Biomedical Engineering Reference
In-Depth Information
1
b
= 3
b
= 10
0.5
0
0
2
4
6
8
10
t
(days)
Fig. 2
Graphs of the antigen function
a
(
t
)
givenbyEq.(
6
)for
b
=
3and
b
=
10 when
c
=
1.
The function
a
(
t
)
represents the time-dependent rate that immature APCs pick up antigen and are
stimulated
T cell population into CD4+ and CD8+ T cells. CD4+ T cells are the primary
producers of positive growth signal, particularly IL-2, and CD8+ T cells are the
main proliferators. Furthermore, iTregs differentiate from effector CD4+ cells and
suppress both effector CD4+ and CD8+ cells [
24
]. The extended model can be
summarized in six steps (illustrated in Fig.
3
):
1. APCs mature, present relevant target antigen, and migrate from the site of
infection to the draining lymph node.
2. In the lymph node, APCs activate naıve CD4+ and CD8+ T cells that enter a
minimal developmental program of
m
1
or
m
2
cell divisions, respectively.
3. Effector CD4+ and CD8+ T cells both secrete positive growth signal at different
rates.
4. CD4+ and CD8+ T cells that have completed the minimal developmental
program become effector cells that keep dividing as long as they are not
suppressed by iTregs.
CD4+ T cells proliferate in response to interactions with APCs (It is assumed
that CD4+ T cells produce enough IL-2 to stimulate their own growth in
an autocrine loop. Hence, we do not explicitly model the secretion and
consumption of IL-2 by CD4+ T cells.).
CD8+ T cells proliferate after consuming free positive growth signal.
5. During the immune response, some effector CD4+ T cells differentiate into
iTregs.
6. The iTregs suppress effector CD4+ and CD8+ T cells and proliferate after
consuming free positive growth signal.
The mathematical model corresponding to Fig.
3
is formulated as the following
system of DDEs:
A
0
(
t
)=
s
A
−
d
0
A
0
(
t
)
−
a
(
t
)
A
0
(
t
)
,
(7)
A
1
(
t
)=
a
(
t
)
A
0
(
t
)
−
d
1
A
1
(
t
)
,
(8)
H
0
s
H
−
δ
0
H
0
H
0
(
t
)=
(
t
)
−
kA
1
(
t
)
(
t
)
,
(9)
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