Proceedings of the 2nd WSEAS International Conference on BIOMEDICAL ELECTRONICS and BIOMEDICAL INFORMATICS
The Simulation Of A Stochastic Model For Tumour-Immune System
RALUCA HORHAT1, RAUL HORHAT2, AND DUMITRU OPRIS3
1 ENT Department
V. Babes University of Medicine and Pharmacy Timisoara
P-ta Eftimie Murgu nr 2, 300041, Timisoara
2 Cmed Research Timisoara
Str Coriolan Brediceanu nr 10,300011, Timisoara
3 Mathematics Department
West University Timisoara
3 Blvd V Parvan nr 4, 300223, Timisoara
ROMANIA
1 rhorhat@umft.ro 2 rhorhat@cmedresearch.com 3 opris@math.uvt.ro
Abstract. In this paper we investigate some stochastic model for tumour-immune system. To describe this
model we used a Wiener process, as the noise has a stabilization effect. Their dynamics are studied in terms
of stochastic stability in the equilibrium points, by constructing the Lyapunov exponent, depending on the
parameters that describe the model. We have studied and analyzed a Kuznetsov-Taylor like stochastic model
for tumour-immune systems. These stochastic models are studied from stability point of view and they were
represented using the Euler second order scheme.
Keywords: tumour-immune system, modelling and simulation, stochastic model, Kuznetsov-Taylor
model,Wiener process, Lyapunov exponent.
the best known finite dimensional models [4,10], we
note that their main features are the following:
existence of a tumor free equilibrium;
depending on the values of parameters, there is
the possibility that the tumor size may tend to + or
to a macroscopic value;
1 Introduction
Millions of people die from cancer every year. And
worldwide trends indicate that millions more will
die from this disease in the future. Great progress
has been achieved in fields of cancer prevention and
surgery and many novel drugs are available for
medical therapies [4] ,[10],[12]. Biophysical models
may prove to be useful in oncology not only in
explaining basic phenomena [1], but also in helping
clinicians to better and more scientifically plan the
schedules of the therapies [12]. An interesting
therapeutic approach
is immunotherapy [4],
consisting in stimulating the immune system in
order to better fight, and hopefully eradicate, a
cancer. In particular, in this paper I will be referring
to generic immunostimulations, for example, via
cytokines, but for the sake of simplicity I will use
the term immunotherapy. The basic idea of
immunotherapy is simple and promising but the
results obtained in medical investigations are
globally controversial, even if in recent years there
has been evident progress.
From a theoretical point of view, a large
body of research has been devoted to mathematical
models of cancer-immune system interactions and to
possible applications to cure the disease . Analyzing
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possible existence of a small tumor size
equilibrium, which coexists with the tumor free
equilibrium.
Stochastic modelling plays an important
role in many branches of science. In many practical
situations perturbations appear and these are
expressed using white noise, modelled by Brownian
motion. We will study stochastic dynamical systems
that are used in medicine, in describing a tumour
behaviour, but still we don't know much about the
mechanism of destruction and establishment of a
cancer tumour, because a patient may experience
tumour regression and later a relapse can occur. The
need to address not only preventative measures, but
also more successful treatment strategies is clear.
Efforts along these lines are now being investigated
through immunotherapy ([10], [12]). This tumourimmune study, from theoretical point of view, has
been done for two cell populations: effectors cells
and tumour cells. It was predicted a threshold above
which there is uncontrollable tumour growth, and
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ISBN: 978-960-474-110-6
Proceedings of the 2nd WSEAS International Conference on BIOMEDICAL ELECTRONICS and BIOMEDICAL INFORMATICS
below which the disease is attenuated with periodic
exacerbations occurring every 3-4 months. There
was also shown that the model does have stable
spirals, but the Dulac-Bendixson criterion
demonstrates that there are no stable closed orbits. It
is consider ODE's for the populations of immune
and tumour cells and it is shown that survival
increases if the immune system is stimulated, but in
some cases an increase in effectors cells increases
the chance of tumour survival.
In the last years, stochastic growth models
for cancer cells were developed, [1], [2], [3], [4],
[5], [6].
Our goal in this paper is to construct
stochastic model and to their behaviour around the
equilibrium point. In these points stability is studied
by analyzing the Lyapunov exponent, depending of
the parameters of the models. Numerical
simulations are done using a deterministic algorithm
with an ergodic invariant measure.
In this paper the authors studied and
analyzed one stochastic model. In Section 2, we
considered a Kuznetsov and Taylor stochastic model
[9]. Beginning from the classical one, we have
studied the case of positive immune response. We
gave the stochastic model and we analyzed it in the
equilibrium points. Numerical simulations for this
new model are presented in Section 3. We wrote this
model as a stochastic model, and we discussed its
behaviour around the equilibrium points. Numerical
simulations were done using the software Maple 12
and we implemented the second order Euler scheme
for a representation of the discussed stochastic
models. Finally, some medical inferences are
proposed.
f2(x; y) = (h3(x) + h4(x))y + h5(x).
The functions h1, h2, h3, h4, h5 are given
such that the system (2) admits the equilibrium
point P1(x1; y1) with x1 = 0; y1 > 0 and P2(x2; y2)
with x2 > 0, y2 > 0.
Particular cases are the following:
1.Volterra-model : h1(x) = a1, h2(x) = a2x, h3(x) =
b3x, h4(x) =-b2, h5(x) = -b1x.
2. Bell-model:h1(x) = a1x; h2(x) = a2x; h3(x) = b1x,
h4(x) =b3, h5(x) = -b2x + b4.
3. Stepanova-model : h1(x) = a1, h2(x) = 1, h3(x) =
b1x, h4(x) =b2 + b3x2, h5(x) = 1.
4. Valdar-Gonzales model: h1(x) = log(k/x), h2(x)
= 1, h3(x) = b1x, h4(x) =b2 + b3x2, h5(x) = 1
5.Exponential-model : h1(x) = 1, h2(x) = 1, h3(x) =
b1x, h4(x) =b2 + b3x2, h5(x) = 1.
6. Logistic model : h1(x) = 1-a1/x, h2(x) = 1, h3(x)
= b1x, h4(x) =b2 + b3x2, h5(x) = 1.
In this paper we will begin our study from
the model of Kuznetsov and Taylor given by (1) if
a3 > 0 that means that immune response is positive.
For the equilibrium states P1 and P2; we study the
asymptotic behaviour with respect to the parameter
a1 in (1). For b1a2 <a1; the system (1) has the
equilibrium states P1(x1; y1) and P2(x2; y2) with x1,
y1, x2, y2 given in [9].
In [9] it is shown that there is an a10 such
that if a1 < a10; the equilibrium state P1 is
asymptotical stable, for a1 > a10 the equilibrium
state P1 is unstable and if a1 < a10 the equilibrium
state P2 is unstable and for a1 > a10 the equilibrium
state P2 is asymptotical stable.
In the following, we associate a stochastic
system of differential equations to the classical
system of differential equations (1). Let us consider
(Omega; Ft>0; P) a filtered probability space and
(W(t))t>0 a standard Wiener process adapted to the
filtration (F)t>0: Let {X(t) =(x(t); y(t))}t>0 be a
stochastic process.
The system of Ito equations associated to
system (1) is given by
2 Problem Formulation
We will begin our study from the model of
Kuznetsov and Taylor [9]. This model describes the
response of effectors cells to the growth of tumour
cells and takes into consideration the penetration of
tumour cells by effectors cells that causes the
interaction of effectors cells. This model can be
represented in the following way:
dx(t) = (a1 - a2x(t) + a3x(t)y(t))dt, (1)
dy(t) = (b1y(t)(1 - b2y(t)) - x(t)y(t))dt.
where initial conditions are x(0) = x0 > 0, y(0) = y0
> 0 and a3 is the immune response to the appearance
of the tumour cells.
A general representation for such models
can be considered in form given in [4].
dx(t) = f1(x(t), y(t)),
dy(t) = f2(x(t), y(t)).
x(0) = x0, y(0) = y0.
where
f1(x; y) = x(h1(x) + h2(x)y),
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x(t ) = x0 + (a1 a 2 x( s ) + a3 x( s) y ( s) )ds +
0
+ g1 ( x( s ), y ( s) )dW ( s)
0
y (t ) = y 0 + (b1 y ( s)(1 b2 y ( s ) ) x( s ) y ( s ) )ds +
0
+ g 2 ( x( s ), y ( s ) )dW ( s )
0
where the first integral is a Riemann integral, and
the second one is an Ito integral. {W(t)}t>0 is a
Wiener process [11].
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Proceedings of the 2nd WSEAS International Conference on BIOMEDICAL ELECTRONICS and BIOMEDICAL INFORMATICS
The functions g1(x(t), y(t)) and g2(x(t), y(t))
are given in the case when we are working in the
equilibrium state. In P1 those functions have the
following form
g1(x(t), y(t)) = b11(x(t)-x1) + b12(y(t)-y1)
g2(x(t),y(t)) = b21(x(t)-x1) + b22(y(t)-y1) .
I
In the equilibrium state P2, the functions
g1(x(t), y(t)) and g2(x(t), y(t)),are given by
g1(x(t), y(t)) = b11(x(t)-x2) + b12(y(t)-y2)
g2(x(t), y(t)) = b21(x(t)-x2) + b22(y(t)-y2) .
The functions g1(x(t), y(t)) and g2(x(t), y(t))
represent the volatilizations of the stochastic
equations and they are the therapy test functions.
Fig. 2 displays the optimal behavior of the tumor
cellules for SDE(2) in P1
3 Problem Solution
Using the formulae from [8], [10] and Maple 12
software, we get the following results, illustrated in
the figures below. For numerical simulations, we
use the following values for the parameters of the
system (1):
a1 = 0.1181; a2 = 0.3747; a3 = 0.01184; b1 = 1.636;
b2 = 0.002:
Using the second order Euler scheme for the
ODE system (1), respectively SDE system (2), we
get the following orbits.
Fig 3: (n; y(n)) in P1 for ODE (1)
Fig. 3 displays the optimal behavior of the
effectors cellules for ODE(1) in P1
Fig 4: (n; y(n,omega)) in P1 for SDE (2)
Fig. 4 displays the optimal behavior of the
effectors cellules for SDE(2) in P1
Fig 1: (n; x(n)) in P1 for ODE (1)
Fig. 1 displays the optimal behavior of the tumor
cellules forODE(1) in P1
Fig 5: (x(n); y(n)) in P1 for ODE (1)
Fig. 5 displays the optimal behavior of the tumor
cellules vis. the effectors cellules for ODE(1) in
P1
Fig 2: (n; x(n; omega)) in P1 for SDE (2)
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Proceedings of the 2nd WSEAS International Conference on BIOMEDICAL ELECTRONICS and BIOMEDICAL INFORMATICS
Fig. 9 displays the optimal behavior of the
effectors cellules for ODE(1) in P2
Fig 6: (x(n,omega); y(n,omega)) in P1 for SDE
(2)
Fig. 6 displays the optimal behavior of the tumor
cellules vis. the effectors cellules for SDE(2) in
P1
Fig 10: (n; y(n,omega)) in P2 for SDE (2)
Fig. 10 displays the optimal behavior of the
effectors cellules for SDE(2) in P2
Fig 7: (n; x(n)) in P2 for ODE (1)
Fig. 7 displays the optimal behavior of the tumor
cellules for ODE(1) in P2
Fig 11: (x(n); y(n)) in P2 for ODE (1)
Fig. 11 displays the optimal behavior of the
tumor cellules vis. the effectors cellules for
ODE(1) in P2
Fig 8: (n; x(n; omega)) in P2 for SDE (2)
Fig. 8 displays the optimal behavior of the tumor
cellules for SDE(2) in P2
Fig 12: (x(n,omega); y(n,omega)) in P2 for SDE
(2)
Fig. 12 displays the optimal behavior of the
tumor cellules vis. the effectors cellules for
SDE(2) in P2
Fig 9: (n; y(n)) in P2 for ODE (1)
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Proceedings of the 2nd WSEAS International Conference on BIOMEDICAL ELECTRONICS and BIOMEDICAL INFORMATICS
general family [4] and [9] may be, of course, further
generalized following Volterras ecological theory,
i.e. by considering that there may be a delay
between the consumption of a prey and the birth of
a predator. This delayed model and stochastic
models will be the subject of further investigations.
In this paper we focused on important
tumour-immune systems, presented from stochastic
point of view: a Kuznetsov-Taylor model, that
belongs to a general family of tumour-immune
stochastic systems. We have determined the
equilibrium points and we have calculated the
Lyapunov exponents. These exponents help us to
decide whether the stochastic model is stable or not.
For numerical simulations we have used the Euler
scheme and the implementation of this algorithm
was done in Maple 12. In a similar way other
models can be studied. The model given by the SDE
(2) allows the control of the model given by ODE
(1) with a stochastic process.
Finally, we would like to illustrate some
qualitative
medical
inferences
from
the
investigations that we have here proposed. The main
problem of immunotherapy is that, as it is clear from
our analysis and simulations, in general, eradication
may be possible but is dependent on the initial
conditions (x0,y0). However, the Ics are in medical
practice unknown or known with very large
confidence intervals ( for the cancer cells at the start
of a radiotherapy and). This makes it impossible to
plan an anticancer therapy based solely on this
therapy. This is a peculiarity of immunotherapy,
since there are other kinds of anticancer cures for
which a globally stable eradication is possible [4].
However, in our simulations we have seen that in
some particular cases the model [9] predicts that
globally stable eradication is possible also in case of
immunotherapy, but that it depends on the degree
of aggressiveness of the cancer.
If in the future it might be possible, the
option to use immunotherapy as main strategy, for
relatively small non-aggressive tumors, could be
seriously considered boli based therapy. This result
may be of interest, since continuous intravenous
infusion may cause major practical problems to the
patients.
The Lyapunov exponent variation, with
b11= a variable parameter, is given in Figure 13 for
the equilibrium point P1, and in Figure 14 for the
equilibrium point P2:
Fig 13 : (; lambda()) in P1
Fig 14 : (; lambda()) in P2
From the figures above, the equilibrium
points P1 and P2 are asymptotically stable for all
such that the Lyapunov exponents lambda() < 0;
and unstable otherwise. So P1 is asymptotically
stable for from (- ,-1.78)U(2.02; ) and P2 is
asymptotically stable for from
(- ;1.62)U(1.88; ):
4 Conclusion
It is interesting to use well established conceptual
frameworks of ecological models to model
competition phenomena in human biology, but it is
important to pay attention to the whole ecological
modeling aspect, such as the basic requirement of
the positivity of the solutions. Even if model [12]
violates the positivity rule, it is valuable because it
may be read as a model which takes into account a
disease-induced depression in the influx of
lymphocytes. Then, instead of proposing another
specific model, we preferred to add this new feature
to a family of equations, and to analyze its
properties. We stressed also that models which do
not allow the possibility to have LAS tumor-free
solutions should be cautiously considered. The
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Proceedings of the 2nd WSEAS International Conference on BIOMEDICAL ELECTRONICS and BIOMEDICAL INFORMATICS
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