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A set-valued approach to control immunotherapy
Khalid Kassara
Department of Mathematics, University of Casablanca 1, P.O. Box 5366 CasaMaarif, Casablanca, Morocco Received 22 November 2005; received in revised form 26 March 2006; accepted 29 March 2006
Abstract
In this paper, we use viability theory and set-valued analysis to investigate destroying cancer cells from a tissue that is treated by immunotherapy. The problem is set as a target control problem under state-control constraints, the mathematical model being a control system of three nonlinear ODEs. The goal of the therapy is to get a decreasing tumor cell density that attains zero at terminal time. We derive a family of continuous protocol laws and prove that the minimal selection of the feedback map, though it is discontinuous, stands for the best protocol law that involves minimum doses to clear tumor cells.
c
2006 Elsevier Ltd. All rights reserved.
Keywords:Immunotherapy; Feedback control; Viability theory; Set-valued analysis
1. Introduction
Three decades ago, modern cancer research began involving mathematicians in order to build models [1–4] for effective therapies for cancer patients. Such models, not solely enable a deeper understanding of the mechanisms associated with tumor initiation and progression, but also they open the field of investigations to control theorists for formulating protocol laws that aim at suppressing tumor cells from the tissue, see [5–8] and the references therein.
Cytokines are small secreted proteins which mediate and regulate immunity, inflammation, and hematopoiesis, letting the immune system compete against the tumor. Responses to cytokines involve increasing or decreasing expression of membrane proteins (including cytokine receptors), proliferation, and secretion of effector molecules.
Numerous studies focus on the mathematical modeling of immunotherapy, leading to different kinds of models such as ODE models in [1,5] or PDE models in [2,3].
In order to control immunotherapy an optimal control theory approach is used in [6,7] after a suitable objective functional be designed. For the study [6] that concerns immunotherapy, the objective functional encompasses the total amounts of tumor cells over the therapy session that are to be reduced as well as possible, but also it involves both the total immune cells and the Interleukin-2 to be maximized as well as the amounts of the dosages of supply effector cells to be minimized. By the use of the Hamilton–Jacobi equation, the study reveals the existence of a unique optimal protocol in bang–bang form. We even refer to the work [8] that uses feedback spreading control techniques [9] in order to investigate the immunotherapy PDE model developed by [3].
Nevertheless the approach we will use in this paper is originated from the key idea that prior to clearing tumor cells from the tissue it may be of interest to let them decrease over the therapy horizon. This leads us to formulate the
E-mail address:k kassara@yahoo.com.
0895-7177/$ - see front matter c2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.mcm.2006.03.016
problem in the framework of Aubin viability theory as expounded in [11], enabling us to determine immunotherapy feedback protocol (control) laws by using the technical aid of set-valued analysis and contingent subsets.
The layout of this paper is as follows: in Section 2 we present the model of immunotherapy and set the immunotherapy control problem we will investigate. Section3 is concerned with backgrounds on viability theory along with some facts from set-valued analysis that we will use in the remainder of the paper. Section4is devoted to seeking immunotherapy protocol laws, while Section5gives simulation results and Section6provides concluding remarks with discussions.
2. The model and statement of the problem
The model [1] under consideration in the present paper is constituted of three coupled nonlinear differential equations which govern the evolution of three populations, namely the effector cells, which are the activated immune- system cells (we denote their density byx(·)), the tumor cells (y(·)stands for their concentration), and the Interleukin- 2 (with concentration denoted byz(·)).
These differential equations are set as follows, x˙= f(x,y,z)+s1u(t),
y˙=g(x,y,z),
z˙=h(x,y,z)+s2v(t),
(1)
with initial data,
x(0)=x0, y(0)=y0, z(0)=z0. (2)
The dynamics f,gandhare given by, f(x,y,z) .=cy−µ2x+ a1x z g1+z, g(x,y,z) .=r2y(1−by)− ax y
g2+y, h(x,y,z) .= a3x y
g3+y−µ3z.
(3)
Here all the parameters are positive constants, seeTable 1in Section5for their values and units.
The control (or protocol for cancer specialists) in system(1)and(2)is represented by the couple(u(·), v(·)), where u(·)stands for the dosage of an external source of effector cells andv(·)denotes the supply rate of Interleukin-2 that are cultured with lymphocytes. In this context we conceive a successful therapy by taking into account the following items:
– The protocol(u, v)must range in the subset C =. [0,umax]× [0, vmax],
whereumax and vmax are positive numbers that are estimated so as to keep the toxicity to the normal tissue acceptable.
– It should be of interest to get a decreasing tumor cell concentration y(·)during the therapy horizon [0,tθ]. One advantage is that one will be continuously acquainted on the progress of the therapy.
– Letyattain zero at timetθin order to clear cancer from the tissue.
Thus the control problem we have to deal with is stated as follows, Find a protocol(uθ, vθ)such that:
(uθ, vθ):[0,tθ]→C, (4)
yθ(t)decreases on[0,tθ], (5)
yθ(tθ)=0, (6)
where(xθ,yθ,zθ)denotes a solution of system(1)–(3)when it is controlled by the protocol(uθ, vθ)on the interval [0,tθ]. Motivated by the health of the cancer patient and the cost of the therapy which have to be optimized, we set the following optimal control problem,
Find a protocol(u?, v?):[0,t?]→Csuch that:
(u?, v?)is a solution of(4)–(6)and, (7)
if(uθ, vθ)is a solution of(4)–(6)then:
t? ≤tθ,u?≤uθandv?≤vθon[0,t?]. (8)
3. On viability theory and contingent cones
We devote this section to giving the main mathematical tools to be used in the remainder of this paper. LetDbe a non-empty subset of the Euclidean spaceRnwithn≥1 and define the contingent cone atx∈ Das in [10] as follows
TD(p)=
q ∈Rn|lim inf
ε↓0
d(p+εq,D)
h =0
where d(r,D)=infq∈Dkq−rkfor eachr ∈Rn. It is useful to note the following properties of the contingent cones:
(i) Ifp∈int(D)thenTD(p)=Rn. (ii) TD(p)is closed for eachp∈ D.
(iii) If Dis convex thenTD(p)is convex for eachp∈ D.
(iv) IfDis closed and convex then the mapTD(·)fromDto subsets ofRnis lower semicontinuous. The subsetDis then said to be a sleek subset.
In (iv) above, lower semicontinuity of the mapTD(·)means that for eachx ∈ Dand any sequence of elementsxkof Dconverging toxthen for eachy∈TD(x), there exists a sequence of elementsyk ∈TD(xk)that converges toy. The result below [11] has a central role in building our control laws. We will use it in the next section.
Lemma 1. Let L ⊂ Rn and M ⊂Rk two closed sleek subsets andϕ : Rn → Rk be a continuously differentiable mapping. If p∈L∩ϕ−1(M)satisfies the transversality condition,
ϕ0(p)TL(p)−TM(ϕ(p))=Rk, (9)
then
TL∩ϕ−1(M)(p)=TL(p)∩ϕ0(p)−1TM(ϕ(p)). (10) One of the most important properties of contingent subsets is provided by their use in viability theory [11]. Specifically letϕ:Rn→Rnand a subsetDofRnbe given, thenDis said to be viable under system
ξ˙=ϕ(ξ), t ≥0.
If for allξ0∈ Dthere existst1 >0 and a solutionξ¯ : [0,t1)→ Dwhich satisfiesξ(0)¯ =ξ0. Notet1= ∞if Dis compact. Such a viability property may be characterized by the tangential condition
ϕ(p)∈TD(p) for eachp∈ D (11)
provided that
Dis locally compact and
ϕis continuous onD. (12)
For a set-valued map Q: D→Rnthe mappings: D→Rnis called a selection ofQifs(p)∈ Q(p)for every p.
We take this opportunity to present the Michael selection theorem [10]: If the map Qhas closed convex values and
is lower semicontinuous then for any couple(p0,q0)such thatq0 ∈ Q(p0)the map Qhas a continuous selections which satisfiess(p0)=q0.
The minimal selection (well defined ifQhas closed convex values) of the mapQis given by smin(p)=πQ(p)(0) for all p∈ D.
Hereπ stands for the operator of best approximation that is defined for a non-empty subsetK ofRnas follows, q0=πK(p0)⇔
q0∈K and kp0−q0k = inf
q∈K
kp0−qk.
It is of interest to notice that the minimal selection is rarely continuous, see [10].
We end this section by providing a result that we will use in the next section. Given a control system, ξ˙ =ϕ(ξ)+G(ξ)w,
w∈W(ξ) (13)
wherew ∈Rm denotes the control andW(·)denotes the set-valued map of constraints. Define the feedback map as follows,
F(p) .= {w∈W(p) | ϕ(p)+G(p)w∈TD(p)},
whereG maps D intoL(Rn,Rm). Assume in addition to(12) that G(·)is continuous on D then any continuous selection of the feedback mapF provides a control law that generates a viable solution to system(13)inD. This is so for the minimal selection whenever it is continuous. Otherwise we may use [11, Theorem 4.3.2] as follows.
Lemma 2. Assume that the feedback mapF is lower semicontinuous with convex values. Then system (13)with feedback controlw=πF(ξ)(0)has a viable solution in D over a finite horizon.
It is useful to notice [11] that whenever the right-hand side of the control system(13)has linear growth with respect toξ then the viable solution inLemma 2evolves in an unbounded horizon.
4. Immunotherapy protocol laws
In this section we investigate Problem(4)–(6)in the context of Section3. Out of(1)–(3)the decrease condition(5) may be written as,
g(x(t),y(t),z(t))≤0 for eacht∈ [0,tθ]. This may be reduced to,
ψ(x(t),y(t),z(t))≤0 for eacht ∈ [0,tθ], where,
ψ(x,y,z) .=r2(1−by)(g2+y)−ax, (14)
for all(x,y,z)∈R3+. This is due to the fact that,
y≥0 and g(x,y,z)≤0⇐⇒y≥0 and ψ(x,y,z)≤0, as we have,
g(x,y,z) .= y
g2+yψ(x,y,z) for allx,y,z. (15)
Thereby condition(5)may be expressed in other words by the viability under system(1)of the subset D0=. n
p∈R3+|ψ(p)≤0o .
Next we need to use the notation, Dν =. n
p∈R3+|ψ(p)≤ −νo
, (16)
whereνstands for a non-negative number. Then we are ready to show the following result.
Proposition 3. Assume that the protocol(uθ, vθ) : [0,tθ] → C leads to a solution of system(1), say(xθ,yθ,zθ), which is viable in Dνwithν >0and satisfying the condition
νtθ =yθ(0), (17)
then that protocol is a solution of Problem(4)–(6).
Proof. It is obvious that(uθ, vθ)and(xθ,yθ,zθ)satisfy both(4)and(5). To show(6)we have, yθ(tθ)=yθ(0)+
Z tθ 0
g(xθ(t),yθ(t),zθ(t))dt. Now, as(15)yieldsg≤ψwe get,
yθ(tθ)≤yθ(0)+ Z tθ
0
ψ(xθ(t),yθ(t),zθ(t))dt.
Consequently, since(xθ,yθ,zθ)is viable inDνit follows that, 0≤yθ(tθ)≤yθ(0)−νtθ,
and(17)implies thatyθ(tθ)=0.
Definition 4. The mappingκ :D0→Cis said to be an immunotherapy protocol law (in short, itp law) if the feedback control(u, v)=κ(x,y,z)is a solution to Problem(4)–(6).
This means that system(1)–(3)with an itp law(u, v)=κ(x,y,z)has a viable solution(x,¯ y,¯ z)¯ inD0defined on an interval
0,t¯
and that the protocol(u,¯ v)¯ =κ(x,¯ y,¯ z)¯ is a solution of Problem(4)–(6).
Now we need to define the feedback map for eachν >0 as follows, Fν(p) .=
(u, v)∈C|(f(p)+s1u,g(p),h(p)+s2v)∈TDν(p) (18) for eachp∈ Dν.
Theorem 5. Letν >0, then any continuous selection of the mapFν provides an itp law on a horizon[0,tθ]for all initial data(x0,y0,z0)which satisfy the two statements below:
(i) (x0,y0,z0)∈Dν (ii) νtθ =y0.
Proof. Let κν : Dν ⊂ D0 → C be a continuous selection of the mapFν, then the right-hand side of(1) with (u, v)=κν(x,y,z)is continuous. Furthermore the subsetDν is locally compact because it is closed. Therefore the assumptions(11)and(12)are satisfied. It follows thatDν is viable under system(1)–(3)with(u, v) =κν(x,y,z). By (i), (ii) andProposition 3any extension ofκνtoD0yields an itp law in the sense ofDefinition 4.
Remark 6. We see that itp laws are provided as continuous selections of the mapFν. However it is of interest to stress that discontinuous selections also lead to an itp law whenever they generate a viable solution inDν.
Remark 7. Condition (ii) lets us consider the numberνas the average speed of the therapy.
Thus we are led to search selectionsζ of the mapFν for which system(1)–(3)with(u, v)=ζ(x,y,z)has a viable solution inDν. For that purpose and due to Eq.(18), we need to express the contingent coneTDν(·). We use for that endLemma 1as follows. First of all, by observing(16)we get,
Dν =L∩ψ−1(Mν) for eachν >0
where
L =. R3+ and Mν =. (−∞,−ν].
SinceLandMνare closed and convex, they are sleek subsets as required byLemma 1. Furthermore the mappingψ in(3)is continuously differentiable on Dν, and its partial derivatives are given by,
∂ψ
∂x(p)= −a,
∂ψ
∂y(p)=r2(1−bg2−2ba3),
∂ψ
∂z(p)=0
(19)
for allp=(p1,p2,p3)∈ Dν. Now a direct calculation of the contingent cones of bothLandMν yields q ∈TL(p)⇐⇒
qi ≥0 if pi =0
fori =1,2 or 3 (20)
and
r ∈TMν(m)⇐⇒r ≤0 ifm= −ν, (21)
for allp=(p1,p2,p3)∈ L,m∈Mνandq=(q1,q2,q3)∈R3. Proposition 8.Let p∈ Dν. Then
q ∈TDν(p)⇐⇒
qi ≥0 if pi =0, fori =1,2 or 3, and ψ0(p)q ≤0 ifψ(p)= −ν.
(22) Proof. By considering(20)and(21), it follows that(22)stands for an equivalent expression of the formula
TDν(p)=TL(p)∩ψ0(p)−1TMν(ψ(p)).
Thereby we may useLemma 1to show(22). For that purpose we have to check the transversality condition(9)which may be written in this case as follows,
ψ0(p)TL(p)−TMν(ψ(p))=R for eachp∈ Dν. (23)
Indeed, letp∈ Dνandr ∈R, then we have to prove that there existq¯ ∈TL(p)andm¯ ∈TMν(ψ(p))which satisfy, r =ψ0(p)q¯− ¯m.
By considering the partial derivatives(19), the last equality reduces to, r = −aq¯1+r2(1−bg2−2ba3)q¯2− ¯m.
Now, by using(20)and(21)and distinguishing the casesr ≥0 andr <0, we merely see thatm¯ andq¯can be taken as,
q¯=. (1,0,0)∈TL(p),
m¯ = −. a−r∈TMν(ψ(p)) (forr ≥0), and
q¯=.
−r a,0,0
∈TL(p),
m¯ =. 0∈TMν(ψ(p)) (forr <0).
Hence(23)is satisfied for all p ∈ Dν. As a consequence we may use formula(10)to characterize the elements of TDν(·).
Next, we proceed to determine a useful expression of the mapFν given by(18). To that end we need to let, f0(p) .=r2
a (1−bg2−2ba3)g(p), (24)
for all p=(p1,p2,p3)∈R3+.
Proposition 9. We have for each(u, v)∈C and p∈ Dν, (u, v)∈Fν(p)⇐⇒
`(p)≤u ≤umax if ψ(p)= −ν,
0≤v≤vmax, (25)
where,
` .= max(f0− f,0)
s1 . (26)
Proof. By(18)andProposition 8we get
(u, v)∈Fν(p)⇐⇒
f(p)+s1u≥0 if p1=0, g(p)≥0 if p2=0, h(p)+s2v≥0 if p3=0, α(p)u+β(p)≤0 ifψ(p)= −ν,
for all p=(p1,p2,p3)∈ Dν and(u, v)∈Cand where the functionsαandβare given onDνby, α .=s1∂ψ
∂x and β .= ∂ψ
∂x f +∂ψ
∂yg. (27)
Thanks to(3), we have f(p)≥0 forp1=0 g(p)=0 forp2=0 h(p)≥0 forp3=0. It follows that,
(u, v)∈Fν(p)⇐⇒
(u, v)∈C and,
α(p)u+β(p)≤0 ifψ(p)= −ν.
Now, by observing(19), the functionsα, βof(27)are given for allp ∈Dν by, α(p)= −as1 and β(p)= −a f(p)+r2(1−bg2−2ba3)g(p).
Consequently,
(u, v)∈Fν(p)⇐⇒
(u, v)∈C and,
u≥`ν(p) ifψ(p)= −ν, where,
`ν(p) .= β(p) as1
forψ(p)= −ν, and by(24)we merely see that
`ν = f0− f s1 . That ends the proof.
Fig. 1. Except over the marked zone that isD¯0of Eq.(30), all initial data(x0,y0)are admissible for an itp law formulation. The data ofTable 1 are used to get this graphic.
The minimal selection of the mapFν is given onDνby:
ζ?ν(p)=
(0,0) ifψ(p) <−ν,
(`(p),0) ifψ(p)= −ν. (28)
It is the best in accord with Problem(8). Although it is not a continuous selection it stands for an itp law as we will see below. To that end we need to define the function,
%(y) .=r2
a(1−by)(g2+y) for ally≥0, (29)
and then we let D¯0=. n
(x,y)∈R2+|x > %(y)o
. (30)
Theorem 10. Let the initial data(x0,y0,z0)be given. Then assume that the following statements are satisfied, (i) x0> %(y0), (seeFig.1)
(ii) There existsνsuch that:
0< ν≤a(x0−%(y0)) (31)
maxn
`(p)| p∈R3+, ψ(p)= −νo
≤umax. (32)
Then the minimal selectionζ?ν given by(28)is an itp law on the horizon 0,yν0
.
Proof. First of all, Eq.(31)implies that(x0,y0,z0) ∈ Dν and condition(32)guarantees that the map Fν is strict (i.e.Fν(p)6= ∅for all p∈Dν).
Since the selectionζ?ν is not continuous thenTheorem 5does not apply to it. Nevertheless we may useLemma 2 to show that system(1)–(3)with control(u, v)=ζ?ν(x,y,z)has a viable solution in Dν. For that purpose we need to show that the feedback mapFν is lower semicontinuous and has convex values. The latter fact is obvious. To get lower semicontinuity of that map, letpnbe a sequence ofDν that converges to p ∈ Dν andq =(q1,q2)∈Fν(p). We have to seek a sequence(qn)nthat satisfies
qn∈Fν(pn) for eachn, andqn→q.
Assume thatψ(p) <−ν, since the functionψis continuous and pn→ pwe may consider the smallest numbern0
such that,
ψ(pn) <−ν for alln≥n0, then the sequence,
qn=.
q ifn≥n0, (`(pn),0) ifn<n0,
stands for a suitable choice due to the fact thatψ(pn)= −νwhenevern <n0.
Now suppose thatψ(p)= −ν, it follows thatq1≥`(p). Let the sequenceqnbe such that, qn1=. `(pn)+q1−`(p),
thenqn1→q1as`is continuous andqn∈Fν(pn)for allnbecauseqn1≥`(pn)for allnasq1≥`(p). To end the proof we applyProposition 3to the viable solution that we have just shown the existence.
To end our analysis, we show in the following result how a suitable continuous protocol law may be designed.
Theorem 11. Let(x0,y0,z0)be given as inTheorem10and assume conditions(i)and(ii)therein are satisfied then a continuous itp law does exist on the horizon
0,yν0 .
Proof. The conditions (i) and (ii) imply that(x0,y0,z0)∈ Dν, then any continuous selection of the mapFνprovides an itp law on the horizon
0,yν0
. Now (ii) yields the existence of such a selection. Indeed the latter may for instance be given as follows,
ζλ(p)=
eλ(ψ(p)+ν)`(p),0
for eachp∈Dν, (33)
whereλis a positive number that is great enough such as, max
p∈Dνeλ(ψ(p)+ν)`(p)≤umax.
Remark 12. Condition (i) inTheorem 11seems to express the admissibility of the initial state of the tumor to be treated by using an itp law, see Fig. 1. It may be stressed that this condition involves the stage of the tumor with respect to the immune system: to be admissible for an itp law treatment the initial immune cell density must be greater than%(y0). Therefore the function%may be used to determine the stage of the tumor in such a manner that an itp law can be formulated to fight it.
Next we turn our attention to study the therapy horizon. For a cancer patient having an admissible initial immune/tumor density (ie.x0> %(y0)), thanks to Eq.(31)the minimal therapy horizon is obtained forν .=a(x0−%(y0)). Then it may be given by the function that we define as follows,
τ(x0,y0) .= y0
a(x0−%(y0)) for all(x0,y0)∈ ¯D0. (34)
Then we may show the result below.
Proposition 13. We have the following statements:
(i) The therapy horizon decreases in x0> %(y0)for all fixed y0.
(ii) It increases in y0such that x0> %(y0)over the interval0≤ y0≤qax
0−r2g2
r2b for all fixed x0. Proof. We examine the variation of the functionτ of Eq.(34), by using its partial derivatives,
∂τ
∂x = − 1
a(x−%(y))2,
Table 1
The parameter values in model(1)–(3)
0≤c≤0.2778 µ2=0.1667 a1=0.6917 g1=0.02
g2=10−4 r2=1 b=1 a=5.5556
µ3=55.5556 a3=27.778 g3=10−6
and
∂τ
∂y =ax−r2by2−r2g2 a2(x−%(y))2 . Therefore we get (i) and (ii).
Thereby we may derive the important natural conclusions:
(i) the more the initial immune system is healthy the less therapy needs time to clear cancer.
(ii) tumors in advance need more time to be cleared.
5. Simulation results
According to [1] the parameters of system (1)–(3)are given as inTable 1. The units for the state variablesx,y andzare billions of cells by volume and a time unit corresponds to 5.5 days. The doses of supply of effector cellsu are in billions of cells by time unit. Now, given a patient with initial tumor/immune data(x0,y0,z0)then according to Section4a protocol law that aims at clearing cancer may be formulated by using the itp laws as in the following scheme:
(i) Compute%(y0)and check the stage conditionx0> %(y0)(seeFig. 1).
(ii) Let:
(a) ν:0< ν≤a(x0−%(y0))(the therapy average speed) (b) tθ =. yν0 (the therapy horizon).
(iii) Make a (continuous) selection of the mapFν given by Eq.(25). For instance one may consider the minimal selection given by Eq.(28)or use the continuous selections of Eq.(33).
We use the Matlab program to solve our differential equations (by using the function ode45). The data that remain to be given in order to run our simulations are taken as follows,
x0=1, y0=10, z0=1.2, s1=600 and c=0.1388.
Therefore
%(y0)= −16.2<x0 and a(x0−%(y0))=95.55. Next we take,
ν=55.556 and tθ =1.
For our simulation results we consider, at the level of step (iii) above, two kinds of itp laws, namely:
– the minimal itp law as given by Eq.(28)that is represented inFig. 2.
– the itp laws expressed by formula(33)withλ=0.5 andλ=0.005 for which results are illustrated respectively in Figs. 3and4.
6. Concluding remarks and discussions
In this paper we have investigated the problem of clearing cancer cells from a tissue that is treated by immunotherapy. The approach by set-valued analysis seems to be satisfactory because it provides the technique by which protocol laws can be formulated, namely by making suitable selections from the feedback mapFν that is well expressed in Eq.(25). Insight has been shed on the minimal selection of that map by proving (by using a
Fig. 2. Simulation of immunotherapy by using the minimal law as expressed in(28). We note as expected discontinuity of the control near time 0.2.
Fig. 3. Simulation of immunotherapy by the itp law(33)withλ=0.5, note continuity of this protocol.
Fig. 4. Simulation of immunotherapy by the itp law(33)withλ=0.005. As expected it is greater than the itp law ofFig. 3. The latter is greater than the minimal law ofFig. 2.
topological argument) that it is the best protocol law that minimizes the doses of supply of effector cells, although it is discontinuous in the state variables. The latter fact recalls the conclusion of papers [6] that derive the optimal protocol in bang–bang form, therefore it is discontinuous. Further conclusions can be drawn from our study as follows:
• We stress that the method may be followed for any dynamics f,gandh, especially it is mainly dependent upon the tumor dynamicsgat the level of generation of itp laws. Even stochastic or PDE [3] models for immunotherapy may be taken into consideration since viability results [12] for such kind of models do actually exist in the literature.
• Immunotherapy using itp laws necessitates that the conditionx0 > %(y0)must be satisfied as demonstrated in Theorems 10and11. This condition may be understood as being related to the stage of the cancer with respect to the immune system. This fact actually corresponds to real situations in cancer medicine.
• Although the minimal itp law provides minimum doses to be administrated in order to clear cancer, its discontinuity may be disadvantageous for the cancer patient. An alternative therapy is to consider a continuous itp law. The one proposed in(33)also leads to acceptable doses due to an exponential factor that quickly decreases to zero.
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