Download A time delay model for control of malignant tumor growth

NATIONAL CONFERENCE ON NONLINEAR SYSTEMS & DYNAMICS
1
A time delay model for control of malignant tumor growth
Ram Rup Sarkar and Sandip Banerjee
Abstract— We present a delay induced three dimensional deterministic
system, consisting of malignant tumor cells, hunting and resting immune
cells. Stability analysis is performed along with numerical simulations. We
obtain certain thresholds which are helpful to control the malignant cell
growth. Estimation of time delay factor from our study gives suitable preventive strategy. Our analysis is supported through experimental results
available from literature.
Keywords—Malignant tumor cells, immune cells, time delay, differential
equation, stability, control
I. I NTRODUCTION
C
ANCER is one of the greatest killers in the world. It comes
as no surprise that scientists around the world are devoted
to cancer research with particular emphasis on experimental and
theoretical immunology. During recent decades, the disciplines
of cybernetics, nonlinear dynamics and stability theory have
emphasized the importance of studies in this direction. Several authors have suggested different mathematical models of
the disseminated cancers, which are used to capture some essential characteristics of its cell kinetics (see, [1], [2]). Evolutionary biologists and ecologists are contributing new perspectives that may significantly affect an understanding of the clinical behaviour of tumors. Efforts along these lines are now being
investigated through tumor-immune interaction or immunotherapy [3]. Cell-mediated immunity involves the production of cytotoxic T-lymphocytes (CTLs), activated macrophages, and release of various cytokines in response to an antigen. Cytokines
activate and deactivate immune defense cells and inhibit a variety of non-specific body defenses. One of the body’s major defenses against viruses and cancer is the destruction of infected
cells and tumor cells by CTLs inducing a programmed cell death
known as apoptosis.
A good summary on the theoretical studies of tumor-immune
dynamics can be found in [4]. Malignant tumors have properties of great clinical significance that are difficult to deduce
and explain from first principles of their cell and molecular biology alone. All cancer cells live in an ecological system that
requires them to interact with immune cells. Recently, [5] have
approached the problem differently. The model they developed
for spontaneous tumor regression is an interaction between the
immune cell (CTLs) with the malignant tumor cells, that is, a
prey-predator like relationship which is often found in ecological systems. They studied the system under external fluctuations
and proposed certain thresholds which are helpful to control the
malignant tumor growth.
Modelling the actual phenomenon in the tumor-immune cell
interactions is very difficult. Most of the resulting cells react
to eliminate the malignant cells, has the antigen as part of its
structure, but some of the resulting cells remain as memory
Dr.
Ram Rup Sarkar (corresponding author) is with Centre for
Cellular and Molecular Biology, Hyderabad, India, e-mail:
[email protected]/[email protected], and Dr. Sandip Banerjee is with
Department of Biological and Environmental Sciences, University of Helsinki,
Finland
cells. Most CTLs also require cytokines from helper (resting)
T-cells in order to be activated efficiently. Helper T-cells release
interleukin-2 (which stimulates CTLs and convert them into natural killer cells or hunting cells). It is interesting to note that this
activation process and conversion of resting (or helper) T-cells to
hunting cytotoxic T-cells are not instantaneous but followed by
some time lag. This occurs due to several reasons, such as, identification of malignant cells by T-cell receptors, storing information as memory cells, processing the cytolytic information to the
T-helper cell for activation and simultaneous co-stimulation etc.
All the above processes requires some time interval to materialize, though small but cannot be ignored (a detail description
of the above mechanisms has been given in [6]). The study of
biological systems with time delays have been of considerable
interest for a long time (see, [7], [8]). Time delays in connection
with tumor growth also appeared in [9], [10].
Using analytical techniques for estimation of system parameters in such nonlinear dynamical models requires special attention. We modified the model of [5] using the discrete time delay
in degradation of resting cells to hunting cells as well as in the
growth/activation of hunting cells. In this paper, we studied the
delay induced modified model for tumor-immune interaction.
We found different thresholds in terms of activation rate of the
CTLs and this will be helpful to control some of the malignant
tumors (Neurogranuloma, Lung Carcinoma etc.) which are really difficult to treat conventionally.
II. T HE M ATHEMATICAL M ODEL
In the present article, we consider two prominent cellular
species to model the tumor-immune interaction. The predator
is CTLs/ natural killer cells of immune system, which attacks,
destroys or ingests the tumor cell. The prey is the tumor cells
which are killed by the immune cells. The predator has two
stages, hunting and resting. CTLs while attacking tumor cells
release series of cytokines, which activates the resting CTLs that
coordinates the counter attack. The resting cells cannot kill tumor cells, but they are converted to a special type of CTLs called
natural killer or hunting cells and begin to multiply and release
other cytokines. This stimulation or conversion between hunting and resting cells result in a degradation of resting cells undergoing natural growth and an activation of hunting cells. We
assumed the growth of both tumor cells and resting T-cells as logistic growth [11]. Since, tumor cells have a proliferative advantage over the normal (resting) cells [2], the carrying capacity of
tumor cells is greater than that of the normal cells. We consider
that the tumor cells are being destroyed at a rate proportional to
the densities of tumor cells and hunting predator cells according
to the law of mass action. Further, there is a loss in the hunting
cell due to encounters of tumor cells following the mass action
law. We assume that the resting predator cells are converted to
the hunting cells either by direct contact with them or by contact with cytokines produced by the hunting cells. The process
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of conversion has been well discussed in the introduction, which
inspire us to consider the time delay factor (τ units of time) in
the conversion term (from resting to hunting state) and in the
growth term of hunting cells. We also consider that once a cell
has been converted, it will never return to the resting stage and
active cells die at a constant probability per unit of time. This
results in the following model:
dM
dt
dN
dt
dZ
dt
= r1 M (1 −
M
) − α1 M N,
k1
= βN Z(t − τ ) − d1 N − α2 M N,
= r2 Z(1 −
(1)
TABLE I
E STIMATED VALUES OF THE PARAMETERS USED FOR SIMULATION .
Parameters
r1
k1
α1
α2
d1
d2
r2
k2
β
Estimated Values
0.18 / day
5.0 × 106 cells
1.101 × 10−7 / cells / day
3.422 × 10−9 / cells / day
0.0412 / day
0.0412 / day
0.1045 / day
3.0 × 106 cells
4.32 × 10−8 / cells / day
Z
) − βN Z(t − τ ) − d2 Z,
k2
where, M, N, Z are the numbers of tumor cells, hunting
and resting predator cells respectively. r1 , r2 (> 0) are the
growth rates of tumor cells and resting predator cells respectively, k1 , k2 (> 0) are the maximum carrying or packing capacities for tumor cells and resting cells respectively. The terms
d1 N and d2 Z are the respective natural deaths of the hunting and resting cells (d1 , d2 > 0). The term α1 M N represent loss of tumor cells due to encounter with the hunting
cells, and α2 M N that of hunting cells due to encounter with
the tumor cells. Again, there is delay in conversion of resting stage to hunting stage of CTLs, which explains the term
Z(t − τ ). This delay in transformation also induce delay in the
growth of hunting T-cells, and this justifies our claim for considering the term βN Z(t − τ ) in the second equation. System
( 1) has to be analyzed with the non-negative initial conditions
(M (0), N (0), Z(0)) for t ∈ [−τ, 0].
planar equilibrium are respectively, E3 (k1 , 0, kr22 (r2 − d2 )) and
2 )−r2 d1 d1
1
E4 (0, βk2 (r2β−d
, β ). E4 exists if β > k2 (rr22d−d
; and
2k
2
2)
∗
∗
∗
(iv) The interior equilibrium is E∗ (M , N , Z ), where,
M∗ =
k1 [r1 k2 β 2 − α1 {βk2 (r2 − d2 ) − r2 d1 }]
,
β 2 k2 r1 − α1 α2 k1 r2
r1
M∗
α2 M ∗ + d1
.
N∗ =
(1 −
), Z ∗ =
α1
k1
β
This interior equilibrium E∗ (M ∗ , N ∗ , Z ∗ ) exists if
x1 < β, α1 < x2 ,
where,
x1
=
x2
=
III. E STIMATION OF S YSTEM PARAMETERS
In order to examine if system ( 1) is adequate, we have used
the results of experiments on the dynamics of growth of highly
malignant B Lymphoma / Leukemic cells (BCL1 ) in the spleen
of chimeric mice [12]. BCL1 has a number of advantages as
a model tumor system. We consider this data, which represent
the logistic growth of the tumor in the absence of an immune
response. In our case, values of r1 = 0.18 / day, k1 = 5 × 106
cells, give a predicted growth curve that closely approximates
the data and these estimated values are used for our analysis.
But, most of the other parameter values are estimated from [13]
by using a direct integral method [14] to obtain a good initial
guess for the parameters followed by a non linear least squares
fitting to the data in the above experiment. With r1 = 0.18 /
day, k1 = 5 × 106 cells, we estimate the other parameters and
put them in table 1.
IV. S TABILITY A NALYSIS
Now, we find all biologically feasible equilibria admitted by
system ( 1) and study the dynamics around each equilibrium
point. The equilibria for ( 1) are as follows: (i) there exists
equilibrium at the origin E0 (0, 0, 0); (ii) there are two equilibrium points on the boundary of the first and third octants respectively, namely, E1 (k1 , 0, 0) and E2 (0, 0, kr22 (r2 − d2 )). E2
exists if r2 > d2 ; (iii) The M-Z planar equilibrium and N-Z
(2)
r2 d1
α2 k1 r2
+
,
k2 (r2 − d2 ) k2 (r2 − d2 )
β 2 k1 2r1
β 2 r1 k2
min{
,
}.
βk2 (r2 − d2 ) − r2 d1 α2 k1 r2
Biological Implication: With our set of parameter values
(from table 1), we get x1 = 3.2087×10−8 , x2 = min(8.6172×
10−7 , 5.6363 × 10−7 ) = 5.6363 × 10−7 . For the existence of
positive interior equilibrium, the parameter β must be greater
than a threshold value x1 and parameter α1 must be less than
a threshold value x2 , which is true with our choice of parameters. It is worth noted, that for this condition, the steady state
value of the malignant tumor cells (M ∗ in E∗ ) in the presence of
hunting cells, is lesser, than that in the absence of hunting cells
(M = k1 in both E1 and E3 ). This result can also be verified
from the parameter set considered in table 1. This justifies the
need of activation of hunting cells above a certain threshold to
control the malignant cell density.
A. Stability Analysis without delay
By computing the variational matrix of system ( 1) around the
respective biologically feasible equilibria, one can easily deduce
the following lemmas:Lemma 1: The steady state E0 of ( 1) is an unstable saddle
point.
Lemma 2: The existence of the steady state E2 of ( 1) implies
that the steady state E1 is an unstable saddle point.
RIASM, UNIVERSITY OF MADRAS, CHENNAI, 6-8 FEB. 2006
3
Lemma 3: The steady state E2 of ( 1) is an unstable saddle
point.
Lemma 4: The existence of the positive interior equilibrium
E∗ of ( 1) implies that the steady state E3 is an unstable saddle
point.
Lemma 5: The existence of the positive interior equilibrium
E∗ of ( 1) implies that the steady state E4 is an unstable saddle
point.
Lemma 6: System ( 1) is locally asymptotically stable around
E∗ if the following condition hold true
α1
β 2 (2 −
>
2r2
k2
−
α2 k1
d1 )
r1 k 2 β 2
r1 β
r2 d1 + k1 α2
= x4 .
(3)
x 10
Malignant cells decay rate (Alpha1)
p1 =
r1 M ∗
k1
− βN ∗ +
r2 Z ∗
k2 ,
∗ ∗
1
∗ ∗
∗ ∗
k1 k2 [r1 r2 M Z − α1 α2 k1 k2 M N − βk2 r1 M N ],
∗
∗
p3 = Mk1 kN2 [α1 α2 k1 (βk2 N ∗ − r2 Z)],
q1 = βN ∗ , q2 = k11k2 [βk2 r1 M ∗ N ∗ + β 2 k1 k2 N ∗ Z ∗ ],
p2 =
M ∗N ∗
2
∗
k1 k2 [β r1 k2 Z
− α1 α2 k1 k2 βN ∗ ].
Further to observe the existence of periodic solutions in
the delayed system, the eigenvalues will be purely imaginary.
Therefore, we substitute λ = iω (where the real part is zero)
in ( 4) and we separate the real and imaginary parts to obtain a
positive real value for ω. We obtain the system of transcendental
equations as below
p1 ω 2 − p 3
ω 3 − p2 ω
= (q3 − q1 ω 2 )cos(ωτ ) + q2 ωsin(ωτ ), (5)
= q2 ωcos(ωτ ) − (q3 − q1 ω 2 )sin(ωτ ). (6)
Squaring and adding ( 5) and ( 6) we get,
−7
x4
beta = x1
x5
3
ω 6 + A1 ω 4 + A2 ω 2 + A3
Unstable zone for delayed system
(alpha 1 > x5)
2.5
=
0,
(7)
where,
2
Stable zone for both delayed
& non−delayed system
(x4 < alpha 1 < x5, beta > x1)
1.5
A1 =
1
Unstable zone for
non−delayed system
(alpha 1 < x4)
0.5
0
(4)
where,
q3 =
Biological Implication: From ( 3), we get a threshold in
terms of the rate at which tumor cells are destroyed by the hunting cells (α1 ) and the conversion rate β along with other system
parameters. With our set of parameters (from table 1), we observe that x4 = 1.0196 × 10−8 . This threshold is biologically
very much important as it gives the idea for the range of β required to activate resting cells into hunting cells so as to enhance
decay of malignant tumor cells (also, see Fig.1).
3.5
λ3 + p1 λ2 + p2 λ + p3 + (q1 λ2 + q2 λ + q3 )e−λτ = 0,
3.1 3.2087
3.4
3.6
3.8
4
4.2
Coversion rate (Beta)
2α1 α2 M ∗ N ∗ +
4.6 4.7
x 10
−8
Fig. 1. Region of stability for increasing activation rate.
B. Stability Analysis with delay
We do not include the equilibria E0 , E1 , E2 and E3 in this
stability analysis as they are clearly unstable. We do not consider E4 because of the simple fact that M ∗ = 0 (i.e. no tumor
cells) and it is not so important from the view point of disease
management. Hence, we solely concentrate on the stability analysis with delay around E∗ . In our subsequent analysis, we try to
obtain some more threshold conditions on the delay factor τ for
which the system enters a bifurcation. That is, there exists an
τ0 such that for τ < τ0 , positive equilibrium is locally asymptotically stable; as τ increases through τ0 , the periodic solution
can occur, and for τ > τ0 , the positive equilibrium is unstable. This result is in accordance with the fact that the models
with delays are less stable than the analogous models without
delays, which is accepted among both mathematicians and biologists. We now perturb the system around E∗ and obtain a set
of linearized system of differential equations. In case of positive
delay, the characteristic equation for the linearized equations is
given by
+
A3 =
+
r2 Z ∗
(r2 Z ∗
k22
− 2βk2 N ∗ ),
α12 α22 k12 k22 (M ∗ )2 (N ∗ )2
+2α1 α2 k12 M ∗ N ∗ r2 Z ∗ (r2 Z ∗ − 2βk2 N ∗ )
A2 =
4.4
(M ∗ )2 r12
k12
Z ∗ ((M ∗ )2 r12 r22 Z ∗ −2βk2 r12 r2 (M ∗ )2 N ∗ −β 4 k12 k22 (N ∗ )2 Z ∗ )
,
k12 k22
−{ (β
{ α1 α2 k1 (r2 Z
∗
2
k2 r1 −α1 α2 k1 r2 )
}×
k1 k2
−2βk2 N ∗ )+β 2 k2 r1 Z ∗
}(M ∗ N ∗ )2 Z ∗ .
k1 k2
The simple assumption that ( 7) will have a positive root is
A1 > 0 and A3 < 0. Since, the existence condition of E∗ holds
true, we have the condition for A1 to be positive as well as A3
to be negative as
r2 Z ∗ − 2βk2 N ∗ > 0 (8)
3β(d1 + α2 k1 )r1 r2 − 2β 2 k2 r1 (r2 − d2 )
⇒ α1 <
= x5 . (9)
α2 k1 (r2 − d2 )r2
This condition for existence of positive ω gives the range of
α1 (< x5 ) for which a purely imaginary eigenvalue of ( 4) can be
obtained and a stable periodic solution of the system can be observed in the presence of time delay. Also it helps us to identify
the region in parameter space for the stability of the delayed system. If α1 is increased beyond this value the system bifurcates
to unstable situation and no more stable dynamics or stable periodic orbits are available. Later, we confirm our claim through
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NATIONAL CONFERENCE ON NONLINEAR SYSTEMS & DYNAMICS
numerical simulations. It is worth noted that α1 changes with
the variation in the conversion rate β. We present the regions of
stability of the system for both delay and non-delayed case in
Fig.1).
Also, we can say that there is a unique positive ω0 satisfying
( 7), that is, the characteristic equation ( 4) has a pair of purely
imaginary roots of the form ± iω0 . From ( 5) and ( 6), we get,
(p1 ω 2 − p3 )(q2 ω) − (ω 3 − p2 ω)(q3 − q1 ω 2 )
.
(p1 ω 2 − p3 )(q3 − q1 ω 2 ) + (ω 3 − p2 ω)(q2 ω)
=
τn∗
=
1
ω0
arctan[
(p1 ω02 −p3 )(q2 ω0 )−(ω03 −p2 ω0 )(q3 −q1 ω02 )
(p1 ω02 −p3 )(q3 −q1 ω02 )+(ω03 −p2 ω0 )(q2 ω0 )
]
η+
(10)
For τ = 0, E∗ is stable, provided condition ( 3) holds. Hence,
by Butler’s lemma [15], E∗ remains stable for τ < τ0 where
τ0 = τ0∗ as n = 0. We observe that there is a stability switch as τ
crosses τ0 and Hopf-bifurcation occurs at ω = ω0 , τ = τ0 after
λ)
]ω=ω0 ,τ =τ0 > 0.
satisfying the transversality condition [ d (Re
dτ
Note: It must be pointed out that the above analysis cannot
determine the stability of bifurcating periodic orbits, that is, the
periodic solutions may exist either for τ > τ0 or for τ < τ0 ,
near τ0 . Hence, we investigate the stability of bifurcating periodic orbits and try to estimate the maximum length of delay
preserving the stability.
C. Estimation of the Length of Delay to Preserve Stability
Following the lines of [16] and using Nyquist criterion, it
can be shown that the conditions for local asymptotic stability
of E∗ are given by
Im H(i η0 ) > 0,
Re H(i η0 ) = 0,
L1
L3
+ 2nπ
ω0 .
(11)
(12)
where, H(s) = s3 + p1 s2 + p2 s + p3 + e−sτ (q1 s2 + q2 s + q3 )
and η0 is the smallest positive root of ( 12).
We have already shown that E∗ is locally asymptotically stable in absence of delay (by virtue of ( 3)). Hence, by continuity, all eigenvalues will continue to have negative real parts for
sufficiently small τ > 0 provided one can guarantee that no
eigenvalues with positive real parts bifurcates from infinity as τ
increases from zero. This can be proved using Butler’s lemma
[15], already stated before.
In our case, ( 11) and ( 12) gives
p3 − p1 η02
> −q2 η0 cos(η0 τ ) + q3 sin(η0 τ )
−q1 η02 sin(η0 τ ),
= q1 η02 cos(η0 τ ) − q3 cos(η0 τ )
−q2 η0 sin(η0 τ ).
2
= p1 p2 − p3 − q3 + q1 η+
+ p1 q2 ,
1
[ q2 + q22 + 4(|p1 | − q1 )(|p3 + q3 |) ],
=
2(|p1 | − q1 )
then for 0 ≤ τ < τ+ , the Nyquist criterion holds true
and τ+ estimates the maximum length of delay preserving the
stability.
Biological Significance: From the parameter values in table
1 we observe that without delay there exists a unique interior
equilibrium point E∗ (3.5674 × 106 , 0.468425 × 106 , 1.23629 ×
106 ). Positive steady state is locally asymptotically stable,
since the eigenvalues, associated with the variational matrix of
the system ( 1) at E∗ , have negative real parts and are found
as (−0.132774, −0.0193583 + 0.0215968i, −0.0193583 −
0.0215968i). Simulation of the model in this situation with
τ = 0 produces stable dynamics. With the same set of parameters, we find that A1 > 0 and A3 < 0 for the delayed system,
which indicates the existence of a positive root ω. Solving ( 7)
numerically, we see that there exists one simple positive root
of ( 7), namely, 0.0297146 (=ω0 ). Hence, we can say that as
τ increases, stability switch may occur. The value of τ where
stability switch occurs (in our case) is τ0 = 35.95, which can
be easily calculated using ( 10). Hence, by Butler’s lemma, E∗
remains stable for τ < τ0 (= 35.95), which can be seen in Fig.
2 (the phase portrait of the system ( 1) for τ = 28.0).
1,255,000
1,250,000
1,245,000
1,240,000
1,235,000
1,230,000
1,225,000
480,000
475,000
470,000
Hunting Cells
(13)
(14)
Now, if ( 13) and ( 14) satisfy simultaneously then they are
sufficient conditions to guarantee stability. We shall utilize them
to get an estimate on the length of delay. Our aim is to find an
upper bound η+ on η0 , independent of τ and then to estimate τ
(15)
1
2
2
|(q3 − q1 η+
− p1 q2 )|η+
,
2
2
= {|(q2 − p1 q1 )| η+
+ |p1 ||q3 |},
460,000
p2 η0 −
L22 + 4L1 L3 ),
=
465,000
η03
where,
L2
Then τn∗ corresponding to ω0 is given by
1
(−L2 +
2L2
=
τ+
Resting Cells
tan(ωτ )
so that ( 13) holds true for all values of η, 0 ≤ η ≤ η+ and
hence in particular at η = η0 . Our analytical study yields that if,
455,000
3,620,000
3,600,000
3,580,000
3,560,000
3,540,000
3,520,000
3,500,000
Malignant Cells
Fig. 2. Phase portrait depicting stable dynamics for τ = 28.0 (the trajectory
is a spiral and approaches to the steady state). Initial values: M (0) =
3.5 × 106 , N (0) = 0.46 × 106 , Z(0) = 1.3 × 106 .
As τ increases through τ = τ0 = 35.95, a small amplitude
periodic solution occurs which is the case of Hopf-bifurcation.
The importance of Hopf-bifurcation in this context is that, at the
bifurcation point a limit cycle (Fig. 3) is formed around the fixed
point, thus resulting in stable periodic solutions. The existence
RIASM, UNIVERSITY OF MADRAS, CHENNAI, 6-8 FEB. 2006
5
of periodic solutions is relevant in cancer models. It implies
that the tumor levels may oscillate around a fixed point even in
absence of any treatment. Such a phenomenon, which is known
as Jeff’s Phenomenon or self-regression of tumor [17], has
been observed clinically.
Hunting Cell density
1,236,310
1,236,300
1,236,290
Resting Cells
1,236,280
1,236,270
468,380
468,400
468,420
468,440
468,460
Hunting Cells
3,570,000
3,568,000
3,566,000
3,564,000
Malignant Cells
Fig. 3. Phase portrait depicting stable limit cycle for τ = 35.95. Initial conditions and the parameter set are same as before.
No more stability switches occur and for τ > τ0 = 35.95, E∗
is unstable (Fig. 4), with increasing oscillations. It is interesting to observe that for sufficiently large τ , system ( 1) remains
unstable with high level of malignant cell density. This helps us
to identify the region in parameter space needed for activating β
and so as to control malignant tumor growth. Also, one should
keep in mind the allowable time lag (τ < τ0 ) to get control over
the indefinite malignant cell growth or periodic oscillations of
the cells.
the activation rate and the tumor decay rate, which may be effective to control the unlimited growth of malignant tumor cells.
In cancer chemotherapy, stability switching is a very important
issue for designing drug protocol. We must keep in mind that in
many cases the drug prevent cells from continuing through their
cell cycle, thus trapping them at some point during interphase,
where the cells die from natural causes. This effect can be interpreted as an increase of the delay factor (τ ). Another important
issue is to find out the allowable time lag for activation of the
immune cell to fight against the malignant tumor cells. The estimation of delay parameter (particularly the length of delay for
preserving stability) gives the idea about the mode of action for
controlling oscillations in malignant cell growth. We consider a
real data set to establish our analytical observations. We hope
this approach will help to chalk out suitable preventive measure
against the greatest killer.
Acknowledgment: The authors are very much grateful to the
anonymous referees of the article for their useful suggestions
and comments for further improvement.
R EFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
1,800,000
[8]
1,600,000
Resting Cells
1,400,000
[9]
1,200,000
1,000,000
[10]
800,000
600,000
500,000
3,800,000
400,000
[11]
3,600,000
300,000
3,400,000
3,200,000
200,000
3,000,000
Hunting Cells
100,000
2,800,000
Malignant Cells
Fig. 4. Phase portrait depicting unstable dynamics for τ = 40.0 (the trajectory
is a spiral and diverges out from the steady state). Initial conditions and the
parameter set are same as before.
[12]
[13]
[14]
V. C ONCLUSION
In this paper we explore the effects and interactions of tumor
cells and immune cells through a system of nonlinear delay differential equations. The model we propose is very simple and
of general kind. The major difference between this work and
that of the others in this direction is the use of delay differential
equations and subsequent analysis, which appear naturally when
one consider the cell interactions. In these dynamics a key role
is played by the activation rate (from resting to hunting stage)
of the immune cells. Our study reveals certain thresholds for
[15]
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