds6A2(t)6

( ∫u eCuf02 (u) du

)× e

∫ t0 A(s) ds

where C= �21�21+ �21�25A05

�21�52A02, which proves Lemma 5.2.


The proof of Proposition 5.2 will be carried out in two steps. The �rst step consists inshowing that

∀K; ∃A05 and ∃t06 1 s:t: A2(t0)¿K

Given any di�erentiable function �(t), let us de�ne

J (t)=∫e�(t)uf1(t; u) du

From Equation (7) we have

ddt

J (t)6 �′(t)∫

u e�(t)uf1(t; u) du+ (�13�(t) + �13)A3(t)∫

u e�(t)u f1(t; u) du



Now we choose � s.t. �′(t)=−(�13�(t) + �13)A03, and this implies

ddt

J (t)6 0 and �(t)=Ce−�13A03t − �13�13

where C is an arbitrary constant. Now we take C s.t. �(1)=1 and consider the followingestimate, ∀t6 1:∫

euf1(t; u) du6∫e�(t)uf1(t; u) du= J (t)6 J (0)

∫e(C−(�13=�13))uf01 (u) du (26)

Now

(i) A3(t)6A03; ∀t6 1, from (17).(ii) A2(t)¿A02 e

∫ t0 (�25A5(s)−�21A1(s)) ds, from (20),

(iii) A1(s)6∫euf1(s; u) du6 1

�21C̃(A03; f

01 ), from (26)

and using property (iii) in (ii), we have

A2(t)¿A02 e∫ t0 �25A5(s) ds−C̃t¿A02 e

−C̃ e∫ t0 �25A5(s) ds ∀t6 1 (27)

Assume now that ∃K s.t. ∀A05; A2(t)6K ∀t6 1.Recalling Equation (19) we get

A5(t)¿A05 e−�52Kt

and

A2(t)¿A02 e−C̃ e�25A

05=�52K×[1−e−�52k ] →+∞ as A05 →+∞

but this is a contradiction and the �rst step is proved.Now, the second step consists in showing that we can choose �6 1 and K s.t. A02¿K

implies A2(t)¿ �K .Let us choose �; K s.t. A02¿K and �6 1. De�ne T the �rst time (if it exists) s.t. A2(t)= �K .

Then, ∀t ∈ [0; T ]; A2(t)¿ �K .By Equation (7) we obtain

ddt

∫e�uf1(t; u) du

= (��13A3(t) + �13A3(t)− ��12A2(t))

×∫

ue�uf1(t; u) du − �12A2(t)∫e�uf1(t; u) du

If we choose � s.t.

A03�13�+ �13

�12�6 �K (28)

then we have ∫e�uf1(t; u) du6 e−�12�Kt

∫e�uf01 (u) du



and we use this estimate to deduce that

A1(t)61�

∫e�uf1(t; u) du6

1�e−�12�Kt

∫e�uf01 (u) du

Using the previous computations and the relation

ddt

A2(t)¿ − �21A1(t)A2(t)

that we obtain from Equation (8), we have the following estimate for A2(t):

A2(t)¿A02 × e−�211� (

∫ t0 e

−�12�Ks ds)×∫e�uf01 (u) du

K × exp(

− �21�12��K

×∫e�uf01 (u) du

)¿�K

provided

exp(

− �21�12��K

×∫e�uf01 (u) du

)¿� (29)

Summarizing, we can choose �; K s.t. (28) and (29) are satis�ed, and in this case we get thecontradiction that A2(t)¿�K , and this proves the second step of the proof of Proposition 5.2.


Take the equation for A3 in Lemma 5.1, multiply it by �43, and subtract it from the equationfor A4 in the same lemma, multiplied by �34, to obtain

ddt(�34A4 − �43A3)=�43(�31 + �31)A1 A3¿ 0

As a consequence �34A4 is larger than K = �34A04 − �43A03, and this constant as high as wewant provided A04 is high enough. Since

ddt

A36−�34 A4 A3

we deduce that

A3(t)6A03 e−Kt (30)

Now let us write down the characteristics for (7), which read

@tU (t; u)= (�13A3 − �12A2)U (t; u); U (0; u)= u

Thanks to (30), we have that

U (t; u)= u × e∫ t0 (�13A3(s)−�12A2(s)) ds6 u × e

∫ t0 �13A3(s) ds6� u (31)

with

�=e�13 A03=K



On the other hand,

@t(f1(t; U (t; u)))6 (�31 A3(t)U (t; u) + �12 A2 − �13 A3)f1(t; U (t; u))

and consequently

f1(t; u)6f01 (u e− ∫ t

0 (�13A3(s)−�12A2(s)) ds)× e�u × e−∫ t0 (�13A3(s)−�12A2(s)) ds

where � is uniformly bounded in terms of A04, provided this last quantity is bounded awayenough from 0 since

�=�13�13A03K

e�13 A03=K

From the inequality on f1 and (31), we may deduce that∫(u+ u2)f1(t; u) du6

∫(U (t; u) + |U (t; u)|2) e�U (t;u) f0(u) du

6 2�2∫(u+ u2) e� �u f0(u)6C (32)

where C does not depend on A04, provided A04 is large enough (so that K ¿ 1 for instance).To conclude the proof, we write that

ddt((�12 + �12)n2 + �21A1)=�21

(�13A3A1 + �13A3

∫u2f1(t; u) du

)

Thanks to (30), we get

(�12 + �12)n2(t) + �21A1(t)6 (�12 + �21)n02 + �12A01 +C̃K

with C̃ uniformly bounded if K ¿ 1. Consequently we may choose A04 large enough such that

lim supt→∞

((�12 + �12)n2(t) + �21A1(t))¡ (�12 + �12)n∗2

Applying again the main result of Reference [7], as in the proof of Proposition 5.1, we knowthat A1 (and n1) converges toward zero whereas A2 is bounded away from 0, since the limitof n2 is strictly less than n∗

2 .Before turning to another proof, we mention that if we had initially that

(�12 + �12)n02 + �21A01¿ (�12 + �12)n∗2

then, no matter what are the values of A3 and A4, but provided f5 = 0, we know that

ddt((�12 + �12)n2 + �21A1)¿ 0

We also know that the limit of A1 is either non-zero or that the limit of n2 is n∗2 . This

guarantees that A2 converges toward 0.




From Equation (19) we have

�52∫ t

0A2(s)A5(s) ds=A05 − A5(t)6A05 ∀t

and Equation (21) is proved. In the same way, we derive Equation (22) from Equation (18).From Equation (17) we obtain

(�31 + �31)∫ t

0A1(s)A3(s) ds=A03 − A3(t)− �34

∫ t

0A3(s)A4(s) ds6Const ∀t

and this proves Equation (23).

6.8. Proof of Theorem 5.2

As it was already observed, from Equation (17) we have two possible cases: either A3 → 0or ∃c¿ 0 s.t. A3(t)¿ c; ∀t.(1) Let us start with the case A3 bounded from below. From Equation (17) we know that∫ +∞

0 A1(t) dt ¡+∞ and, as a consequence,∫ t

0A(s) ds¿ − �21

∫ +∞

0A1(t) dt

Using this inequality in the left-hand side of Equation (20), we have

A2(t)¿A02 e−�21

∫ +∞0 A1(t) dt

and this proves that A2 is bounded from below. From Equation (21), we have that∫ +∞0 A5(t) dt ¡+∞.

(2) Let now consider the case A3 → 0. From Equation (17) we know that∫ +∞

0A1(t) dt=+∞ (33)

We analyse the following three subcases:

Case a:∫ t0 A(s) ds bounded from below, i.e. ∃c ∈ R s.t.

∫ t

0A(s) ds¿ c; ∀t ∈ R+

This implies that A2(t)¿A02ec, so from Equation (19) we deduce the inequality

A5(t)6A05e−�52A02e

ct and as consequence∫ +∞0 A5(s) ds¡+∞.

Therefore, from Equation (33),∫ t0 A(s) ds→ −∞; as t →+∞, but this is a contra-

diction, and∫ t0 A(s) ds cannot be bounded from below.

Case b:∫ t0 A(s) ds converges to −∞, as t →+∞: from Equation (20) this directly implies

A2(t)→ 0; as t →+∞



Case c:∫ t0 A(s) ds be neither bounded from below nor converging to −∞.

We may construct two sequences {tn} and {t̃n} s.t.

tn6 t̃n6 tn+1;∫ tn

0A(s) ds6 − n and

∫ t̃n

0A(s) ds¿ c

with c constant.We show that A5(t)→ 0, as t →+∞. Let sn ∈ [tn; t̃n] be the �rst time s.t.

∫ sn

0A(s) ds= c − 1 and

∫ t

0A(s) ds¿ c − 1; ∀t ∈ [sn; t̃n]

Asddt

∫ t

0A(s) ds=A(t)6 �25A5(t)6 �25A05

we know that t̃n − sn¿ 1=�25A05. Moreover, ∀t ∈ [sn; t̃n]; A2(t)¿A02ec−1 and so

A5(t̃n)6A5(sn)× e�52A02ec−1(�25A05)−1

Consequently, A5(t̃n)→ 0 and this implies A5(t)→ 0, since A5 is decreasing.We now show that A2(t)→ 0, as t →+∞. From Equation (20) we know that:

A2(t)6( ∫

u eCuf02 (u) du)

× e∫ t0 A(s) ds6K e�25

∫ t0 A(s) ds

Given any t16 t2, we have

A2(t1)¿A02 e∫ t10 A(s) ds and A2(t2)6K e

∫ t20 A(s) ds

so

A2(t2)6KA02

A2(t1) e∫ t2t1

A(s) ds

6 K̃A2(t1) exp(�25A5(t1)

∫ t2

t1e−�52

∫ st1

A2(�) d� ds)

Now we prove the following inequality:

A2(t2)6 K̃A2(t1) exp(�25A5(t1)�52A2(t1)

)(34)

for t1 �xed and ∀t¿ t1. If it is not true, then ∃t2 with

A2(t2)¿K̃A2(t1) exp(�25A5(t1)�52A2(t1)

)

De�ne t0 as the largest time t6 t2 with A2(t)=A2(t1).



Clearly t16 t06 t2 and ∀t ∈ [t0; t2] : A2(t)¿A2(t1). Therefore

A2(t2)6 K̃A2(t0) exp(�25A5(t0)

∫ t2

t0e−�52(s−t0)A2(t1) ds

)

6 K̃A2(t1) exp(�25A5(t0)�52A2(t1)

)

6 K̃A2(t1) exp(�25A5(t1)�52A2(t1)

)

as A5 is decreasing, but this is a contradiction, and Equation (34) is proved.Now we know that A5 → 0 and take tn6 t̃n6 tn+1 such that A2(tn)→ 0 and A2(t̃n) is bounded

from below.We can �nd a sequence rn s.t. A2(rn)→ 0 and (A5(rn)=A2(rn))6 (�52=�25).Then ∀t¿ rn, from Equation (34), A2(t)6 K̃A2(rn) e and A2 → 0, but this contradicts the

fact that A2(t̃n) is bounded from below. So it is proved that A2 → 0.


We are in the case ∃ c¿ 0 s.t. A3(t)¿ c; ∀t. From Equation (18) we have

A4(t)6A04 e−�43ct hence

∫ +∞

0A4(t) dt ¡+∞

and because of Equation (17) this implies∫ +∞

0A1(t) dt ¡+∞

We also have, by de�nition,∫ t

0A(t) dt¿ − �21

∫ t

0A1(s) ds¿ − �21

∫ +∞

0A1(s) ds

and from Equation (20)

A2(t)¿A02 e−�21

∫ +∞0 A1(s) ds

so A2 is bounded from below. Finally, from Equation (21) we have∫ +∞0 A5(t) dt ¡+∞ and

this concludes the proof.

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Mathematical models of therapeutical actions related to tumour and immune system competition

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