Biomedical Engineering Reference
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migrated to the lymph node. The variable
H
0
is the concentration of naıve CD4+
killer T cells,
H
is the concentration of effector CD4+ cells,
K
0
is the concentration
of naıve CD8+ (killer) T cells,
K
is the concentration of effector CD8+ cells, and
R
is the concentration of iTregs. In addition,
P
is the concentration of positive growth
signal (e.g., IL-2).
Equations (
7
)and(
8
) are identical to Eqs. (
1
)and(
2
) in Sect.
2.1
. Equations (
9
)
and (
11
) pertain to naıve CD4+ and CD8+ T cells, respectively. The CD4+ and
CD8+ populations are replenished at constant rates
s
H
and
s
K
, respectively, and die at
a proportional rate
0
. The third terms in Eqs. (
9
)and(
11
) are the rates of stimulation
of naıve CD4+ and CD8+ T cells by mature APCs. The bilinear form of this term
follows the law of mass action where
k
is the proportionality constant (or kinetic
coefficient). We assume that all cell-cell or cell-signal interactions follow the same
law of mass action.
Equation (
10
) pertains to effector CD4+ cells. The first term gives the rate at
which activated naıve CD4+ T cells enter the effector state after finishing the
minimal developmental program of
m
1
cell divisions. The time delay
δ
1
is the
duration of the minimal developmental program. The second term is the rate at
which effector CD4+ cells are stimulated by mature APCs for further division, and
the third term is the rate in which cells reenter the effector CD4+ population after
having divided once. The time delay
σ
1
is the duration of one CD4+ cell division.
The fourth term is the rate at which effector CD4+ cells exit the population through
death at rate
ρ
H
or differentiation into iTregs at rate
r
. The final term is the rate at
which effector CD4+ cells are suppressed by iTregs.
Equation (
12
) pertains to effector CD8+ cells. The first term gives the rate at
which activated naıve CD8+ T cells enter the effector state after finishing the
minimal developmental program of
m
2
cell divisions. The time delay
δ
2
is the
duration of the minimal developmental program. The second term is the rate at
which effector CD8+ cells are stimulated by positive growth signal for further
division, and the third term is the rate at which cells reenter the effector CD8+
population after having divided once. The time delay
σ
2
is the duration of one CD8+
cell division. The fourth term is the rate at which effector CD8+ cells die at rate
ρ
δ
K
.
The final term is the rate at which effector CD8+ cells are suppressed by iTregs.
Equation (
13
) pertains to positive growth signal. The first two terms are the
rates at which positive growth signal is secreted by effector CD4+ and CD8+ cells,
respectively. The third term is the decay rate of positive growth signal. The fourth
and fifth terms are the rates at which positive growth signal is consumed by effector
CD8+ cells and iTregs, respectively.
Equation (
14
) pertains to iTregs. The first term is the rate at which effector
CD4+ cells differentiate into iTregs The second term is the rate at which iTregs
are stimulated by positive growth signal for further division, and the third term is
the rate at which cells reenter the iTreg population after having divided once. The
time delay
1
is the duration of one CD4+ cell division. The fourth term is the rate
at which iTregs die. We assume that iTregs have the same division time and death
rate as CD4+ cells.
ρ
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