Biomedical Engineering Reference
In-Depth Information
Tabl e 1
Parameter values for the model (
5
)
Name
Definition
Baseline (units)
Range
μ
2
Half-life of effector cells
E
0.03 (1/time)
0.03
p
3
Proliferation rate of
E
0.1245 (1/time)
0.1245
10
−
3
10
−
5
f
Half-sat for
E
proliferation term
(cells)
[
,
1
]
10
−
2
10
2
s
1
(
t
)
Immunotherapy term
1 (cells/time)
[
,
]
10
−
3
c
(
t
)
Cancer antigenicity
0.05 (1/time)
[
,
0
.
5
]
10
−
1
r
2
(
t
)
Cancer growth rate
0.18 (1/time)
[
,
2
]
10
−
9
10
−
9
b
Cancer cell capacity (logistic growth)
(1/cells)
10
−
2
10
2
a
(
t
)
Cancer clearance term
1 (1/time)
[
,
]
10
5
10
5
g
2
Half-saturation, for cancer clearance
(cells)
and a small percentage of patients had complete tumor regression. In this study,
Rosenberg and colleagues took a blood sample from each patient and transferred
genes into T cells inducing each cell to produce specialized T-cell receptors (TCR).
These cells are then transferred back into the patient. In the body, T cells produce
TCRs on their outer membrane and the TCRs recognize and attach to certain
molecules found on the surface of the tumor cells. Finally, signaling through the
TCRs activates T cells to attack and kill the tumor cells. To explore these studies
further we will build on the KP model.
First, to simplify the model we can remove the IL-2 Eq. (3c). We replace the IL-2
saturation term in Eq. (3a) with a self-proliferation term, i.e.
p
1
E
.Theidea
that the proliferation rate of effectors may be a decreasing function of effectors
has been explored by d'Onofrio et al. [
19
]. To capture the effects of gene therapy
(see Fig.
1
) we must allow for the immune parameters of the model, i.e.
a
and
c
,to
be step functions. Antigenicity,
c
, will signal stronger to the immune system during
gene therapy and the clearance of tumor cells,
a
, will be strongly enhanced after
gene therapy. Finally the source term representing TIL cells,
s
1
(
/
(
E
+
f
)
should be time
dependent. We can also combine this with a self-limiting gene therapy treatment for
tumors, which affects the growth rate of the tumor,
r
2
, by allowing it to be a step
function that decreases its growth rate. The new equations are:
⎧
⎨
t
)
E
E
=
c
(
t
)
T
−
μ
2
E
+
p
3
f
+
s
1
(
t
)
,
E
+
(5)
⎩
ET
T
=
r
2
(
t
)
T
(
1
−
bT
)
−
a
(
t
)
g
2
.
+
T
It is this model that we analyze both analytically and numerically in the next
sections. We define the parameters of system (
5
), their values as well as their
ranges of variation in Table
1
. They are mostly based on previously published data
(cf.,[
5
,
7
]).
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