Citation: Alexander P. Krishchenko, Konstantin E. Starkov. The four-dimensional Kirschner-Panetta type cancer model: How to obtain tumor eradication?[J]. Mathematical Biosciences and Engineering, 2018, 15(5): 1243-1254. doi: 10.3934/mbe.2018057
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The Kirschner-Panetta equations [2] have a great influence on the modelling tumor dynamics under immunotherapy. These equations describe dynamics of interactions between tumor cells, effector immune cells and interleukin-2 (IL-2). One of the promising generalizations of this model is obtained by incorporating in these equations yet another differential equation characterizing the dynamics of the suppressor cytokine, transforming growth factor-
˙x=rx(1−xk)−awx1+x+p2xz1+z,˙y=−y+p3xw(g4+x)(1+αz)+s2,˙z=−z+p4x2x2+τ2,˙w=cx1+γz−μ1w+wy1+y(p1−q1zq2+z)+s1. | (1) |
In equation (1)
All parameters are supposed to be positive excepting
We notice that if we put
Dynamics of the KP-model has been studied in [1,2,3,14] and some others. In particular, ultimate upper and lower bounds for state variables of the KP- model have been derived in different cases. The main result of [14] consists in global asymptotic tumor clearance conditions obtained under various assumptions imposed on the ratio between the proliferation rate of the immune cells and their mortality rate. To the best of the authors' knowledge, up to now there have not been published any results concerning rigorous dynamical analysis of (1). We notice that the system (1) introduced for the case
In this paper we consider the tumor growth system pertained to the broad class of life sciences models which possess the following characteristic feature: there is a tumor-free equilibrium point, which is the most preferable state of the system. From the biological point of view some deviations from this equilibrium point can be dangerous and cause fatal outcomes.
Therefore, the control goal is to return the system to the indicated equilibrium point and keep it in a sufficiently small neighborhood of this state. It can be done by different treatment injections.
This article establishes the existence conditions for the positively invariant polytope that has a biological meaning. Further, two types of conditions are found: the first one is the tumor persistence (impossibility to achieve the control goal), another one is the tumor eradication (possibility of global asymptotical stabilization at the tumor-free equilibrium point).
The goal and the novelty of this work consists in studies of ultimate dynamics (1) in case of applied treatments
In other words, the three most important results for understanding of any dynamical system's behavior are actually proved: the existence conditions of the global system's attractor; the coincidence of this attractor with the tumor-free equilibrium point; the presence of a local attractor of the system that does not contain the tumor-free equilibrium point, but attracts almost all ("perturbed") trajectories of the system.
Our approach is based on the localization method of compact invariant sets in which the first order extremum conditions are utilized, see [4,5,6]. We also mention that earlier this method has been successfully utilized in studies of various cancer tumor growth models, see e.g. [7,9,10,11,12,13,14,15] and references therein.
The remainder of the paper is organized as follows. In Section 2 we briefly present useful results. In Section 3 under some condition we obtain formulae for a polytope containing all compact invariant sets. This polytope provides us ultimate upper and lower bounds for all variables of the system (1), see Theorem 3.4. In Section 4 under the same condition as in Theorem 3.4 we show in Theorem 4.1 that this polytope contains the attracting set of the system (1). In Section 5 we present the nonexistence conditions of compact invariant sets in
In what follows, we examine dynamics of (1) in the positive orthant
R4+={(w,x,y,z)T∈R4,w;x;y;z>0}; |
let
We consider a nonlinear system
˙x=v(x) | (2) |
where
Assume that we are interested in the localization of all compact invariant sets contained in the set
S(h;U):=S(h)∩U={x∈U∣Lvh(x)=0};hinf(U):=inf{h(x)∣x∈S(h;U)};hsup(U):=sup{h(x)∣x∈S(h;U)}. |
Assertion 1 [5,6] For any
K(h;U)={x∈U∣hinf(U)≤h(x)≤hsup(U)} |
as well. If
Assertion 2. Let
H={x∈U∣hsup(U)<h(x)}. |
Then for any
ˆK(h;U;τ0):={x∈U∣h(x)≤hsup(U)+τ0} | (3) |
is positively invariant.
Assertion 3. Let
H1={x∈U∣hsup(U)+τ0≤h(x)≤hsup(U)+τ0+τ1}, |
then every trajectory of the system (2) goes into the set
Here we find localization sets for the system (1). Let
Lemma 3.1. All compact invariant sets in
K1=K(x,R4+,0)={0≤x≤xmax:=k(p2+r)r}∩R4+,0. | (4) |
Proof. We apply the function
S(h1)={r(1−xk)−aw1+x+p2z1+z=0}∪{x=0} |
and
r(1−xk)+p2z1+z=aw1+x≥0 |
holds. Using the last inequality to calculate the supremum we obtain
h1,sup(R4+,0)=k(p2+r)r. |
Let
Lemma 3.2. All compact invariant sets in
K2={ymin:=s2≤y≤ymax:=(1+r+p2)2k4rη+s2}∩R4+,0. | (5) |
Proof. We apply the function
Lfh2=xw(p3η(g4+x)(1+αz)−a1+x)−ηy+rx−rx2k+p2xz1+z+ηs2=xw(p3η(g4+x)(1+αz)−a1+x)+x−h2+rx−rx2k+p2xz1+z+ηs2. |
In
Lfh2≤x−h2+rx−rx2k+p2x+ηs2 |
because
p3η(g4+x)(1+αz)−a1+x≤0. |
After calculating the supremum we obtain that
h2|S(h2,R4+,0)≤h2,sup(R4+,0)=supx≥0((1+r+p2)x−rkx2)+ηs2=(1+r+p2)2k4r+ηs2. |
Now let us use the function
h3,inf(K∗)=s2;h3,sup(K∗)≤supK∗y≤(1+r+p2)2k4rη+s2. |
Therefore,
K(h3,K∗)⊂{s2≤y≤(1+r+p2)2k4rη+s2}∩K∗=K2 |
and we come to the desirable conclusion.
Lemma 3.3. All compact invariant sets in
K3=K(z,R4+,0)={0≤z≤zmax:=p4}∩R4+,0. | (6) |
Proof. We apply the function
S(h4)={z=p4x2x2+τ2},h4,sup(R4+,0)=p4 |
which leads to the desirable conclusion.
Let
Theorem 3.4. If
μ1>μpol1=μpol1(s2):=p1ymax1+ymax,ymax:=(1+r+p2)2k4rη+s2 | (7) |
then all compact invariant sets are located in the polytope
Π={0≤x≤xmax;ymin≤y≤ymax;0≤z≤zmax;wmin≤w≤wmax}, |
where
wmin=s1μ1+q1p4q2+p4⋅ymax1+ymax−p1s21+s2,wmax=cxmax+s1μ1−μpol1(s2). |
Proof. We apply the function
w(μ1+y1+y(q1zz+q2−p1))=cx1+γz+s1. |
Therefore,
h5|S(h5,M)(μ1−p1ymax1+ymax)≤cxmax+s1;h5,sup(M)≤wmax |
and
s1≤h5|S(h5,M)(μ1+y1+y⋅q1zq2+z−p1y1+y)≤≤h5|S(h5,M)(μ1+ymax1+ymax⋅q1zmaxq2+zmax−p1ymin1+ymin); |
h5|S(h5,M)≥h5,inf(M)≥wmin, |
because
μ1+ymax1+ymax⋅q1zmaxq2+zmax−p1ymin1+ymin>p1ymax1+ymax−p1ymin1+ymin≥0, |
and all compact invariant sets are located in the set
Corollary 1. If
μ1>μM1=μM1(s2):=−q1p4q2+p4⋅ymax1+ymax+p1s21+s2 | (8) |
then all compact invariant sets are located in the set
M1:={wmin≤w}∩M=={0≤x≤xmax;ymin≤y≤ymax;0≤z≤zmax;wmin≤w}. |
Below we shall establish conditions under which the system (1) is dissipative in the sense of Levinson. Here we recall that the system (2) is called dissipative in the sense of Levinson if there exists
limt→∞sup|φ(x,t)|<r; |
here
In this case there exists a bounded set which attracts any trajectory in
Theorem 4.1. If condition (7) is fulfilled then the system (1) is dissipative in sense of Levinson in
Proof. Firstly, we note that extended localization sets
ˆK1={h1=x≤ˆxmax:=xmax+τ1}∩R4+,0; |
ˆK2={h2=x+ηy≤ˆh2,sup(R4+,0):=h2,sup(R4+,0)+τ2}∩R4+,0; |
ˆK3={h4=z≤ˆzmax:=zmax+τ3}∩R4+,0, |
where
Next, if condition (7) is fulfilled then for some sufficiently small
μ1>p1ˆymax1+ˆymax,ˆymax=ymax+τ2η. |
We fix such value of
K:={w≤h5,sup(ˆM)≤¯wmax}∩ˆM,¯wmax=cˆxmax+s1μ1−p1ˆymax1+ˆymax |
(see the proof of Theorem 3.4). By Assertions 2, 3 (see remark below), the bounded set
ˆK4:={h5=w≤ˆwmax:=¯wmax+τ4}∩ˆM,τ4>0, |
is a positively invariant set and every trajectory goes into this set in finite time. As a result, the polytope
Remark 1. The conditions of Assertions 2, 3 are fulfilled for localizing functions
in the set
˙h1=(ˆxmax+Δ1){r−rk(xmax+τ1+Δ1)−aw1+x+p2z1+z}≤−(xmax+τ1+Δ1)rk(τ1+Δ1)≤−(xmax+τ1)rkτ1<0; |
in the set
˙h2≤−τ2−Δ2≤−τ2<0; |
in the set
˙h4≤−τ3−Δ3≤−τ3<0; |
in the set
˙h5≤cˆxmax+s1+(¯wmax+τ4+Δ4)(−μ1+p1ˆymax1+ˆymax)=(τ4+Δ4)(−μ1+p1ˆymax1+ˆymax) |
≤−τ4(μ1−p1ˆymax1+ˆymax)<0. |
Under condition (8) all compact invariant sets lying in the set
Let us denote
C1:=μ1−p1s2s2+1+q1ymaxp4(ymax+1)(q2+p4);C2:=r+rk+p2p41+p4. |
Theorem 5.1. Suppose that (8) and
s1>satt1=satt1(s2):=w0aC1, | (9) |
where
w0={r+p2p41+p4,ifr(1−1k)+p2p41+p4≤0,kC224r,ifr(1−1k)+p2p41+p4>0, |
hold. Then there are no compact invariant sets in the set
Proof. Let us apply the function
r(1−xk)−aw1+x+p2z1+z<0, |
i.e.
awmin>maxx∈[0;xmax](1+x)(r−rxk+p2zmax1+zmax)=w0. |
In order to find
ˆη(x)=(1+x)(r−rxk+p2p41+p4) |
and get
ˆη(0)=r+p2p41+p4>0;ˆη(xmax)<0;ˆη′(0)=r(1−1k)+p2p41+p4; |
ˆη′(x∗)=0 if x∗=k2−12+kp2p42r(1+p4);ˆη(x∗)=k4rC22. |
Therefore if
If
μ1>μTFEP1=μTFEP1(s2):=p1s21+s2 | (10) |
the system (1) has the TFEP
E1=(0,s2,0,w1)T, |
where
w1=s1(1+s2)μ1+μ1s2−p1s2=s1μ1−p1s21+s2. |
The TFEP is asymptotically stable if
s1>sst1=sst1(s2):=ra(μ1−p1s21+s2). | (11) |
Theorem 6.1. If conditions (7); (9) and (11) hold then the TFEP attracts all trajectories in
Proof. If conditions of this theorem are fulfilled then all trajectories of the system (1) go into bounded positively invariant set
The system (1) has no compact invariant sets in
˙y=−y+s2,˙z=−z,˙w=−μ1w+wy1+y(p1−q1zq2+z)+s1. | (12) |
Next, applying localizing functions
K(y,R3)=Y:={y=s2};K(z,R3)=Z:={z=0};K(w,Y∩Z)={(s2,0,w1)}. |
Now let us prove that the system (1) has no compact invariant set
Theorem 7.1. Suppose that condition (7) holds and
s1<sper1=sper1(s2):=ra(μ1−p1ymax1+ymax)−ck(p2r+1). | (13) |
Then in
P:=ˆK4∩{x≥x+−τ5} |
where sufficiently small
x+:=k−12+√(k−1)24+k−akrwmax |
in finite time.
Proof. In the set
˙x≥x(r−rkx−aˆwmax1+x)=−rxk(1+x)Q(x),Q(x)=x2+x(1−k)+akrˆwmax−k. |
We note that
{0<x≤x+−τ5}∩ˆK4. |
For any
Now we describe how some features of ultimate dynamics depend on values
μTFEP1(s2)=p1s21+s2(see (10));μM1(s2)=−q1p4q2+p4⋅ymax1+ymax+p1s21+s2(see (8));μpol1(s2)=p1ymax1+ymax(see ( 7)). |
It is easy to see that if
μM1(s2)<μTFEP1(s2)<μpol1(s2) |
is fulfilled. We notice that the value
(ⅰ) the localization set
(ⅱ) TFEP, with
(ⅲ) bounded localization set
Let us consider the case when our system has the TFEP, i.e.
sper1(s2)=ra(μ1−p1ymax1+ymax)−ck(p2r+1)==ra(μ1−μpol1(s2))−ck(p2r+1)(see (13)); |
sst1(s2)=ra(μ1−s21+s2)=ra(μ1−μTFEP1(s2))(see (11)); |
satt1(s2)=w0a(μ1−p1s2s2+1+q1ymaxp4(ymax+1)(q2+p4))=w0a(μ1−μM1(s2))(see (9)). |
It is easy to see that
sper1(s2)<sst1(s2)<satt1(s2), s2>0. |
If
s1<sper1(s2);μpol1(s2)<μ1 |
we have the tumor persistence.
If
s1>satt1(s2);μpol1(s2)<μ1 |
all trajectories in
The main contribution of the present paper lies in the rigorous dynamical analysis of the four-dimensional system (1) and in obtaining global tumor clearance conditions via the localization method of compact invariant sets. We have studied various aspects of the ultimate dynamics of (1) describing interactions of cancer cells, TGF-
1. Under condition (7) we have found all upper bounds for variables of the state vector of the system (1). Moreover, in this case it was shown that (1) has the property of the dissipativity in the sense of Levinson, because there exists the positively invariant polytope.
2. Further, we provide conditions (8) and (9) under which there are no compact invariant sets in the set
3. We find conditions (7); (9) and (11) under which the TFEP attracts all trajectories in
4. Tumor eradication and tumor persistence bounds are compared in Section 8. One can point to the following essential difference of dynamics of (1) in cases
All assertions are formulated in terms of simple algebraic inequalities imposed on parameters of the model and treatments. These inequalities are stable for sufficiently small perturbations caused by imprecise knowledge of parameters' values which is convenient in applications.
1. We thank anonymous referees for the careful reading of this manuscript and helpful comments. 2. The work of the first author is supported by the Ministry of Education and Science of the Russian Federation (project Scientific Research Organization no. 1.4769.2017/6.7) and by the Russian Foundation of Basic Research (projects 16-07-00927 and 16-07-00902). The work of the second author is supported by the CONACYT project N 219614 "Analisis de sistemas con dinamica compleja en las areas de medicina matematica y fisica utilizando los metodos de localizacion de conjuntos compactos invariantes", Mexico.
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