Biomedical Engineering Reference
In-Depth Information
⎧
⎨
ET
g
E
=
s
+
p
T
−
mET
−
dE
,
+
(2)
⎩
T
=
aT
(
1
−
bT
)
−
nET
,
where
s
b
are positive parameters.
Here the exponential growth of
T
in the second equation was replaced by a more
realistic one in logistic form:
aT
,
p
,
g
,
m
,
d
,
a
,
n
,
(originally due to Verhulst, 1838), where
b
−
1
is the maximal carrying capacity for tumor cells and
a
is the maximal growth
rate. The term
(
1
−
bT
)
nET
describes the loss of tumor cells due to the presence of immune
cells. In the first equation
s
is normal immune cell growth, which is n cell death with
d
the loss rate;
−
mET
describes the decay of
E
cells due to interacting with tumor
cells in a mass action way. The term
p
−
ET
g
+
T
represents Mchaelis-Meten growth of
the immune response in response to tumors.
The Kuznetsov equations describe several important features and allow us to
make predictions that are relevant for understanding cancer immunotherapy. The pa-
per by Kuznetsov et al. [
31
] establishes existence of long period oscillations of
tumor that agrees with recurrent clinical manifestations of certain human leukemias.
In addition, the model predicts the existence of a critical level of
E
-cells in the
body below which the tumor growth cannot be controlled by the immune response.
It describes qualitatively the “escape” phenomena in which low doses of tumor cells
can escape immune defenses and grow into a large tumor, whereas larger doses of
tumor cells are eliminated.
The Kuznetsov model was generalized by Kirschner and Panetta in 1998 [
28
].
The idea was to introduce a third population (concentration) of effector molecules
known as
cytokines
, which are information signaling molecules used extensively
in intercellular communication by the immune system. Below we describe briefly
the Kirschner-Panetta equations. Tumor cells are tracked as a continuous variable
as they are large in number and are generally homogeneous; their concentration
is denoted by
T
. Immune cells (called effector cells) are also large in number
and represent those cells that have been stimulated and are ready to respond to
the foreign matter (known as antigen); their concentration is denoted by
E
(
t
)
(
t
)
.
Finally, effector molecules are represented as a concentration
C
. These are self-
stimulating, positive feedback proteins for effector cells that produce them. The
equations that describe the interactions of these three state variables are referred
herein as the Kirschner-Panetta (KP) system:
⎧
⎨
⎩
(
t
)
d
E
d
t
=
p
1
EC
g
1
+
−
μ
+
C
+
(
)
cT
2
E
s
1
3a
d
T
d
t
=
aET
g
2
+
r
2
T
(
1
−
bT
)
−
(
3b
)
T
d
C
d
t
=
p
2
ET
g
3
+
T
+
s
2
−
μ
3
C
(
3c
)
Search WWH ::
Custom Search