Mathematical Models of Therapeutical Actions Related to Tumour and Immune System Competition


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1 Mathematical Moels of Therapeutical Actions Relate to Tumour an Immune System Competition Elena De Angelis (1 an PierreEmmanuel Jabin (2 (1 Dipartimento i Matematica, Politecnico i Torino Corso Duca egli Abruzzi 24, 1129 Torino, Italy (2 École Normale Supérieure Département e Mathématiques et Applications, CNRS UMR rue Ulm, 7523 Paris Ceex 5, France Abstract This paper eals with the qualitative analysis of a moel relate to the escription of two meical therapies which have been intensively evelope in recent years. In particular, we refer to the moeling of the actions applie by proteins, to activate the immune efense, an to the control of angiogenesis, to contrast the growth of tumour cells by preventing the feeing actions of enothelial cells. The therapeutical actions which are object of the moeling process evelope in this paper have to be regare as applie within the framework of the competition between the immune system an tumour cells. We prove the existence of solutions to the Cauchy problem relate to the moel. The efficiency of the therapies an the asymptotic behaviour in time of our solutions is also investigate. Keywors: Vlasov kinetic theory, Cauchy problem, cell population, tumourimmune competition 1 Introuction Methos of the mathematical kinetic theory have been applie in the last few years to moel the competition between tumour an immune cells. The literature in the fiel an a critical analysis on the existing results an open problems can be foun in the review papers [2] an [3]. Mathematical moels are expecte to escribe the interactions an competition between tumours an the immune system. The evolution of the system may en up either with the blowup of the host (with inhibition of the immune system, or with the suppression of the host ue to the action of the immune system. The mathematical structure of the equations suitable to eal with the moeling of the above system have 1
2 been evelope an critically analyze in [1], while motivations from scientists operating in the sciences of immunology in favor of evelopment of methos of nonequilibrium statistical mechanics can be recovere, among others, in [12] an [13]. The mathematical methos are those typical in nonequilibrium statistical mechanics an generalize kinetic theory. The general iea, as ocumente in [4], consists in eriving an evolution equation for the first istribution function over the variable escribing the microscopic internal state of the iniviuals. Generally, this variable may inclue position an velocity, but it can also refer to some aitional specific microscopic features. Interactions between pairs have to be moele taking into account not only mechanical rules but also moifications of the nonmechanical physical (internal state. Specifically we are intereste in a evelopment of the moel propose in [4], whose analytical properties have been stuie in [5], towar the escription of meical therapies which have been intensively evelope in recent years. More precisely a moel of competition between the immune system an tumour cells was propose in [4], this moel i not inclue the effect of any therapy. In [5], it was prove that there exists global solutions to this moel an the possible asymptotic behaviours in time were etaile. The aim of this paper is first to present a general framework (in which the moel of [4] fits which is then use to buil a new moel. The ifference with the previous moel of [4] is that the effect of two ifferent therapies is now taken into account. The last part of the paper stuies the properties of the new moel; Existence of global solutions is a consequence of the proof given in [5] an consequently we only state the result. However the analysis of the asymptotic behaviour in time is more complicate for the new moel than what it was for the one of [4], thus requiring new ieas an new proofs which are presente. Concerning the therapies, we refer to the moeling of the actions applie by proteins to activate the immune efense [15] an to the control of angiogenesis [9], [1], to contrast the growth of tumour cells by preventing the feeing actions of enothelial cells. All above therapeutical actions which are object of the moeling process evelope in this paper have to be regare as applie within the framework of the competition between the immune system an tumour cells. Referring to the literature on the immune competition, evelope within meical sciences, the intereste reaer is aresse to the survey [6]. Another evelopment of the moel introuce in [4] is presente in [7], where the capacity of the boy to repair cells amage is taken into account. The iea is that for a long time range the boy prouces new cells to replace the ea ones in orer to try to reach its normal healthy state. After this introuction, the contents of this paper are organize in five more Sections: Section 2 eals with the erivation of a mathematical framework for the esign of specific moels suitable to escribe the therapeutical actions inicate above. This means eriving a class of integroifferential equations to escribe, by methos of the mathematical kinetic theory, the evolution over the biological state of the cells; 2
3 Section 3 eals with a short escription of the two therapies we will consier: angiogenesis control an activation of the immune efense; Section 4 eals with some phenomenological assumptions about microscopic interactions, by which we obtain, within the general framework, the specific moels object of our interest; Section 5 eals with the presentation of the analytical results relate to the qualitative analysis of the solution to the initial value problem introuce in Section 4. This Section is also eicate to the biological interpretation of the analytical results; Section 6 eals with the technical proofs of the results liste in Section 5. 2 Mathematical framework towars moeling This section eals with the esign of the general framework within which the specific moels propose in the next section will be evelope. The objective is presente through three sequential steps. First we eal with the characterization of cell or particles population, then with the moeling of microscopic interaction, an finally with the erivation of a class of evolution equation. Specific moels can be erive, as we shall see, by specializing the above mentione microscopic interactions. 2.1 Cell populations an statistical representation Consier a large system of cells or particles homogeneously istribute in space. Moeling the immune competition between tumour an immune cells uner the meical action evelope by particles which are artificially inserte into a vertebrate, nees the efinition of the various populations which play the game. In particular, we aopt the following assumptions: Assumption 2.1 Cells an particles are homogeneously istribute in space. The system is constitute by the following populations: cells or particles of the aggressive host, immune cells, environmental cells, an particles of the therapeutical host (with possibly several ifferent populations in this last class. Each population is labele, respectively, by the ineces i = 1,..., n. Assumption 2.2 Each cell is characterize by a certain state, a real variable u I = [, + escribing its main properties: progression for the host cells, activation for the immune cells, feeing ability for the environmental cells, an therapeutical ability for the particles of the therapeutical host. Remark 2.1 The therapeutical ability nees to be specialize accoring to the specific meical action which is being moele. In the cases which will be stuie in the next section, we will consier two populations of particles with therapeutical effect an 3
4 the therapeutical ability can hence be either the control of the activation ability of the immune system (for the first population, or the control of the feeing ability of the environmental cells (for the secon population. Assumption 2.3 The statistical escription of the system is escribe by the number ensity functions N i = N i (t, u, which are such that N i (t, u u enotes the number of cells per unit volume whose state is, at time t, in the interval [u, u + u]. Moreover, if n is the number per unit volume of environmental cells at t =, n = N 3 (, u u, I the escription of the system can be given by the following istributions, normalize such as to have total ensity 1 at t = of environmental cells, f i = f i (t, u = 1 n N i (t, u. Other normalizations are of course possible. We opte for this one as the ensity of environmental cells represents in some sense the normal healthy state of the boy. Remark 2.2 If the istribution function f i is given, it is possible to compute, uner suitable integrability properties, the size of the population still referre to n n i (t = f i (t, u u, (2.1 I an first orer moments such as the activation A i (t = A i [f i ](t = u f i (t, u u, (2.2 of each population. 2.2 Moeling microscopic interactions This section eals with the moeling of microscopic interactions. Specifically we refer to the framework propose in [4], which may be classifie as mean fiel moeling, accoring to the fact that a test cell feels the presence an interacts with the surrouning fiel cells localize in a suitable volume aroun the test cell. Assumption 2.4 Interactions are homogeneously istribute in space an can be ivie into three types of encounters: mass conservative interactions, which moify the state of the pair, but not the size of the population, proliferative an estructive interactions, which prouce proliferation or estruction of the interacting subjects, an population shifting interactions which generate iniviuals into a thir population out of interactions within iniviuals of two ifferent populations. I 4
5 Assumption 2.5 Conservative encounters are interactions moele by the term P ik = P ik (u, u which efines the action on the cell of the ith population with microscopic state u ue to the cell, with state u, of the kth population, so that the resultant action is F ik [f](t, u = P ik (u, u f k (t, u u, (2.3 I where f = {f 1,..., f n }. Aitional conitions shoul be consiere on the term P ik for u = to guarantee the conservation property of the term (2.3 on the interval [, [. We will see, in Section 4, that uner the phenomenological assumptions we consier, such conitions are automatically satisfie. Assumption 2.6 The term escribing proliferation an/or estruction phenomena, within the same population, in the state u ue to pair interactions between cells of the ith population with microscopic state u with cells of the kth population, with microscopic state u, is moele by the source/sink term: S ik [f](t, u = σ ik (u, u ; u f i (t, u f k (t, u u u, (2.4 I I where σ ik is a suitable proliferation estruction function. Assumption 2.7 The term escribing proliferation an/or estruction phenomena, within the ith population, in the state u relate to pair interactions between cells of the jth population with microscopic state u ue to the subject of the kth population, with microscopic state u is given by: where ψ (i jk Q (i jk [f](t, u = I I ψ (i jk (u, u ; u f j (t, u f k (t, u u u, (2.5 is a suitable proliferation estruction function. Assumption 2.8 The above framework refers to the evolution in absence of source/sink terms. This means that cells are containe in a vessel an the system is close. Tumour cells can then replicate exploiting the existing environmental cells. On the other han, the above mentione supply or consumption can be moele for the ith population by suitable source/sink terms I i (t, u 2.3 Evolution equations The mathematical moel consists in an evolution equation for the istribution functions f i corresponing to the above mentione cell populations. The mathematical structure of the moel propose in Section 5 of [4] is as follows: t f i(t, u + F i [f](t, u = S i [f](t, u + Q i [f](t, u + I i (t, u, (2.6 5
6 where, accoring to the framework for microscopic moeling escribe in Eqs. (2.3, (2.4, (2.5, we have: an F i [f](t, u = [ f i (t, u u S i [f](t, u = Q i [f](t, u = n k=1 n S ik [f](t, u, k=1 n n j=1 k=1 ] F ik [f](t, u, Q (i jk [f](t, u. This general framework can generate specific moels after a etaile moeling of microscopic cell interactions, as it will be shown in Section 4. 3 On the control of angiogenesis an immune activation The framework escribe in Section 2 can be exploite to erive specific moels after having properly moele all microscopic interactions among the various subjects playing the game. Of course the specific therapeutical actions which are applie has to be properly specifie. A large variety of therapeutical actions are known in the fiel of meicine: a brief account is given in this section with reference to the following two specific actions (we refer the intereste reaer to [3] for more etails Moeling of the actions applie by proteins to activate the immune efense, e.g. [11] or [15], thus preventing the ability of tumour cells to inhibit immune cells. Control of angiogenesis phenomena, that is the formation of new bloo vessel from preexisting vasculature, e.g. [9] an [1], thus preventing tumour growth by limiting the feeing ability from bloo vessels. There is an experimental evience that immunotherapy have the potential to treat many tumour types, see [8]. The immunotherapy approach consists in the activation of specific tumour antigen combine with incorporation of an immunological ajuvant into a vaccine regime. In etail, cancer vaccines involve the inuction of an active immune response that may lea to the subsequent estruction of tumour tissue. On the other han, as reporte in [16], an aoptive cancer immunotherapy involves the use of tumourkilling lymphocytes an lymphokines engaging in a search an estroy anticancer activity. Following [11], tumour vaccine an cytokine therapy are two methos of promoting an anticancer immune response an these techniques are highly effective when combine. In cancer prevention using cancer vaccines the target is not the tumour mass but the potential risk of cancer (the socalle primary prevention, a preneoplastic lesion (the socalle seconary prevention or a small number of isolate 6
7 neoplastic cells remaining after a temporarily successful therapeutical treatment (the tertiary prevention. Moreover, vaccination after the removal of a tumour mass can stop the formation of minimal resiual isease or metastatic iffusion. Referring to the secon therapy, e.g. the control of angiogenesis phenomena, tumour progression an growth cannot occur without angiogenesis, which supplies the necessary oxygen an nutrients to the growing tumour. Various angiogenesis inhibitors have been evelope to target enothelial cells an stop the process. Referring to [1], a new class of rugs is represente by ifferent type of angiogenic inhibitors an they are extremely important in cases for which the general rules involving conventional chemotherapy might not apply. Inhibitors like angiostatin prevent vascular enothelial cells from proliferating an migrating, while inirect angiogenesis inhibitors can prevent the expression of the activity of one of the tumour proteins which rive the angiogenic switch. Another feature of the angiogenesis process is the evient abnormal vasculature as a hallmark of soli tumour, see [14], an the normalization of this abnormal vasculature can facilitate rug elivery to tumours an it represents an other important goal in the antiangiogenic therapy. 4 Moeling the immune competition an the therapeutical actions The general principles followe towar moeling are precisely the same we have seen in Section 2. Interactions with cells of the other populations moify the biological state an may generate proliferation an/or estruction phenomena. In etail, referring to interactions between host, immune an enothelial cells, we shall essentially exten the moel propose in [4] an [5]. For i = 1, 2, 3 we will inicate, respectively, the aggressive host, the immune system an the environmental cells. Concerning the angiogenesis control an the activation of the immune system, let the relate population of particles be enote respectively by the subscript i = 4 an i = 5. The assumptions which efine the microscopic interactions can be state as follows. Consier first conservative encounters escribe by Eq. (2.3: Assumption 4.1 The progression of neoplastic cells is not moifie by interactions with other cells of the same type. On the other han, it is weakene by interaction with immune cells (linearly epening on their activation state an it is increase by interactions with environmental cells (linearly epening on their feeing ability. The effect increases with increasing values of the progression. Moreover, the progression of the aggressive host is not moifie by interactions with both particles of the antiangiogenesis an of the immune activation therapeutical actions: P 11 =, P 12 (u, u = α 12 uu, P 13 (u, u = α 13 uu, P 14 = P 15 = Assumption 4.2 The efense ability of immune cells is weakene by interactions with tumour cells (linearly epening on their activation state ue to their ability to inhibit the immune system. On the other han, it is not moifie by interactions with other 7
8 cells of the same type, with environmental cells an with the angiogenic therapeutical host. Moreover, it is increase by interaction with the immune activation therapeutical host an the effect is linearly epening on their activation state: P 21 (u, u = α 21 uu, P 22 = P 23 = P 24 =, P 25 (u, u = α 25 uu. Assumption 4.3 The feeing ability of the enothelial cells is weakene by interaction with tumour cells linearly epening on their activation state. On the other han, it is not moifie by interactions with immune cells an with other cells of the same type. Moreover, it is weakene by interaction with the therapeutical host an we assume that this action epens linearly on their activation state. Finally it is not moifie by interactions with the immune activation therapeutical host: P 31 (u, u = α 31 uu, P 32 P 33 =, P 34 (u, u = α 34 uu, P 35 =. Assumption 4.4 The therapeutical ability of the angiogenic therapeutical host is not moifie by interactions with all the other populations an with cells of the same type: P 41 = P 42 = P 43 = P 44 = P 45 =. Assumption 4.5 The therapeutical ability of the immune activation host is not moifie by interactions with all the other populations an with cells of the same type: P 51 = P 52 = P 53 = P 54 P 55 =. Consier now the nonconservative encounters escribe by Eq. (2.4. A simple moeling can be base on the assumption that the terms σ ij are elta functions over the state u of the interacting test cell: for all i, j = 1,..., 5. σ ij (u, u ; u = s ij (u, u δ(u u, Assumption 4.6 No proliferation of neoplastic cells occurs ue to interactions with other cells of the same type. On the other han, interactions with immune cells generate a estruction linearly epening on their activation state, while interactions with environmental cells generate a proliferation epening on their feeing ability an the progression of tumour cells. Moreover, no proliferation arises ue to the interactions with the therapeutical hosts: s 11 =, s 12 (u, u = β 12 u, s 13 (u u = β 13 u u, s 14 = s 15 =. Assumption 4.7 Proliferation of immune cells occurs ue to interactions with tumour cells, linearly epening on their efense ability an on the activation state of tumour cells. On the other han, no proliferation of immune cells occurs ue to interactions with other cells of the same type, with environmental cells an with cells of the therapeutical hosts: s 21 (u, u = β 21 u u, s 22 = s 23 = s 24 = s 25 =. 8
9 Assumption 4.8 A estruction of the environmental cells occurs ue to interactions with tumour cells, linearly epening on the activation state of tumour cells. On the other han, no proliferation of environmental cells occurs ue to interactions with immune cells, with other cells of the same type an with cells of the therapeutical hosts: s 31 (u, u = β 31 u, s 32 = s 33 = s 34 = s 35 =. Assumption 4.9 No proliferation of cells of the angiogenic therapeutical host occurs ue to interactions with tumour cells an with immune cells. On the other han, a estruction occurs ue to interactions with environmental cells, linearly epening on the activation state of environmental cells. No proliferation occurs ue to interactions with cells of the same type an with cells of the immune activation therapeutical host: s 41 = s 42 =, s 43 (u, u = β 43 u, s 44 = s 45 =. Assumption 4.1 No proliferation of cells of the immune activation therapeutical host occurs ue to interactions with tumour cells. On the other han, a estruction occurs ue to interactions with immune cells, linearly epening on the activation state of the cells of the host. Moreover, no proliferation occurs ue to interactions with the immune cells, with cells of the angiogenic therapeutical host an with other cells of the same type: s 51 =, s 52 = β 52 u, s 53 = s 54 = s 55 =. Assumption 4.11 For all the actions we consier, the terms Q (i jk in Eq. (2.5 are equal to zero. This means that the possibility of shift between populations is not relevant. Assumption 4.12 No source terms are consiere in the evolution equations for all the istribution functions f i, expressing the absence of source/sink terms. Base on the above moeling of cell interactions, we are now able to erive the evolution equations (2.6 for each f i : [ ( ] f 1 (t, u = uf 1 (t, u α 12 A 2 [f 2 ](t α 13 A 3 [f 3 ](t t u ( +f 1 (t, u β 12 A 2 [f 2 ](t + β 13 ua 3 [f 3 ](t, (4.1 [ ( ] f 2 (t, u = uf 2 (t, u α 21 A 1 [f 1 ](t α 25 A 5 [f 5 ](t t u +β 21 uf 2 (t, ua 1 [f 1 ](t, [ ( ] f 3 (t, u = uf 3 (t, u α 31 A 1 [f 1 ](t + α 34 A 4 [f 4 ](t t u β 31 f 3 (t, ua 1 [f 1 ](t, (4.2 (4.3 9
10 f 4 t (t, u = β 43f 4 (t, ua 3 [f 3 ](t, (4.4 f 5 t (t, u = β 52f 5 (t, ua 2 [f 2 ](t, (4.5 for all t, u R +, where the activations A i [f i ] are efine in Eq. (2.2. The Cauchy problem for the system (4.1  (4.5 is efine given the initial conitions f i (t =, u = f i (u, i = 1,..., 5, (4.6 for all u R +. All the parameters α an β, which appear in the above Assumptions, have to be regare as positive, small with respect to one, constants, to be ientifie by suitable experiments. 5 Analytic Results an Biological Interpretation In the first part of this section we present the analytical results relate to the qualitative analysis of the solution to the Cauchy problem (4.1  (4.6 an we refer to the next Section 6 for the technical proofs. The following theorem states the existence of the solutions of the Cauchy problem ( Theorem 5.1 Assume that the initial conitions fi, for i = 1,..., 5, satisfy respectively the following assumptions e λu f 1 (uu <, λ >, (5.1 (1 + u e λu f 2 (u u < +, λ >, (5.2 (1 + uf i (uu <, i = 3, 4, 5. (5.3 Then there exists at least one solution (f 1, f 2, f 3, f 4, f 5 C([,, L 1 ((1 + uu to the initial value problem (4.1(4.6 (in the istributional sense which satisfy e λu f 1 (t, uu L ([, T ], λ, T >, (1 + u e λu f 2 (t, uu L ([, T ], λ, T >, (1 + uf i (t, uu (1 + uf i (uu, i = 3, 4, 5. (5.4 1
11 Notice that to obtain the existence of solutions, we woul only require to ask that e λu f2 (u u be finite for all λ >. The stronger hypothesis (5.2 is however useful in what follows, which concerns the asymptotic behaviour in time. We also point out that the equation for f 4 an f 5 are not really necessary. Inee the only interesting quantities are A 4 an A 5 an the relative equations, together with the equation for A 3, are liste in the following lemma. Lemma 5.1 For the solutions to the initial value problem (4.1(4.6 given by Theorem 5.1 we have t A 3(t = [(α 31 + β 31 A 1 (t + α 34 A 4 (t]a 3 (t. (5.5 Remark 5.1 From now on we will use the notation A i := A i ( = i = 1, t A 4(t = β 43 A 3 (ta 4 (t. (5.6 t A 5(t = β 52 A 2 (ta 5 (t. (5.7 u f i (u u, an n i := n i ( = + f i (u u, In general we are not able to preict exactly the asymptotic behaviour given a set of initial ata, except in the case where only the host an the immune system are present where we have the following result. Proposition 5.1 For f3 = f 4 = f 5 =, we efine n 2 = e β 21 u/α 21 f2 (u u an we have that i If (α 12 +β 12 n 2 +β 21 A 1 < (α 12+β 12 n 2, then A 1 as t an A 2 is uniformly boune from below. ii If (α 12 + β 12 n 2 + β 21 A 1 (α 12 + β 12 n 2, then A 2 as t. Before presenting the partial results which we are able to obtain in the general case, we state the next lemma concerne with an estimate for the istribution f 2. Lemma 5.2 Assume fi satisfy (5.1(5.3, i = 1,..., 5. Then we have for the activation A 2 (t the following estimate ( A t 2eR A(s s A 2 (t u e Cu f2 t (u u er A(s s, (5.8 where C = β 21 α 21 + β 21α 25 A 5 α 21 α 52 A 2 an A(s = α 25 A 5 (s α 21 A 1 (s. The next two propositions will show that it is always possible to have a therapeutical action sufficiently high in orer to have, asymptotically, the estruction of the tumour cells. 11
12 Proposition 5.2 It is always possible to choose A 5 sufficiently high so that the activity of the immune system A 2 together with A 3 are boune from below by a positive constant an A 1 converges towar as time tens to infinity. Concerning the first therapy, the result is a bit weaker as it reas: Proposition 5.3 If A 5 =, an (α 12 + β 12 n 2 + β 21 A 1 > (α 12 + β 12 n 2, it is possible to choose A 4 such that A 2 is boune from below an both A 1 an A 3 converge towar zero as time tens to infinity. Notice that this result is optimal in the sense that if A 5 =, an 2n 2 + A 1 < 2n 2, then for any A 4, it is A 2 which converges to since we have convergence towar the asymptotic behaviour prescribe by Prop We o not know the precise conition if A 5 is non zero. We stress the point that the previous two propositions are inepenent from each other. Let us now investigate the asymptotic behaviour in time of the solution. We will use the following lemma. Lemma 5.3 For the solutions to the initial value problem (4.1(4.6 given by Theorem 5.1 we have + A 2 (ta 5 (t t < +, ( From Eq. (5.5, A 3 (t is ecreasing, an hence A 3 (ta 4 (t t < +, (5.1 A 1 (ta 3 (t t < +. (5.11 either A 3 or c > s.t. A 3 (t c, t. First, we consier the asymptotic behaviour in the case A 4 =. Theorem 5.2 Assume that A 5 an A 4 =. Then as t +, there are only the following two possibilities: i If A 3 (t is boune from below by a positive constant, then also A 2 (t is boune from below by a positive constant, + A 1 (t t < + an + A 5 (t t < + ; ii If A 3, then also A 2 (t an + A 1 (t t = +. We are not able to inicate all possible asymptotical features of the system, if A 5 an A 4, but we can still give some partial answers. In particular, property i of Theorem (5.2 remains true. Proposition 5.4 If A 3 (t is boune from below by a positive constant, then also A 2 (t is boune from below by a positive constant, + A 1 (t t < +, + A 4 (t t < + an + A 5 (t t < +. 12
13 An interesting biological interpretation can be given to the main results escribe in this section. First we observe that conitions (5.1(5.3 neee to prove the existence theorem are consistent with the biological requirements of cell populations. Specifically the ecay to zero at infinity implie by conitions (5.1(5.3 can be regare as natural, since the istribution functions f i must effectively ecay to zero at infinity to have a boun to the initial ensity an activation as given by Eqs. (2.1 an (2.2. Exponential might nevertheless seem too emaning but we recall the result obtaine in [5], which also applies here an which tells that, without some exponential ecay, no solution exists even for a short time. The theorem states in (5.4 the L bouneness of the weighte istributions f 1 an f 2 an a boun of f 3 to f 5 with respect to the corresponing quantities at t =. The theorem provies a useful backgroun for simulations, while the results which are useful for a biological interpretation are state in Propositions (5.1(5.4 an Theorem (5.2. In etail, Proposition (5.1 concerns the simplest case where only the tumour cells an the immune system are present an it inicates how the system chooses between the two asymptotic behaviours, alreay escribe in [5], in terms of the initial ata an of the parameters of the moel. We recall that in the general case stuie in [5] we i not have such a result. Propositions (5.2 an (5.3 show the efficiency of the treatment in both the cases of the two therapies consiere, even if the analytical results obtaine in the case of the angiogenesis control is a bit weaker. The two propositions state how in principle is always possible to reach the situation where the immune system wins an completely eliminates the tumour cells of the organism. As a consequence of the activation of the immune system or of the weakening of the feeing ability of the environmental cells, respectively, the activation of the tumour system evolves towar lower egrees of malignity. Unfortunately, these mathematical results o not always correspon to the reality, in the sense that the amount of the initial conitions of the treatments can be not realistic (if a high ose of treatment is angerous for other reasons. Theorem (5.2 shows the complete asymptotic analysis in the presence of the secon therapy. Recalling that this is the therapy referring to the actions applie by proteins to activate the immune efense, the analysis correspons to a well efine meical motivation relate to the action of cytokine signals, [11] an [15]. The competition between the tumour cells an the immune system may en up with the regression of progresse cells, ue to the action of the immune system, or with the blow up of progresse cells an 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 estruction of the host is still possible. In that case, some more complicate asymptotic behaviours are probably possible, corresponing to chronical iseases for instance (with oscillations for the numbers of immune cells an agressive hosts. 13
14 6 Estimates an Proofs This section is evote to the technical proofs of the qualitative properties of the solution to the Cauchy problem (4.1  ( Existence theorem 5.1 The proof of the existence Theorem 5.1 is the same as that of Theorem 2.1 in [5]. We just list here the main steps of the proof: A priori estimates: any solution f i, for i = 1,..., 5, to the system which is the weak limit of compactly supporte solutions in L t L 1 u, satisfies the estimates (5.4. Continuity in time: given A i L ([, T ], for i = 1,..., 5, the weak solutions f i, for i = 1,..., 5, to the system, satisfying the conitions (5.4, also belong to Stability result: C([, T ], L 1 ((1 + uu, T >. Let fi n in C([,, L 1 ((1 + uu, for i = 1,..., 5, be a sequence of solutions in the istributional sense. Assume that they satisfy the a priori estimates (5.4 uniformly in n. Then any weak limit f i, for i = 1,..., 5, of converging subsequences also belongs to C([,, L 1 ((1 + uu an solves the system in istributional sense with initial ata the weak limits of the initial ata. Existence: we construct sequences of approximate solutions of our system, satisfying every a priori estimate. Taking weak limits, we obtain solutions for any initial ata satisfying the hypotheses (5.1  (5.3 of the theorem, on the interval [, T ]. Since the solutions satisfy the estimates (5.4 uniformly in t, we may exten inefinitely the time of existence. 6.2 Proof of Proposition 5.1 If f3 = f 4 = f 5 =, they remain so at any latter time, which obviously simplifies the equations on f 1 an f 2. Multiplying Eq. (4.1 by u an integrating we formally obtain in this case t A 1(t = (α 12 + β 12 A 1 A 2. This formal computation can easily be mae rigorous (see [5] for more etails. Integrating Eq. (4.2 we get t n 2(t = β 21 A 1 A 2. Then with these two relations, we conclue that ((α 12 + β 12 n 2 + β 21 A 1 =. (6.1 t 14
15 We now recall the main result from [5] which states that either A 1 converges towar, A 2 is boune from below an n 2 is boune away from n 2, or A 2 converges towar an n 2 converges towar n 2. The first case means that lim t an the secon that lim t ( (α 12 + β 12 n 2 (t + β 21 A 1 (t < (α 12 + β 12 n 2, ( (α 12 + β 12 n 2 (t + β 21 A 1 (t (α 12 + β 12 n 2. These are exactly the two conitions given in Prop Proof of Lemma 5.1 It is easy to see that the estimates given in the Lemma 5.1 are formally true. For a rigorous proof we refer to Lemma 5.5 in [5]. 6.4 Proof of Lemma 5.2 Let us consier equation (4.2 for the istribution f 2 an the characteristics R t U(t, u = u e A(s s where A(s = α 25 A 5 (s α 21 A 1 (s. Then the equation for f 2 along the characteristics is given by so that [ ] f 2 (t, U(t, u t f 2 (t, U(t, u + t U(t, u u f 2 (t, U(t, u t = t f 2 (t, U(t, u + A(tU(t, u u f 2 (t, U(t, u = [β 21 U(t, ua 1 (t + A(t]f 2 (t, U(t, u, f 2 (t, U(t, u = f2 R t (u e [β 21U(s,uA 1 (s+a(s] s an hence Uf 2 (t, U U t = u er A(s s f2 t (u er [β 21U(s,uA 1 (s+a(s] s t er A(s s u. Consequently we get for A 2 the expression ( A 2 (t = u f2 t (u er [β 21U(s,uA 1 (s] s t u er A(s s. (6.2 15
16 Let us now consier an α 21 I(t = t t For the first integral we have β 21 U(s, ua 1 (s sβ 21 u t er s A(τ τ s =: β 21 u I(t R s t e A(τ τ s A(s s + α 25 er A(τ τ A 5 (s s =: I 1 (t + I 2 (t. I 1 (t = t ] R s s [e A(τ τ s s1 er A(τ τ 1, while from (6.2 we euce that ( A 2 (t u f2 t (u u er A(τ τ A t 2 er A(τ τ. Therefore I 2 (t α 25 A 2 t A 2 (s A 5 (s s α 25 α 52 A (A 5 A 5 (t, 2 where the last equality is obtain from Eq. (5.7. Finally, an so that we obtain ( A t 2eR A(s s A 2 (t I 2 (t α 25A 5 α 52 A, 2 α 21 I(t 1 + α 25A 5 α 52 A 2 u e Cu f2 t (u u er A(s s, where C = β 21 + β 21α 25 A 5 α 21 α 21 α 52 A, which proves Lemma Proof of Proposition 5.2 The proof of Prop. 5.2 will be carrie out in two steps. The first step consists in showing that K, A 5 an t 1 s.t. A 2 (t > K. Given any ifferentiable function λ(t, let us efine J(t = e λ(tu f 1 (t, u u. 16
17 From Eq. (4.1 we have t J(t λ (t u e λ(tu f 1 (t, u u + (α 13 λ(t + β 13 A 3 (t u e λ(tu f 1 (t, u u. Now we choose λ s.t. λ (t = (α 13 λ(t + β 13 A 3, an this implies t J(t an λ(t = Ce α 13A 3 t β 13 α 13, where C is an arbitrary constant. Now we take C s.t. λ(1 = 1 an consier the following estimate, t 1, ( e u f 1 (t, u u e λ(tu C f 1 (t, u u = J(t J( e β 13 u α 13 f 1 (u u. (6.3 Now i A 3 (t A 3, t 1, from (5.5. ii A 2 (t A R 2 e t ( α 25 A 5 (s α 21 A 1 (s s, from (5.8, iii A 1 (s e u f 1 (s, u u 1 α 21 C(A 3, f 1, from (6.3, an using property iii in ii, we have A 2 (t A 2 er t α 25A 5 (s s Ct A 2 e C er t α 25A 5 (s s, t 1. (6.4 Assume now that K s.t. A 5, A 2(t K, t 1. Recalling Eq. (5.7 we get A 5 (t A 5 e α 52Kt an A 2 (t A 2 e C α 25 A 5 e α 52 K [1 e α 52 k ] +, as A 5 + but this is a contraiction an the first step is prove. Now, the secon step consists in showing that we can choose α 1 an K s.t. A 2 K implies A 2 (t αk. Let us choose α, K s.t. A 2 K an α 1. Define T the first time (if it exists s.t. A 2 (t = αk. Then, t [, T ], A 2 (t αk. By Eq. (4.1 we obtain e λu f 1 (t, u u = t ( λα13 A 3 (t + β 13 A 3 (t λα 12 A 2 (t ue λu f 1 (t, u u β 12 A 2 (t e λu f 1 (t, u u 17
18 If we choose λ s.t. then we have A α 13 λ + β 13 3 α 12 λ e λu f 1 (t, u u e β 12αKt αk, (6.5 e λu f 1 (u u, an we use this estimate to euce that A 1 (t 1 e λu f 1 (t, u u 1 λ λ e β 12αKt e λu f 1 (u u. Using the previous computations an the relation t A 2(t α 21 A 1 (ta 2 (t, that we obtain from Eq. (4.2, we have the following estimate for A 2 (t provie A 2 (t A 2 e α 21 1 λ (R t e β 12 αks s R e λu f1 (u u ( K exp α 21 β 12 λαk e λu f1 (u u > αk ( exp α 21 β 12 λαk e λu f1 (u u > α. (6.6 Summarizing, we can choose α, K s.t. (6.5 an (6.6 are satisfie, an in this case we get the contraiction that A 2 (t > αk, an this proves the secon step of the proof of Proposition Proof of Proposition 5.3 Take the equation for A 3 in Lemma 5.1, multiply it by β 43, an subtract it from the equation for A 4 in the same lemma, multiplie by α 34, to obtain t (α 34A 4 β 43 A 3 = β 43 (α 31 + β 31 A 1 A 3. As a consequence α 34 A 4 is larger than K = α 34 A 4 β 43A 3, an this constant as high as we want provie A 4 is high enough. Since we euce that t A 3 α 34 A 4 A 3, A 3 (t A 3 e Kt. (6.7 Now let us write own the characteristics for (4.1, which rea t U(t, u = (α 13 A 3 α 12 A 2 U(t, u, U(, u = u. 18
19 Thanks to (6.7, we have that U(t, u = u er t (α 13A 3 (s α 12 A 2 (s s u er t α 13A 3 (s s Γ u (6.8 with On the other han Γ = e α 13 A 3 /K. t (f 1 (t, U(t, u (β 31 A 3 (t U(t, u + α 12 A 2 α 13 A 3 f 1 (t, U(t, u, an consequently f 1 (t, u f 1 (u e R t (α 13A 3 (s α 12 A 2 (s s e Λu e R t (α 13A 3 (s α 12 A 2 (s s, where Λ is uniformly boune in terms of A 4, provie this last quantity is boune away enough from since α 13 A 3 Λ = β 13 K eα 13 A 3 /K. From the inequality on f 1 an (6.8, we may euce that (u + u 2 f 1 (t, u u (U(t, u + U(t, u 2 e ΛU(t,u f (u u 2 Γ 2 (u + u 2 e Λ Γu f (u C, (6.9 where C oes not epen on A 4, provie A 4 is large enough (so that K > 1 for instance. To conclue the proof, we write that t ((α 12 + β 12 n 2 + β 21 A 1 = β 21 (α 13 A 3 A 1 + β 13 A 3 Thanks to (6.7, we get u 2 f 1 (t, u u. (α 12 + β 12 n 2 (t + β 21 A 1 (t (α 12 + β 21 n 2 + β 12 A 1 + C K, with C uniformly boune if K > 1. Consequently we may choose A 4 such that limsup t ((α 12 + β 12 n 2 (t + β 21 A 1 (t < (α 12 + β 12 n 2. large enough Applying again the main result of [5], as in the proof of Prop. 5.1, we know that A 1 (an n 1 converges towar zero whereas A 2 is boune away from, since the limit of n 2 is strictly less than n 2. Before turning to another proof, we mention that if we ha initially that (α 12 + β 12 n 2 + β 21 A 1 (α 12 + β 12 n 2, 19
20 then, no matter what are the values of A 3 an A 4, but provie f 5 =, we know that t ((α 12 + β 12 n 2 + β 21 A 1. We also know that the limit of A 1 is either non zero or that the limit of n 2 is n 2. This guarantees that A 2 converges towar. 6.7 Proof of Lemma 5.3 From Eq. (5.7 we have t β 52 A 2 (sa 5 (s s = A 5 A 5 (t A 5, t an Eq. (5.9 is prove. In the same way, we erive Eq. (5.1 from Eq. (5.6. From Eq. (5.5 we obtain (α 31 + β 31 t an this proves Eq. ( Proof of Theorem 5.2 t A 1 (sa 3 (s s = A 3 A 3 (t α 34 A 3 (sa 4 (s s Const, t, As it was alreay observe, from Eq. (5.5 we have two possible cases: either A 3 or c > s.t. A 3 (t c, t. 1 Let us start with the case A 3 boune from below. From Eq. (5.5 we know that + A 1 (t t < + an, as a consequence, t + A(s s α 21 A 1 (t t. Using this inequality in the left han sie of Eq. (5.8, we have A 2 (t A 2 e α R + 21 A 1 (t t an this proves that A 2 is boune from below. From Eq. (5.9, we have that + A 5 (t t < +. 2 Let now consier the case A 3. From Eq. (5.5 we know that + We analyze the following three subcases A 1 (t t = +. (6.1 2
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