Biomedical Engineering Reference
In-Depth Information
Tabl e 1
Estimates for model parameters
Parameter
Description
Estimate
A
0
(
0
)
Initial concentration of immature APCs
10
K
0
(
0
)
Initial concentration of naıve T cells
0.04
d
0
Death/turnover rate of immature APCs
0.03
d
1
Death/turnover rate of mature APCs
0.8
Death/turnover rate of naıve T cells
δ
0
0.03
δ
1
Death/turnover rate of effector T cells
0.4
s
A
Supply rate of immature APCs
d
0
A
0
(
0
)=
0
.
3
δ
0
K
0
s
K
Supply rate of naıve T cells
(
0
)=
0
.
0012
k
Kinetic coefficient
20
m
# of divisions in minimal developmental program
7
n
Maximum number of antigen-dependent divisions
3 to 10
ρ
Duration of one T cell division
1/3
σ
Duration of minimal developmental program
1
+(
m
−
1
)
ρ
=
3
a
(
t
)
Rate of APC stimulation
Eq. (
6
)
b
Duration of antigen availability
10
c
Level of APC stimulation
1
r
Rate of differentiation of effector cells into iTregs
0.01
Concentrations are in units of k/
μ
L, and time is measured in days
stimulation, we assume that it starts at 0, remains positive for some time, and
eventually returns to 0. To generate a smooth function for
a
(
t
)
,welet
e
−
1
/
x
2
,
if
x
≥
0
,
φ
(
x
)=
,
<
,
0
if
x
0
and set
c
φ
(
t
)
φ
(
b
−
t
)
a
(
t
)=
,
(6)
2
φ
(
b
)
where
b
0. The variable
t
is defined such that mature APCs begin appearing in
the lymph node at
t
,
c
>
0, although the infection may have begun slightly earlier. We
estimate that the duration of antigen availability,
b
, is about 10 days. Furthermore,
we estimate that the level of APC stimulation,
c
, is around 1. (See Fig.
2
for graphs
of
a
=
(
t
)
for
b
=
3and
b
=
10 when
c
=
1.)
2.2
Extended Model of Adaptive Regulation: Helper and Killer
T Cells
The mathematical model presented in this section is an extension of the model
in Sect.
2.1
. The main extension of the model is to separate the nonregulatory
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