—————
*Corresponding author
J. Math. Biol. (1998) 37: 235
—252
Modeling immunotherapy of the tumor
– immune
interaction
Denise Kirschner1, John Carl Panetta2,
*
1 Department of Microbiology and Immunology, The University of Michigan Medical
School, Ann Arbor, MI 48109-0620, USA
2 School of Science, Penn State Erie, The Behrend College, Station Road, Erie,
PA 16563-0203, USA.
e-mail: panetta@wagner.bd.psu.edu
Received: 22 October 1997 / Revised version: 27 November 1997
Abstract. A number of lines of evidence suggest that immunotherapy with the
cytokine interleukin-2 (IL-2) may boost the immune system to fight tumors.
CD4` T cells, the cells that orchestrate the immune response, use these
cytokines as signaling mechanisms for immune-response stimulation as well
as lymphocyte stimulation, growth, and differentiation. Because tumor cells
begin as ��self��, the immune system may not respond in an effective way to
eradicate them.
Adoptive cellular immunotherapy can potentially restore or
enhance these effects. We illustrate through mathematical modeling the dy-
namics between tumor cells, immune-effector cells, and IL-2. These efforts are
able to explain both short tumor oscillations in tumor sizes as well as
long-term tumor relapse. We then explore the effects of adoptive cellular
immunotherapy on the model and describe under what circumstances the
tumor can be eliminated.
Key words: Immunotherapy
— Tumor
— Cytokine
— Modeling
— Interleukin-2
—
Ordinary differential equations
1 Introduction
Cancer is still a leading cause of death in the world yet much is still not known
about its mechanisms of establishment and destruction. While surgery and/or
chemo- and radiotherapies have played key roles in treatment, it is clear that
in many cases they do not represent a cure. Even when patients experience
tumor regression, later 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
[4, 10, 11, 14, 15, 23, 24, 29
—33]. Immunotherapy refers to the use of
cytokines
usually together with
adoptive cellular immunotherapy (ACI).
Cytokines are protein hormones that mediate both natural and specific
immunity. They are produced mainly by activated T cells (lymphocytes)
during cellular-mediated immunity. Interleukin-2 (IL-2) is the main cytokine
responsible for lymphocyte activation, growth and differentiation. It is pro-
duced by CD4` T cells, and in lesser quantities by CD8` T cells (cytotoxic
T cells or CTLs). IL-2 acts on the same cells that produce it. Therefore it is
referred to as an
autocrine growth factor although it can also act on nearby
T lymphocytes (hence a
paracrine growth factor). Clinical trials have shown
that there are immune-stimulation effects from treatment with interleukins [4,
10, 11, 14, 15]. IL-2 has been shown to enhance CTL activity at different
disease stages [29
—31, 33]. Also, there is a restoration of defective natural
killer (NK) cell activity as well as enhancement of polyclonal expansion of
CD4` and CD8` T cells [32, 35].
ACI refers to the injection of cultured immune cells that have anti-tumor
reactivity into tumor bearing host. This is usually done in conjunction with
large amounts of IL-2. This can take two approaches:
1. ¸
AK-(
lymphokine-
activated killer cell)
therapy: These cells are derived from
the
in vitro culturing with high concentrations of IL-2 of peripheral blood
leukocytes removed from patients. The LAKs are then injected back at the
cancer site. These are thought to be mainly natural killer cells.
2. ¹
I¸-(
tumor infiltrating lymphocyte)
therapy: These cells are derived from
lymphocytes recovered from the patient tumors. They are then incubated
with high concentrations of IL-2
in vitro and are comprised of activated
NK cells and CTL cells. They are then injected back into the patient at the
tumor site.
The theoretical study of tumor-immune dynamics has a long history.
A good summary can be found in Adam and Bellomo [2]. We attempt to add
to the existing literature by exploring the role of cytokines in the disease
dynamics as well as address the topics of long-term tumor recurrence and
short-term tumor oscillations. Presently, there is an ongoing search for more
efficacious and less toxic results from treatment with immunotherapy [33].
This study aides in addressing this issue. In this paper we will formulate
a model for tumor-immune dynamics (Sect. 2), explore the analysis of the
model for mathematical and biological implications (Sect. 3), and in Sect. 4, we
enhance the model to explore immunotherapy treatment.
2 Model
To begin a model of tumor-immune dynamics, we first examine some existing
models. Kuznetsov et al. [18] define an ordinary differential equation (ODE)
model for two main populations: effector cells and tumor cells. They
predict a threshold above which there is uncontrollable tumor growth, and
below which the disease is attenuated with periodic exacerbations occurring
every 3-4 months. They also show the model does have stable spirals, but the
236
D. Kirschner, J. C. Panetta
Dulac-Bendixson criterion demonstrates there are no stable closed orbits. In
DeLisi and Rescigno [6] and Adam [1], they again consider ODE��s for the
populations of immune and tumor cells. They show that survival increases if
the immune system is stimulated. They also show in some cases that an
increase in effector cells increases the chance of tumor survival. Furthermore,
they give a threshold for the chance of uncontrolled tumor growth. The
probability will increase if the effector-cell death rate or growth rate of the
tumor cells increase, or if the growth rate of effector cells or the killing of
tumor cells decreases. Nani and Og\uzto��reli [28] developed a model of ACI
based on work by Rosenberg et al. [30, 31]. Their model incorporates
stochasticity effects on the immune
—cancer interactions. Results of their model
are that success of treatment is dependent on the initial tumor burden. Also,
simulation of immunotherapy treatment indicated that more aggressively
growing tumors will overpower the ACI treatment. They do not consider
sensitivity, bifuraction, or stability analysis of the model. Finally, there is
a very detailed model given by DeBoer et al. [5] with 10 (plus) differential
equations coupled with 3
—5 algebraic equations describing most of the players
in the tumor-immune dynamics. They are able to show both tumor regression
(with a highly antigenic tumor) and uncontrolled tumor growth (for a low
antigenic tumor).
Our goal is to use some of the best ideas in these systems, but to keep the
model as simple as possible while incorporating the most important concepts
of tumor-immune dynamics together with the feature of IL-2 dynamics.
Therefore, we define three populations. These include:
E(
t), the activated
immune-system cells (commonly called
effector cells) such as cytotoxic T-cells,
macrophages, and natural killer cells that are cytotoxic to the tumor cells;
¹(
t), the tumor cells; and
I
L
(
t), the concentration of IL-2 in the single
tumor-site compartment we are modeling. Our model describing the interac-
tion between the effector cells, tumor cells, and the cytokine (IL-2) is:
dE
dt
"
c¹!k2
E#
p
1
EI
L
g
1#
I
L
#
s
1
,
(1)
d¹
dt
"
r
2
(¹)¹!
aE¹
g
2#¹
,
(2)
dI
L
dt
"
p
2
E¹
g
3#¹
!k3
I
L#
s
2
,
(3)
with initial conditions:
E(0)"
E
0
,
¹(0)"¹0,
I
L
(0)"
I¸0
,
(4)
and where the model terms are described as follows. The first equation
describes the rate of change for the effector-cell population. Effector cells are
stimulated to grow based on two terms. One is a recruitment term (term 1) due
to the direct presence of the tumor, where the parameter
c models the
antigenicity of the tumor. Antigenicity can be thought of as a measure of how
Modeling tumor immunotherapy
237
Table 1. Parameter values
Eq. (1)
06
c60.05
k
2"0.03
p
1"0.1245
g
1"2]107
Eq. (2)
g
2"1]105
r
2"0.18
b"1]10~9
a"1
Eq. (3)
k
3"10
p
2"5
g
3"1]103
different the tumor is from ��self��. The other growth/source term (term 3) is
a proliferation term whereby effector cells are stimulated by IL-2 that is
produced by effector cells in both an autocrine and paracrine manner. This
term is of Michaelis-Menten form to indicate the saturated effects of the
immune response. Effector cells have a natural lifespan of an average 1/k
2
days. Lastly,
s
1
is a treatment term that represents an external source of
effector cells such as LAK or TIL cells. Equation (2) marks the rate of change
of the tumor cells. This can be described by a linear growth term (if
r
2
(¹) is
constant) or as a type of limiting-growth such as logistic or Gompertz. We
chose the logistic growth function
r
2
(¹)"
r
2
(1!
b¹) .
(5)
The loss of tumor cells is represented by an immune-effector cell interaction at
rate
a. This rate constant,
a represents the strength of the immune response
and is modeled by Michaelis
—Menten kinetics to indicate the limited immune
response to the tumor. (This form could also account for the effects of a solid
tumor, i.e. only a portion of the tumor mass comes in contact with the immune
system cells [6].) Equation (3) gives the rate of change for the concentration of
IL-2. Its source is the effector cells that are stimulated by interaction with the
tumor and also has Michaelis-Menten kinetics to account for the self-limiting
production of IL-2. The next term (k
3
) represents loss/degraded rate of IL-2.
Finally,
s
2
is a treatment term that represents an external input of IL-2 into
the system.
2.1 Parameter estimation
To complete the development of the mathematical model, (1)
—(5), we must
define values for the parameters and initial conditions.
As terms in our model are somewhat similar in the first two equations
(1)
—(2) to that of the models in [5] and [18], we explored the choices for
parameters as presented in those studies. In many cases there were large
ranges in the parameter choices between studies and we chose values most
appropriate for this model. The values are given in Table 1 (units are in
days~1
except for
g
1
,
g
2
,
g
3
, and
b whose units are volume). Values for the rate
constants in equation (3), for which no previous study was done, were found
from the current medical literature and from sensitivity analyses. For example,
according to Rosenberg and Lotze [30] the half-life for IL-2 is between
30
—120 min. We use the average value of this half-life for k
3
in our simulations.
The value of
c, defined as the antigenicity of the tumor varies greatly from
238
D. Kirschner, J. C. Panetta
patient to patient and cancer to cancer. Larger values of
c represent tumor
cells that present a well recognized antigen and smaller values represent tumor
cells that present a weak antigen. We explore a wide range of values for
c.
After carrying out a sensitivity analysis on the parameters
c, the tumor
antigenicity, as well as
a, the immune response, it is clear they play key roles in
the dynamics. We explore these rate constant through bifurcation analyses in
the next sections.
3 Analysis and numerical results
Equations (1)
—(3), together with initials conditions (4) represent the tumor-
immune model in the absence of treatment. For ease of analysis, we first
non-dimensionalize the system, and then carry out a steady-state analysis on
the scaled system. We also discuss the biological relevance of our results.
3.1 Scaling
We non-dimensionalize the model (1)
—(3) using the following scaling:
x"
E
E
0
,
y"
¹
¹0
,
z"
I
L
I¸0
,
q"
t
s
t,
cN "
c¹0
t
s
E
0
,
pN
1"
p
1
t
s
,
gN
1"
g
1
I¸0
,
kN
2"
k
2
t
s
,
gN
2"
g
2
¹0
,
bM "
b¹0,
rN
2"
r
2
t
s
,
aN "
aE
0
t
s¹0
,
kN
3"
k
3
t
s
,
pN
2"
p
2
E
0
t
s
I¸0
,
gN
3"
g
3
¹0
,
s
1"
s
1
t
s
E
0
,
sN
2"
s
2
t
s
I¸0
.
Then, dropping the over-bar notation for convenience, we obtain the follow-
ing scaled model:
dx
dq"
cy!k2
x#
p
1
xz
g
1#
z
#
s
1
(6)
dy
dq"
r
2
y(1!
by)!
axy
g
2#
y
(7)
dz
dq "
p
2
xy
g
3#
y!k3
z#
s
2
(8)
with initial conditions:
x(0)"
x
0
,
y(0)"
y
0
,
z(0)"
z
0
.
(9)
One possible choice for the scaling is to define:
E
0"¹0"
I¸0
"1/
b and
t
s"
r
2
. These scalings need to be chosen to help adjust for the fact that this is
Modeling tumor immunotherapy
239
—————
1All the numerical bifurcation analysis in this and the following sections was carried out
using ����XPPAUT3.0���� by Ermentrout [8] that incorporates the bifurcation analysis program
����AUTO���� by Doedel [7]
Fig. 1. A bifurcation diagram varying the antigenicity (
c). The numbers along the maximal
amplitude limit-cycle curve represent the corresponding period of the limit cycle in days.
The carrying capacity of the tumor is scaled to 1
a numerically ����stiff���� system. That is, without scaling, or inappropriate scal-
ings, the numerical routines used to solve these equations will fail. This is due
to very large changes in some of the variables over very short ranges of time.
3.2 Stability analysis — no treatment case
The dynamics of this model (6)
—(9) are very rich. We begin by exploring the
steady-states when there are no treatment terms, i.e. both
s
1
and
s
2
are zero.
The first equilibrium is the trivial state where all the populations are zero,
namely
E
0"(0, 0, 0). The eigenvalues of the Jacobian matrix for
E
0
are !k
2
,
r
2
, and !k
3
. Therefore,
E
0
is
always a locally unstable saddle point. There
exists multiple, positive, non-trivial steady states, depending on the choice of
parameters, namely
E
i"(
E*
i
, ¹
*
i
,
I*¸i
) where
i can range from 1 to 3. To aid in
explaining these different outcomes, we present a bifurcation diagram1 for the
parameter
c, the tumor antigenicity (see Fig. 1). There are three categories for
the dynamics as
c is varied. These can each be seen in Figs. 1 and 2.
240
D. Kirschner, J. C. Panetta
Fig. 2. Effector cells (solid lines), tumor cells (dotted lines), and IL-2 (long dashed lines) vs.
time. Time is in days. The carrying capacity of the tumor is scaled to 1e#5. (A)
c"5
e!5,
(B)
c"0.01, (C)
c"0.02, (D)
c"0.035
1. Region 0(
c(
c
0
(Insert at top of Fig. 1 and Fig. 2A. Note that
c
0"8.55]10~5
for the parameter values listed in Table 1.) In this region
there is one stable, positive steady state and two unstable, positive steady
states. The steady-state (
E
1
) is a locally stable, improper node such that the
tumor is very large, almost at the size of its carrying capacity (1/
b). The
unstable steady-states (
E
2
and
E
3
) are respectively, an unstable spiral that
persists until
c"
c
1
, and an unstable saddle.
2. Region
c
0(
c(
c
1
. (Main Fig. 1 and Figs. 2B and C. Note that
c
1"0.032
for the parameter values listed in Table 1.) At
c"
c
0
, the states
E
1
and
E
3
are lost but a stable limit cycle is ��born��. This cycle persists for the entire
interval, but as the value of
c increases towards
c
1
(the Hopf bifurcation)
the amplitude and period of the limit cycles decreases. At a value close to
c
0
(e.g.
c"0.001) near where the limit cycle is born, the period of the limit
cycle is 3900 days (using the parameters in Table 1) and the cancer is
detectable (cancer mass near 90% of carrying capacity 1/
b) for about 100
days. However, for a larger choice of
c (e.g.
c"0.01) the period of the limit
cycle is 3250 days and the cancer mass is detectable for about 45 days
reaching a maximum mass of about 10% of the carrying capacity of the
Modeling tumor immunotherapy
241
cancer. For values of
c close to
c
1
(e.g.
c"0.031) the period of the limit
cycle is 107 days, with a very small maximum tumor size. The steady state
E
2
is still present, and unstable.
3. Region
c
1(
c (Insert at bottom of Fig. 1 and Fig. 2D.) At
c
1
, the state
E
2
bifurcates from an unstable, to a stable, spiral node via a Hopf bifurca-
tion. The oscillations are small and damp out quickly. The masses of the
tumors in this region are small (about 5 orders of magnitude less than that
of the
E
1
size for
c(
c
1
).
3.3 Biological implications
The implications of these bifurcation results are provocative. In case 1, the
model allows for the possibility of a large tumor mass when the tumor
antigenicity,
c, is extremely small. Case 2, however, presents a very different
outcome. It implies that for a very low antigenic tumor (small
c), the tumor
mass cycles with a large period (as long as 11 years). The tumor will spend
a portion of the cycle (as much as 2 or 3 months) near its carrying capacity and
then the remainder of the 11 years with a mass near zero, i.e. dormant. (A small
stable tumor that does not change in size is referred to as
dormant). This could
explain long term recurrence of tumors within-host. There is clinical evidence
to support this recurrent phenomenon [3, 12, 13, 34]. Also, Tsao et al. [36] has
found recurrence of melanoma more than 15 years after the patients initial
contact with the disease. As the tumor antigenicity increases, the magnitude
and period of these oscillations shorten, as well as the length of time the tumor
remains at its peak size. This progresses until the oscillations decrease down to
a monthly basis and the tumor size is very small with a quick recovery time.
There is experimental evidence of these short term oscillations in cases like
Chronic Myeloid Leukemia as well as others [9, 16, 17, 25, 37]. Also Gause
et al. [10] observe short-term oscillations in the lymphocyte counts with
a monthly period. Finally, these oscillations give way to a stable spiral with
very quick damping, leading to a small, persistent tumor, that could be
described as dormant.
A natural question that arises from the above study is, what affects the
period of these periodic solutions? We are thus interested in how key para-
meters, namely antigenicity (
c) and immune response (
a), affect the period of
the limit cycles discussed above; and we present a 2-parameter bifurcation
diagram. In particular, Fig. 3 shows how the parameters
a and
c affect the
period of the limit cycle. Each curve in Fig. 3 represents a fixed period of the
limit cycle. Note that for fixed
c, an increase in
a (the immune response) causes
a decrease in the period (and thus the amplitude) of the limit cycle. If
a is
increased enough then we pass through the Hopf Bifurcation and the limit
cycle disappears. In this case the tumor volume is small and dormant.
In the non-treatment case (i.e.
s
1"
s
2"0), the model does not allow for
complete clearance of the tumor. To this end, we now study the role of
enhancing the immune response through immunotherapy.
242
D. Kirschner, J. C. Panetta
Fig. 3. Fixed period curves of antigenicity vs. the immune response (
c vs
a). The period of
the limit cycles are constant for values of
a and
c on each curve. HB"Hopf Bifurcation
curve
4 Immunotherapy
To date, most treatment of cancer focuses on chemo- and radiotherapies
directed at the tumor. Here we explore treatment that serves to boost
the immune-system��s capacity to fight the cancer. Immunotherapy attempts
to use cytokines, the communication/stimulation proteins produced, released,
and used by cells, to enhance cellular activity. The cytokine most effective
in this regard is interleukin-2, as it is the key interleukin responsible for
T-cell growth and differentiation; the cells that orchestrate the immune
response. We will consider immunotherapy to be adoptive cellular
immunotherapy and/or IL-2 delivery at the tumor site either separately or in
combination. Thus, in model (1)
—(4) we consider either or both
s
1
and
s
2
to be
non-zero where rate
s
1
represents the addition of LAK or TIL cells to the
tumor site and rate
s
2
represents the addition of interleukin-2 to the site. We
explore each of these therapies separately and then the combined effects of
both.
4.1 Adoptive cellular immunotherapy (
s
1'0,
s
2"0)
In the absence of
s
1
the trivial non-cancer steady state,
E
0
, is always unstable.
In the presence of
s
1
this state is lost, but there is another state that is a more
Modeling tumor immunotherapy
243
Fig. 4. A two-parameter bifurcation diagram of antigenicity versus ACI (
c vs.
s
1
).
HB"Hopf Bifurcation; LP"Limit Point; BP"Bifurcation Point. The regions are de-
scribed in the text
realistic non-tumor state,
E
0"(
E*
0
, 0, 0). This implies that effector cells can
clear the tumor if this equilibrium is stable. From analyzing the eigenvalues,
this state is locally stable if
s
1'
s1
crit
and unstable if
s
1(
s1
crit
where
s1
crit"
r
2
g
2
k
2
a
.
(10)
(Note that the value of
s1
crit"540 for the parameter values in Table 1.) Hence,
when
s
1"
s1
crit
there is a bifurcation that, with the analysis to follow, can be
shown to be a transcritical bifurcation.
In the cases where a tumor can exist the dynamics are again very rich.
Here, the positive steady states are of the form
E
i"(
E*
i
, ¹
*
i
,
I*¸i
),
i ranging
from 1 to 3. These vary with the treatment input
s
1
. We examine regions in the
c vs
s
1
parameter space to explore how
s
1'0 alters the earlier results. The
Roman numerals (below) correspond to the regions of the bifurcation diagram
in Fig. 4.
A. If
s
1(
s1
crit
:
I. There is only one steady-state,
E
1"(
E*
1
, ¹
*
1
,0), and it is a stable
improper node.
II. A stable limit cycle is born. These cycles persist for the entire region,
but as the parameters approach the value of the Hopf Bifurcation, the
amplitude and the period of the limit cycles decreases.
244
D. Kirschner, J. C. Panetta
III. There is one steady state
E
3"(
E*
3
, ¹
*
3
,
I*¸3
) that is born as a result of
a Hopf bifurcation. It is a stable spiral node.
B. If
s
1'
s1
crit
:
IV. This is a region of bistability. Depending on the initial condition we
start with either the solution will tend either to
E
1
(the tumor survives)
or to
E
0
(the immune system succeeds). This phenomenon of bi-
stability has been previously studied with regard to tumor immunology
[20, 21],
V. In this region
E
0
is stable. Thus, the immune system succeeds in clearing
the tumor. Note that this is the largest area of parameter space.
4.2 Interleukin-2 (
s
1"0,
s
2'0)
We now explore the input of IL-2 into the system. The only non-tumor state is
E
0"(0, 0,
S2
k
3
). This state is always an unstable saddle point. This may imply
that administering IL-2 alone, without ACI, cannot clear the tumor. If
s
2
is
small, we expect the dynamics to be similar to that of the untreated case. If
s
2
is large,
s
2'
k
2
k
3
g
1
p
1!k2
"
s2
crit
(11)
then the effector cells will grow uncontrolled and the only stable ��state�� is
(R, 0,
s
2
/k
3
). To see this we must show that
dE/
dt'0 when (11) holds.
Plugging
I
L"
s
2
/k
3
into Eq. (1) and simplifying we find:
dE
dt
"
EA(
p
1!k2
)
s
2!k2
k
3
g
1
s
2#k3
g
1
B.
(12)
Therefore, for
dE/
dt'0, the numerator of equation (12) must be greater
than zero; this is equivalent to equation (11). Thus, if (11) holds
then (R, 0,
s
2
/k
3
) is stable. The bifurcation diagram in Fig. 5 shows the
different dynamics involved with varying
s
2
with respect to
c. The regions
of Fig. 5 are described as follows, with the biological relevance presented in
Sect. 4.4 (Again, note that the Roman numerals below correspond to those in
Fig. 5):
I. The cancer grows to near its carrying capacity. In this case both the
antigenicity and the IL-2 concentration is small, so the result parallels the
non-treatment case.
II. Stable limit cycles exist (as in Fig. 4).
III. Stable spirals leading to a stable, small tumor (as in Fig. 4).
IV. (R, 0,
s
2
/k
3
) is the only stable ��equilibria��. This state is a new dynamic
acquired from the introduction of the non-zero constant source,
s
2
, in
equation (3).
Modeling tumor immunotherapy
245
Fig. 5. Two-parameter bifurcation diagram of antigenicity versus IL-2 (
c vs
s
2
). HB"Hopf
Bifurcation; LP"Limit Point; BP"Bifurcation Point. The regions are described in the
text
4.3 Immunotherapy with both ACI and IL-2 (
s
1'0,
s
2'0)
In this case we have both treatment terms non-zero in (6)
—(8). The tumor free
equilibrium is now
E
0"(
E*
0
, 0,
I*
L0
) where
E*
0"
s
1
(k
3
g
1#
s
2
)
k
2
(k
3
g
1#
s
2
)!
s
2
p
1
,
(13)
I*
L0"
s
2
k
3
.
(14)
The eigenvalues of the Jacobian matrix are:
p
1
I*
L0
g
1#
I*
L0
!k2,
r
2!
aE*
0
g
2
,
!k3.
(15)
Therefore, for this equilibria to be locally stable we require
s
2(
k
2
k
3
g
1
p
1!k2
"
s2
crit
(16)
and
s
1'
g
2
r
2
a C
s
2
(k
2!
p
1
)#k
2
k
3
g
1
k
3
g
1#
s
2
D.
(17)
Note that for the parameters given in Table 1,
s2
crit"63492063. Varying
the antigenicity (
c) affects the
s
1
vs
s
2
bifurcation diagram. Table 2 and
246
D. Kirschner, J. C. Panetta
Table 2. Summary of results for Fig. 6. KEY: CF"cancer free state; CCC"Cancer grows
to carrying capacity (
E
3
); LC"Limit cycles are present; N/A not applicable; CFI"cancer
free with immune system growing without bounds to (R, 0,
I*
L0
)-stable; SCSS"small
cancer state from a stable spiral; BS"Bistability to either CF or CCC
Figure
Region I
Region II
Region III
Region IV
c"0
5A
CCC
BS
CF
CFI
c"8.5
e~5
5B
CCC
LC
CF
CFI
c"0.0025
5C
N/A
LC
CF
CFI
c"0.006
5D
LC
SCSS
CF
CFI
c"0.0325
5E
LC
SCSS
CF
CFI
c"0.04
5F
N/A
SCSS
CF
CFI
Fig. 6 summarize the results. The biological relevance is discussed below
in section (4.4).
4.4 Biological significance
The results in Sects. 4.1, 4.2, and 4.3 again present some interesting biological
interpretations. For the treatment results, the most significant correlations are
between antigenicity of the tumor together with the type of treatment being
administered. We focus our discussion on those correlations in detail.
In Sect. 4.1 the two key concepts explored are the antigenicity (
c) and the
administration of adoptive cellular immunity (
s
1
). The bifurcation diagram in
Fig. 1 for antigenicity,
c, is indicative of the same general effects of ACI
treatment for different tumors shown in Fig. 4. For low antigenic tumors and
a low input of treatment, the tumor remains large and stable. For low
antigenic tumors but larger amounts of treatment bistability exists where
either the large-tumor state or the tumor-free state accepts stability. If the
treatment is increased to very high levels then the tumor can be cleared. For
tumors that exhibit a medium to high antigenicity there are two cases. If little
immunotherapy is input into the system, then the situation remains basically
the same as with no treatment. But, with medium to high immunotherapy, the
tumor can be cleared.
In Sect. 4.2, the two key concepts we explored are the antigenicity (
c) and
the administration of IL-2 (
s
2
). The bifurcation diagram in Fig. 5 reveals the
comparative effects. Administering low concentrations of IL-2 (below
s2
crit
)
yields qualitatively the same results as with no IL-2 treatment (see Fig. 1).
However, large amounts of administrated IL-2 (above
s2
crit
) together with any
degree of antigenicity yields an interesting result. Here, the tumor is cleared
but the immune system grows unbounded as the IL-2 concentration reaches
a steady-state value. This uncontrolled growth of the immune system repres-
ents a situation that is detrimental to the host. A well-documented side effect
from treatment with dose escalations of IL-2 is capillary leak syndrome (or
vascular leakage syndrome) [19, 22, 26, 32]. This syndrome has been
Modeling tumor immunotherapy
247
Fig. 6. Two-parameter bifurcation diagram of IL-2 versus ACI (
s
1
vs
s
2
) for different
antigenicity
levels: (A)
c"0. (B)
c"8.5
e!5. (C)
c"0.0025. (D)
c"0.006.
(E)
c"0.0325. (F)
c"0.04. HB"Hopf Bifurcation; LP"Limit Point; BP"Bifurcation
Point. The regions are described in the text
attributed to a variety of IL-2-induced activation events [22, 32]. This could
imply that for values of IL-2 administered above
s2
crit
the unbounded growth of
the immune effector cells may indeed reflect the detrimental side effects seen in
cancer patients receiving immunotherapy.
In section (4.3) we combine the treatment strategies of ACI and IL-2.
A summary of the effects of simultaneous treatments is presented in Table 2
248
D. Kirschner, J. C. Panetta
and Fig. 6. The results are a combination of the separate mono-treatment
regimens. For concentrations of IL-2 administered below
s2
crit
, the effects of
administering ACI concurrently make a difference for tumor clearance; and,
the greater the antigenicity of the tumor the more likely the treatment will
succeed. For values of IL-2 administered above
s2
crit
the tumor can be cleared,
but the side effects from an over-activated immune system may out way the
benefits of tumor clearance.
5 Discussion
In this paper we explore the effects of the cytokine interleukin-2 on tumor-
immune dynamics. In these dynamics a key role is played by the antigenicity
of the tumor. For low antigenic tumors the immune system is not able to clear
the tumor; while, for highly antigenic tumors, reduction to a small dormant
tumor is the best case scenario. An interesting intermediate result (for tumors
exhibiting average antigenicity) is the presence of stable limit cycles which
imply that the tumor and the immune system undergo oscillations. If the
tumor has a low to medium antigenicity these cycles are relatively long (on the
order of 8 to 10 years) with large amplitudes and the tumor is in a dormant
state during most of the period. This may explain long term recurrence of
tumors. If the tumor has medium to high antigenicity the periods of the cycles
are short with small amplitudes (Figs. 1 and 2). The health of the immune-
system activation plays a role in the periodicity of these cycles (Fig. 3). For
example, as the immune response improves (i.e.
a increases), the period and
amplitude of the limit cycles decrease, leading to a dormant-tumor state. It
should be noted that these stable limit cycles have not been observed in
previous models of the interaction between tumors and the immune system
(e.g., [1,5,18]).
Next we studied treatment of the tumor-immune system with immuno-
therapy. Two types of treatments are explored individually and together
—
doptive cellular immunotherapy and administration of the cytokine IL-2.
Unlike in the no treatment case (Section 2), the effects of ACI therapy can yield
a tumor-free state. This happens when the treatment concentration is above
a given critical level (Fig. 4). This is true for tumors of almost any antigenicity.
But for tumors with small antigenicity bistability can occur with the tumor-
free state and the near-carrying-capacity state both stable. This indicates the
need for starting treatment early, while the tumor is small, so that the tumor
can be controlled.
Treatment with IL-2 alone does not offer a satisfactory outcome; if IL-2
administration is low there is no tumor-free state. However, if IL-2 input is
high the tumor can be cleared but the immune system grows without bounds
causing problems such as capillary leak syndrome (Fig. 5). Finally, it is
treatment with ACI and IL-2 that gives the combined effects obtained from
the monotherapy regimes (Fig. 6). For any antigenicity, there is a region of
tumor clearance.
Modeling tumor immunotherapy
249
These results indicate that treatment with ACI may be a better option
either as a monotherapy or in conjunction with IL-2. With IL-2 treatment
alone, too little does not boost the immune system enough to clear the tumor;
however, large amounts can have pathologic effects. IL-2 can augment the
ACI treatment indicating that the combined effects may be the best-case
scenario. In either case, it is clear that cytokine-enhanced immune function
can play a significant role in treatment of cancer.
Natural extensions of this model are as follows. First, a more mechanistic-
based method of modeling immunotherapy is needed. An improved version
may include a time dependent source term for immunotherapy. Also, IL-2
is just one of a dozen cytokines involved in the cellular dynamics of
the immune system response to tumor invasion. We plan to investigate how
other cytokines such as Interferon-c, IL-10, and IL-12 affect the
dynamics of the system. These dynamics may include both positive and
negative feedback mechanisms since, for example, IL-10 down-regulates
cytokine production by activated macrophages. Finally, a model including the
combined effects of immunotherapy with chemo- and/or radio-therapies is
needed to better elaborate the results obtained here for the purpose of guiding
clinical trials.
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