Download Non-Dimensional System for Analysis Equilibrium Point

Applied Mathematical Sciences, Vol. 8, 2014, no. 2, 91 - 98
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2014.310602
Non-Dimensional System for Analysis Equilibrium
Point Mathematical Model of Tumor Growth
Subiyanto1, 2, Mustafa Mamat1, 3, Mohammad Fadhli Ahmad2
and 4Noor Maizura Mohamad Noor
1
Fellow Research, Department of Mathematics
4
Department of Computer Science
2
Department of Maritime Technology
Universiti Malaysia Terengganu, Malaysia
3
Faculty of Informatics and Computing
Universiti Sultan Zainal Abidin, Terengganu, Malaysia
Copyright © 2014 Subiyanto et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
Abstract. In this paper we presented analysis equilibrium point of mathematical
model tumor growth use non-dimensional technique. This technique is useful to
simply equation that complicated as a mathematical model that present in this
paper. This mathematical model describes the effect of tumor infiltrating
lymphocytes (TIL) and interleukin-2 (IL-2) on the dynamics of tumor cells. The
steps of this technique are presented. With this technique we can be to determine
more accurately the equilibria of our system which form nonlinear dynamics of
system ordinary differential equation that coupled.
Mathematical Subject Classification: 35F25, 37N25, 92C50
Keywords: equilibrium point, interleukin, non-dimensional, tumor infiltrating
lymphocytes, tumor growth
1. Introduction
There are many models of mathematical tumor growth that formulated in
couple of system ordinary differential equations [1, 2, 3, 4, 6, 7, 8, and 11]. All of
92
Subiyanto et al.
these equations in the form nonlinear dynamics, so that to solve these equations
necessary numerical method. In this paper we present our mathematical model as
previous work [5, 9 and 10]. This model is also described in the form nonlinear
dynamics of system ordinary differential equation that coupled. Moreover, many
variable and units that use in this equation. Therefore, we need specific technique
to solve equations like this. One of the techniques to partial or full removal of unit
from an equation is non-dimensional. Non-dimensional is substitution of suitable
variables for partial or full removal of the unit from an equation in involving
physical quantities.
2. Mathematical Model
The mathematical model is a system of ordinary differential equations
(ODEs) whose state variables are populations of tumor cells based on previous
work [9] as follows:
  T  
dT
(1)
 aT 1      cNT  DT
 b 
dt


dN
T2
 eC  fN  g
N  pNT
(2)
dt
h T 2
dL
D 2T 2
(3)
 mL  j
L  qLT  r1 N  r2 C T  uNL2
dt
k  D 2T 2
dC
   C
(4)
dt
(L / T )l
(5)
Dd
s  (L / T )l
Table 1: Variables and their associated units
Variable
t
T
N
L
C
D
Units
day
cell
cell
cell
cell
day-1
Description
Time
Population tumor cells
Population NK cells
Population CD8+T cells
Population circulating lymphocyte cells
Interaction tumor and CD8+T cells
3. Result and Discussion
3.1. Non-dimensional system
Based on the variables present in Table 1 to non-dimensionalized variable t, we
would necessary to multiply it with some combination of constants that has units
Mathematical model of tumor growth
of
93
. From equations (1)–(5) we obtain that the following combinations have
units of
:
a, cb, d, e, f, g, etc
Any one of these would be absolutely correct choices. The main reason we select
one over the other is because we hope to be able to factor out similar term in the
end. For this problem, we decide to use:
~ T ~ N ~ L ~ C ~ D
~
t  a t and T  , N 
, L
,C
, D
T0
N0
L0
C0
D0
as the non-dimensional variables.
Substitution these variables in the equation (1)-(5) we obtained:
~ 
~
dT ~  T0T   c
~~ D ~ ~
(6)
T 1 
 N 0 NT  0 DT
~



dt
b 
a
a



~2
~
dN e C 0 ~ f ~ g T0 T
~ pT ~ ~
C N
N  0 NT
(7)
~ 
2
~
dt
a N0
a
a h T T
a
0
~2 ~2
~
dL
m ~ j D0 D T0T
~ ~
~ qT ~ ~ T
~
L  0 L T  0 r1 N 0 N  r2 C0 C T 2
~  L
2
2
~
~
dt
a
ak D D TT
a
aL0
 
 
  
  
~ ~
N L 
0


0
uL0 N 0
(8)
a
~
dC

 ~
(9)

 C
~
dt
aC0 a
~ l
d  L0 L 
 ~


~ D0  T0T 
(10)
D
~ l
 L0 L 
s   ~ 
 T0T 
Based on the equation (6)-(10), we obtained a new set of non-dimensional
~ ~ ~ ~
~
variables of T , N , L , C , and D as follow:
~ T ~ L ~ Ca ~ Na 2 ~ D
, N
, D
T  , L , C
b
b

a
e
2

Substitute these variables in the equation (6)-(10) we obtained:

 
~
dT ~
~  ce ~ ~ ~ ~
 T 1  T  3 NT  DT
~
dt
a
~
~
dN ~ f ~ g T 2
~ pb ~ ~
C N
N
NT
~
dt
a
a h ~2
a
T
b2
(11)
(12)
94
Subiyanto et al.
~
dL
m~ j
 L
~
dt
a
a
~ ~
D 2T 2
~ qb ~ ~  r1e ~ r2 ~  ~ ube ~ ~2
L
L T   3 N  2 C T  3 NL
k
~2 ~2
a
a
a
 a

D T
2 2
a b
(13)
~
dC
 ~
(14)
 1 C
~
dt
a
~ l
dL
 ~
a  T 
~
(15)
D
~ l
L
s   ~ 
T 
To determine a new set of non-dimensional constants from the equation (11)-(15)
as follows:
ce
d
f
g
h
j
k
m
c   3 , d   , f   , g   , h  2 , j   , k   2 2 , m  ,
a
a
a
a
a
a
b
a b
r

e
r

pb
qb
ub

e



, q 
, r1  1 3 , r2  2 2 , u   3 ,   
p 
a
a
a
a
a
a
Substitute these constants in equation (11)-(15) we obtained simple system
equations:
~
dT ~
~
~~ ~ ~
(16)
 T 1  T  c  NT  DT
~
dt
~
~
dN ~
T2 ~
~
~~

 C  f  N  g  ~ 2 N  p  NT
(17)
~
dt
h T
~
~ ~
dL
D 2T 2 ~  ~ ~
~~
 ~
~ ~
~



m
L

j
L

q
L
T

r
N

r
C T  u  NL2
(18)
~
~
1
2
~

2 2
dt
k D T
~
dC
~
(19)
 1   C
~
dt
~ l
 L 
d  ~ 
~
T 
D
(20)
~ l
L
s   ~ 
T 
By dropping the embellishment and star for notational clarity in equations (16)(20), the non-dimensionalized system is given by:
dT

 T 1  T   cNT  DT
(21)
dt
dN
T2
(22)
 C  fN  g
N  pNT
dt
h T 2
dL
D 2T 2
(23)
 mL  j
L  qLT  r1 N  r2 C T  uNL2
2 2
dt
kD T

 




Mathematical model of tumor growth
95
dC
 1  C
(24)
dt
l
L
d 
T
(25)
D   l
L
s 
T 
These system equations have a similar term with initial equations (1)-(5), yet
these system equations have one less constants or more simple to analysis.
3.2. Determination of equilibria
In order to examine the behavior of these cell populations according to our model,
we note first that equation (24) decouples from equations (21) through (23), so
that we reduce the system down to three equations by allowing the number of
circulating lymphocytes in the body remain to constant at its equilibrium:
1
(26)
C

Since the differential equation for circulating lymphocytes is independent of the
other cell populations, this is its only stable state. Thus, this leaves system
equations (21) through (23), which we set simultaneously equal to zero to obtain
their equilibria. By setting the derivatives in each of these equations to zeros, the
equilibrium point T, N and L can be found as follows:
dT

0
0  T 1  T   cNT  DT
dt


T  1  cN  D 
1
T=0
or
dN
0
dt
N

0  C  fN  g


C h T2
fh  phT   f  g T 2  T 3

(27)
2
T
N  pNT
h T2
(28)
dL
D 2T 2
0
0  mL  j
L  qLT  r1 N  r2 C T  uNL2
2 2
dt
kD T
2
 b  b  4ac
L
(29)
2a
where:


D 2T 2
a  uN  , b   m  j
 qT  and c  r1 N  r2 C T
2 2
kD T


Equation (27) has one zero at the “tumor-free” equilibrium at T  0 , and possibly
several non-zero tumor equilibria. Let the change in tumor population be set to
1
zero by forcing T = 0. Then substitute T = 0 and C  to equation (28) and (29).

96
Subiyanto et al.
C
1
m
or N 
and (uN ) L2  mL  0 or L2  
uN
f
f
We are only interested in equilibria that real and positive, since these problems are
a biological system.
Then tumor free equilibrium exists when:
TE , N E , LE , C E    0, 1 ,0, 1 

 f
In the case where T  0 , the determined are still determine by finding the
simultaneous of solutions (27) through (29), but in this case value must be
determine numerically.
(a)
(b)
N
200
0.07
180
0.06
160
0.05
140
120
L1
L2
0.04
0.03
100
80
60
0.02
40
0.01
20
0
0
0
0.1
0.2
0.3
0.4
0.5
T
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
T
0.6
0.7
0.8
0.9
1
Figure 1. Plot variable T and variable L. (a) Equation (29). (b) Equation (31)
0.1
L1
L2
0.08
0.06
T: 0.5641
L: 0.06418
0.04
L
0.02
T: 0.9875
L: 0.02096
0
-0.02
-0.04
-0.06
-0.08
-0.1
0
0.1
0.2
0.3
0.4
0.5
T
0.6
0.7
0.8
0.9
1
Figure 2. Intersection equations (29) and (31) in non-dimensional system.
Stage to found these solutions as follows:
Solving for N yields in equation (28) for determine solution T  0 in equation
(27)
(30)
D 1  cN  T 
Using this expression in equation (25) gives an expression for L:
Mathematical model of tumor growth
97
1
 sDT l  l

L2  
(31)
d

D


Finally, Equilibrium points of the system (21) through (25) are found by
intersecting equations (29) and (31). These solutions are shown in Figure 1.
From Figure 2 obtained equilibrium points for TE and L E as follows:
TE , LE  0.5641, 0.06418
TE , LE  0.9875, 0.02096
and
These values substitutes in equations (27) and (28) found equilibrium points in the
case T  0 as follows: TE , N E , LE , C E   0.5641, 0.0082, 0.06418, 35.9167 and
TE , N E , LE , CE   0.9875, 0.0047, 0.02096, 35.9167 .
4. Conclusion
There are times when numerically integrating a given equation might take longer
due to the sizes of some of the constants presented in [5]. This can be simplified
significantly by reducing what seems like a completely different system to the
system is essentially governed by the same rules of motion. While nondimensional might seem like a lot of work for very little gain, it is actually a very
useful tool. Likewise, if we can find a constant with units of
, then the
numerical equation can be solved in much less time, and is also less prone to
instability. In this research we find the constant of time with units of
is
~ T ~ N ~ L ~ C
~
t  a t and constant of population cells is T  , N 
, L
, C
,
T0
N0
L0
C0
~ D
that simplify our system ordinary differential equation dimensional to
D
D0
system ordinary differential equation non-dimensional. This system ordinary
differential equation non-dimensional can be used for more accurate determines
the equilibria of our system which form nonlinear dynamics of system ordinary
differential equation that coupled. We obtained three equilibrium points in this
research that only found two in the previous work [5]. For tumor free equilibrium
 1
1
in case T=0 we obtained one equilibrium point is TE , N E , LE , C E    0, ,0, .

 f
Then for tumor high equilibrium in case T  0 we obtained two equilibrium point
as picture in Figure 2.
Acknowledgment. We would like to thank the financial support from Department
of Higher Education, Ministry of Higher Education Malaysia through the
Fundamental Research Grant Scheme (FGRS) Vot. 59256.
98
Subiyanto et al.
References
[1]
N. Bellomo,and L. Preziosi, Modelling and mathematical problems related
to tumor evolution and its interaction with the immune system, Math.
Comput. Model, 32 (2000), 413-452.
[2] N. Bellomo, A. Bellouquid and M. Delitala, Mathematical topics on the
modelling of multicellular systems in competition between tumor and
immune cells, Math. Mod. Meth. Appl. S., 14 (2004), 1683-1733.
[3] L. G. de Pillis and A. E. Radunskaya, A mathematical tumor model with
immune resistance and drug therapy: an optimal control approach, J. Theor.
Med., 3(2001), 79-100.
[4] L. G. de Pillis, W. Gu and A. E. Radunskaya, Mixed immunotherapy and
chemotherapy of tumor: modeling, application and biological
interpretations, J. Theor. Biol., 238 (2006), 841-862.
[5] A. Kartono and Subiyanto, Mathematical Model of The Effect of Boosting
Tumor Infiltrating Lymphocytes in Immunotherapy, Pakistan Journal of
Biology Sciences, 16(20) (2013), 1095-1103.
[6] D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumorimmune interaction, J. Math. Biol., 37(3) (1998), 235-252.
[7] V. Kuznetsov and I. Makalkin, Bifurcation-analysis of mathematical model
of interactions between cytotoxic lymphocytes and tumor cells effect of
immunological amplification of tumor growth and its connection with other
phenomena of oncoimmunology, Biofizika, 37(6) (1992), 1063-1070.
[8] V. Kuznetsov, I. Makalkin, M. Taylor and A. Perelson, Onlinear dynamics
of immunogenic tumors: parameter estimation and global bifurcation
analysis, Bull. Math. Biol., 56(2) (1994), 295-321.
[9] M. Mamat, Subiyanto and A. Kartono, Mathematical Model of Tumor
Therapy Using Biochemotherapy, Journal of Applied Science Research,
8(1) (2012), 357-370.
[10] M. Mamat, Subiyanto and A. Kartono, Mathematical Model of Cancer
Treatment Using Immunotherapy, Chemotherapy and Biochemotherapy,
Applied Mathematical Sciences, 7(5) (2013), 247-261.
[11] E. Rosenbaum and I. Rosenbaum, Everyone’s Guide to Cancer Supportive
Care: A Comprehensive Handbook for Patients and Their Families,
Andrews McMeel Publishing, 2005.
Received: October 21, 2013