Biomedical Engineering Reference
In-Depth Information
A
1
(
)=
(
)
(
)
−
(
)
,
t
a
t
A
0
t
d
1
A
1
t
(21)
H
i
(
s
H
,
i
−
δ
0
H
i
(
H
i
(
t
)=
t
)
−
k
i
A
1
(
t
)
t
)
,
(22)
H
i
(
2
m
1
k
i
A
1
(
H
i
(
t
)=
t
−
σ
1
)
t
−
σ
1
)
−
k
i
A
1
(
t
)
H
i
(
t
)+
2
k
i
A
1
(
t
−
ρ
1
)
H
i
(
t
−
ρ
1
)
−
(
δ
H
+
r
)
H
i
(
t
)
−
kR
total
(
t
)
H
i
(
t
)
,
(23)
K
i
(
s
K
,
i
−
δ
0
K
i
(
K
i
(
t
)=
t
)
−
k
i
A
1
(
t
)
t
)
,
(24)
K
i
(
2
m
2
k
i
A
1
(
K
i
(
t
)=
t
−
σ
2
)
t
−
σ
2
)
−
kP
(
t
)
K
i
(
t
)+
2
kP
(
t
−
ρ
2
)
K
i
(
t
−
ρ
2
)
−
δ
K
K
i
(
t
)
−
kR
total
(
t
)
K
i
(
t
)
,
(25)
P
(
t
)=
r
1
H
total
(
t
)+
r
2
K
total
(
t
)
−
δ
P
P
(
t
)
−
kP
(
t
)
K
total
(
t
)
−
kP
(
t
)
R
total
(
t
)
,
(26)
R
i
(
t
)=
rH
i
(
t
)
−
kP
(
t
)
R
i
(
t
)+
2
kP
(
t
−
ρ
1
)
R
i
(
t
−
ρ
1
)
−
δ
H
R
i
(
t
)
.
(27)
n
. As in Sect.
2.2
,
A
0
is the concentration of APCs at the site of infection and
A
1
is the concentration of
APCs that have matured, started to present target antigen, and migrated to the lymph
node. For each clone
i
,thevariable
H
i
is the concentration of naıve CD4+ (helper)
T cells,
H
i
is the concentration of effector CD4+ cells,
K
i
is the concentration of
naıve CD8+ (helper) T cells, and
K
i
is the concentration of effector CD8+ cells, and
R
i
is the concentration of iTregs. Finally,
P
is the concentration of positive growth
signal.
Equations (
20
)and(
21
) are identical to Eqs. (
7
)and(
8
). Equations (
22
), (
24
),
and (
27
) describe the dynamics of the naıve CD4+ T cells, naıve CD8+ T cells,
and regulatory cells, respectively, for each clone
i
. These equations are identical to
Eqs. (
9
), (
11
), and (
14
).
The assumption about the nonspecific suppression of the activated CD4+ and
CD8+ T cells is encoded into the model in Eqs. (
23
)and(
25
). The last term in both
equations shows that the suppression of the activated cells is done using the iTregs
that originated from all clones.
Finally, the dynamics of the positive growth signal is proportional to the total
population sizes of the activated CD4+ and CD8+ T cells, as well as the total number
of iTregs in the system.
Here
H
total
=
∑
H
i
,
K
total
=
∑
K
i
,and
R
total
=
∑
R
i
for
i
=
1
,...,
4R su s
In this section we present results obtained by simulating the mathematical
model from Sects.
2
and
3
. We start in Sect.
4.1
with simulations of the basic
model of adaptive regulation. We focus our attention on demonstrating the
robustness of the model to large variations in precursor frequencies. Additional
simulations of this model can be found in [
17
]. In Sect.
4.2
we present simulations
of the immunodominance models. Most of the simulations are of the basic model
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