Biomedical Engineering Reference
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immunodominance. This extended immunodominance model is the model
published in [
16
]. The basic immunodominance model is new. While the basic
model is less accurate from a biological point of view, it still captures the main
principles on which the extended model is based.
3.1
A Basic Model
We extend the model from Sect.
2.1
to polyclonal T cell responses. The model
includes
n
T cell clones that react to mature antigen-bearing APCs at different rates,
k
i
. The model is formulated as the following system of DDEs:
A
0
(
t
)=
s
A
−
d
0
A
0
(
t
)
−
a
(
t
)
A
0
(
t
)
,
(15)
A
1
(
t
)=
a
(
t
)
A
0
(
t
)
−
d
1
A
1
(
t
)
,
(16)
K
i
(
0
K
i
(
K
i
(
)=
s
K
,
i
−
δ
)
−
(
)
)
,
t
t
k
i
A
1
t
t
(17)
K
i
(
2
m
k
i
A
1
(
K
i
(
t
)=
t
−
σ
)
t
−
σ
)
−
k
i
A
1
(
t
)
K
i
(
t
)+
2
k
i
A
1
(
t
−
ρ
)
K
i
(
t
−
ρ
)
(18)
−
(
δ
1
+
r
)
K
i
(
t
)
−
kR
total
(
t
)
K
i
(
t
)
,
R
i
(
t
)=
rK
i
(
t
)
−
δ
1
R
i
(
t
)
,
(19)
where
R
total
=
∑
n
. As before,
A
0
is the concentration of immature
APCs at the site of infection, and
A
1
is the concentration of mature antigen-bearing
APCs in the lymph node. The variables
K
i
,
K
i
,and
R
i
are the concentrations of
naıve, effector, and regulatory T cells with specificity #
i
.
Equations (
15
)and(
16
) for the APCs are identical to Eqs. (
1
)and(
2
).
Equations (
17
)-(
19
) are analogous to Eqs. (
3
)-(
5
), except that each T cell clone
is supplied at a different rate
s
N
,
i
, has its own kinetic coefficient
k
i
, and effector
cells can be suppressed by any regulatory cell, independent of their origin. The
supply rate,
s
K
,
i
, of T cell clones is related to the initial concentration of that clonal
population by
s
K
,
i
=
R
i
and
i
=
1
,...,
d
1
K
i
(
0
)
. From the estimates in [
17
], the kinetic coefficient
k
i
=
40 and
p
i
is the probability that T cells of the
i
th clone react
to antigens presented on the APCs. All other parameters are taken from Table
1
.
p
i
k
0
,where
k
0
=
3.2
An Extended Immunodominance Model: Including
the Helper T Cells
Following the basic principle of the model in Sect.
3.1
, we extend the mathematical
model of Sect.
2.2
to polyclonal T cell responses. The model includes
n
clones that
react to mature antigen-bearing APCs at different rates,
k
i
, and is formulated as the
following system of DDEs:
A
0
(
t
)=
s
A
−
d
0
A
0
(
t
)
−
a
(
t
)
A
0
(
t
)
,
(20)
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