INTRODUCTION
Immunotherapy is also called as biologic therapy or biotherapy. Immunotherapy
become an important component in the multi-pronged treatment approach, it was
developed to treat several types of tumor within a short time frame (De
Pillis et al., 2006). Immunotherapy grouped into three categories:
immune response modifiers or cytokines, monoclonal antibodies and vaccines (Rosenbaum
and Rosenbaum, 2005). Cytokines are chemical that mediated both natural
and specific immunity. Cytokines have an important role in responsible for lymphocyte
activation, growth and differentiation (Kirschner and Panetta,
1998). Most commons of cytokines are interleukins-2 (IL-2) and interferon
alpha (INF-α). Immune system has important role in fighting tumor that
has been verified in the laboratory as well as with clinical experiment (Farrar
et al., 1999; O’Byrne et al., 2000;
Morecki et al., 1996; Muller
et al., 1998; Stewart, 1996). Basic idea
behind immunotherapy is boosting the immune system in vitro, so the body
can eradicate tumor on its own. There are many ways in which the immune system
can be boosted, including vaccine therapy, IL-2 and INF-α growth
factor injections, as well as the direct injection of highly activated specific
immune cells, such Tumor Infiltrating Lymphocyte (TIL) into the bloodstream.
Tumor Infiltrating Lymphocytes (TIL) are white blood cells that have left the
bloodstream and migrated into tumor. They are an important prognostic factor
in melanoma (Spatz et al., 2007; Gallon
et al., 2006) higher levels being associated with a better outcome.
In this study, the mathematical modeling of ordinary differential equations
is based on that originally developed by de Pillis (De
Pillis et al., 2006), but the model of tumor growth without therapy
use generalized logistic equation (Spratt et al.,
1993) while in the model of de Pillis using logistic equation (De
Pillis et al., 2006). The generalized logistic equation is based
on that originally developed by Spratt et al. (1993),
where in this work, they observed in 448 patients suffering from tumor for 564
days. Therefore, they obtained a generalized logistic equation more accurate
than a logistic equation to describe of model tumor growth without therapy.
Moreover, we add the effect INF-α on the immunotherapy based Isaeva and
Osiopov’s model (Isaeva and Osipov, 2009). Through
mathematical modeling we analyzed the effect of boosting tumor infiltrating
lymphocyte in immunotherapy. The hypothesis explains that boosting tumor infiltrating
lymphocyte played an important role on immunotherapy. Some researchers have
never explained the important role this boosting tumor infiltrating lymphocyte.
Therefore, this study will describe the important role of boosting tumor infiltrating
lymphocyte in immunotherapy.
MATHEMATICAL MODELING
The model is a system of Ordinary Differential Equation (ODE) whose state variables
are populations of tumor cells, specific and non-specific immune cells and concentrations
of therapeutic interventions. In this research, the model describes the kinetics
of population tumor cells and three types of immune cells (NK cells, CD8+T cells,
circulating lymphocytes), as well as two drug concentrations in the bloodstream,
the equations are expressed by:
The populations are denoted by:
| T(t) |
= |
Tumor cell population at time t |
| N(t) |
= |
Total NK cell effectiveness at time t |
| L(t) |
= |
Total CD8+T cell effectiveness at time t |
| C(t) |
= |
Number of circulating lymphocytes (or white blood cells) at time t |
| I(t) |
= |
Immunotherapy interleukin 2 drug concentration in the bloodstream at time
t |
| Iα(t) |
= |
Immunotherapy interferon alpha drug concentration in the bloodstream at
time t |
Term VL(t) represent unction of boosting tumor infiltrating lymphocyte in immunotherapy.
While VI(t) and ,
respectively represent drug intervention term are functions of time denoted
of interleukin and interferon.
PARAMETER DERIVATION
To complete the simulation and analysis, it necessary to obtain accurate parameters.
The model is very sensitive to the choice of parameters. Tumor size was measured
as volume (in mm3) while this model considers population of cells.
In the following we will assume that 1 mm3 corresponds to 106
cells and will consider cells. Most of parameters in this model obtained from
Pillis’s model (De Pillis et al., 2006)
and also several parameters were taken from Isaeva and Osipov’s model (Isaeva
and Osipov, 2009), as well as from Spratt' work (Spratt
et al., 1993). Table 1 describes all parameters
to run simulation our model.
NUMERICAL RESULTS
Here, we simulated the effect boosted Tumor Infiltrating Lymphocyte (TIL) in immunotherapy. In this simulation, we denoted initial tumor burden of 2x106 cells, since a tumor consisting of less than 106 cells is considered to be undetectable. In this simulation we also denoted as an initially value with 103 NK cells, 10 CD8+T cells and 6x108 circulating lymphocytes.
Figure 1-4 shown that immunotherapy without boosting tumor infiltrating cannot effectively kill tumor cells, where in these simulations IL-2 is administered in 6 pulses at strength 5x106, 5x1010, 5x1020 and 5x1050 from day 8-12, respectively.
Figure 5-8 show that the effect boosting
tumor infiltrating lymphocyte in immunotherapy. In these simulations, tumor
infiltrating lymphocyte at strength 6.6440x107 boosted in the body
in variation in day. Tumor infiltrating lymphocyte at strength 6.6440x107
boosted every weeks until four weeks, respectively, these simulations shown
that at strength 6.6440x107 is not effective to kill tumor cells.
Figure 9-12 shown that simulation with
boosting tumor infiltrating lymphocyte higher is at strength 6.6439x106.
Figure 9 show that tumor initially decreased at day 10 then
increased at day 28 leading to a dangerous level. Tumor effective kills at day
22 and never relapses again with boosting tumor infiltrating lymphocyte at strength
6.6439x106 in the second, third and fourth week as shown in Fig.
10-12, respectively. The last results shown the effect
boosting tumor infiltrating lymphocyte higher is at strength 6.6440x108
boosted in every week.
Figure 13-16 show that tumor effective
kills at day 28 and never relapses. Figure 14-16,
respectively show that tumor effective kills at day 22 and never relapses.
DISCUSSION AND CONCLUSION
In this study, we analyzed the effect boosting tumor infiltrating lymphocyte
using our mathematical model which has never been done in previous researchers'
work. From our result, we can show the effect of boosting tumor infiltrating
lymphocyte in immunotherapy.
| Table 1: |
Parameter values used for numerical simulation |
|
|
|
| Fig. 1: |
Simulation of immunotherapy in absence boosting tumor infiltrating
lymphocyte, IL-2 is administered in 6 pulses at strength 5x106
from day 8-12 and INF-α is administered in 4 pulses at strength 5 from
day 1-34 |
Immunotherapy without boosting tumor infiltrating cannot effectively kill tumor
cells. In these simulations, IL-2 is administered in 6 pulses at strength 5x106,
5x1010, 5x1020 and 5x1050 from day 8-12 as
pictured in Figure 1-4, respectively.
|
| Fig. 2: |
Simulation of immunotherapy in absence boosting tumor infiltrating
lymphocyte, IL-2 is administered in 6 pulses at strength 5×1010
from day 8-12 and INF-α is administered in 4 pulses at strength 5
from day 1-34 |
|
| Fig. 3: |
Simulation of immunotherapy in absence boosting tumor infiltrating
lymphocyte, IL-2 is administered in 6 pulses at strength 5×1020
from day 8-12 and INF-α is administered in 4 pulses at strength 5
from day 1-34 |
|
| Fig. 4: |
Simulation of immunotherapy in absence boosting tumor infiltrating
lymphocyte, IL-2 is administered in 6 pulses at strength 5×1050 from
day 8-12 and INF-α is administered in 4 pulses at strength 5 from day 1-34 |
|
| Fig. 5: |
Simulation of immunotherapy in present boosting tumor infiltrating
lymphocyte with IL-2 is administered in 6 pulses at strength 5×106
from day 8-12, INF-α is administered in 4 pulses at strength 5 from day
1-34 and 6.6440×107 of TILs boosted from day 7 through 8 |
|
| Fig. 6: |
Simulation of immunotherapy in present boosting tumor infiltrating
lymphocyte with IL-2 is administered in 6 pulses at strength 5×106
from day 8-12, INF-α is administered in 4 pulses at strength 5 from day
1-34 and 6.6440×107 of TILs boosted from day 7 through 8 and day 14
through 15 |
|
| Fig. 7: |
Simulation of immunotherapy in present boosting tumor infiltrating
lymphocyte with IL-2 is administered in 6 pulses at strength 5×106
from day 8-12 and INF-α is administered in 4 pulses at strength 5 from day
1-34 and 6.6440×107 of TILs boosted from day 7 through 8, day 14 through
15 and day 20 through 21 |
|
| Fig. 8: |
Simulation of immunotherapy in present boosting tumor infiltrating
lymphocyte with IL-2 is administered in 6 pulses at strength 5×106
from day 8-12 and INF-α is administered in 4 pulses at strength 5 from day
1-34 and 6.6440×107 of TILs boosted from day 7 through 8, day 14 through
15, day 20 through 21 and day 27 through 28 |
|
| Fig. 9: |
Simulation of immunotherapy in present boosting tumor infiltrating
lymphocyte with IL-2 is administered in 6 pulses at strength 5×106
from day 8-12, INF-α is administered in 4 pulses at strength 5 from day
1-34 and 6.6439×108 of TILs boosted from day 7 through 8 |
|
| Fig. 10: |
Simulation of immunotherapy in present boosting tumor infiltrating
lymphocyte with IL-2 is administered in 6 pulses at strength 5×106
from day 8-12, INF-α is administered in 4 pulses at strength 5 from day
1-34 and 6.6439×108 of TILs boosted from day 7 through 8 and day 14
through 15 |
|
| Fig. 11: |
Simulation of immunotherapy in present boosting tumor infiltrating
lymphocyte with IL-2 is administered in 6 pulses at strength 5×106
from day 8-12, INF-α is administered in 4 pulses at strength 5 from day
1-34 and 6.6439×108 of TILs boosted from day 7 through 8, day 14 through
15 and day 20 through 21 |
|
| Fig. 12: |
Simulation of immunotherapy in present boosting tumor infiltrating
lymphocyte with IL-2 is administered in 6 pulses at strength 5×106
from day 8-12, INF-α is administered in 4 pulses at strength 5 from
day 1-34 and 6.6439×108 of TILs boosted from day 7 through
8, day 14 through 15, day 20 through 21 and day 27 through 28 |
|
| Fig. 13: |
Simulation of immunotherapy in present boosting tumor infiltrating
lymphocyte with IL-2 is administered in 6 pulses at strength 5×106
from day 8-12, INF-α is administered in 4 pulses at strength 5 from day
1-34 and 6.6440×108 of TILs boosted from day 7 through 8 |
|
| Fig. 14: |
Simulation of immunotherapy in present boosting tumor infiltrating
lymphocyte with IL-2 is administered in 6 pulses at strength 5×106
from day 8-12, INF-α is administered in 4 pulses at strength 5 from day
1-34 and 6.6440×108 of TILs boosted from day 7 through 8 and day 14
through 15 |
|
| Fig. 15: |
Simulation of immunotherapy in present boosting tumor infiltrating
lymphocyte with IL-2 is administered in 6 pulses at strength 5×106
from day 8-12 and INF-α is administered in 4 pulses at strength 5 from day
1-34 and 6.6440×108 of TILs boosted from day 7 through 8, day 14 through
15 and day 20 through 21 |
|
| Fig. 16: |
Simulation of immunotherapy in present boosting tumor infiltrating
lymphocyte with IL-2 is administered in 6 pulses at strength 5×106
from day 8-12, INF-α is administered in 4 pulses at strength 5 from
day 1-34 and 6.6440×108 of TILs boosted from day 7 through
8, day 14 through 15, day 20 through 21 and day 27 through 28 |
Figure 5-8 shown that the effect boosting
tumor infiltrating lymphocyte be analyzed in immunotherapy, these simulations
tumor infiltrating lymphocyte at strength 6.6440x107 boosted in the
body in variation in day. Tumor infiltrating lymphocyte at strength 6.6440x107
boosted every weeks until four weeks as pictured in Fig. 5-8,
respectively, these simulations shown that at strength cannot effective to kill
tumor cells. In Fig. 9-12, simulation with
boosting tumor infiltrating lymphocyte higher is at strength 6.6439x108.
In Fig. 9, tumor cells decreased at day 10 then increased
at day 28 leading to a dangerous level. Tumor effective kills at day 22 and
never relapses again with boosting tumor infiltrating lymphocyte at strength
6.6439x106 in the second, third and fourth week as shown in Fig.
10-12, respectively. The last results shown the effect
boosting tumor infiltrating lymphocyte higher is at strength 6.6440x108
in every week. Figure 13 show that tumor effective kills
at day 28 and never relapses. Figure 14-16
show that tumor effective kills at day 22 and never relapses.
Based on our simulation results, we can conclude that the boosting of tumor infiltrating lymphocyte on immunotherapy has a very significant role in killing tumor cells. Boosting tumor infiltrating lymphocyte in immunotherapy is more effective in two week at strength 6.6439x108.
ACKNOWLEDGMENT
The authors are grateful to the anonymous reviewers for their invaluable comments that improve the presentation of this manuscript.