Biomedical Engineering Reference
In-Depth Information
The soluble factors are the fMLP-like factor secreted by bacteria and
chemotactic to phagocytes; the proinflammatory cytokine, modeled on tumor
necrosis factor, produced by activated phagocytes, and chemotactic to them, and
bound and sequestered by their soluble receptor, and the soluble receptor itself.
The system of interest to us for this chapter consists of two cells types and
three soluble factors. The cells are bacteria and phagocytes. The bacterial agents
have just the three obligate states: alive (nonmitotic), mitotic, and dead. The
age-dependent hazard function for transition from nonmitotic to mitotic is
h
(
a
) =
B
H
(
a
-
T
CYC
), where
a
is the age, B is a positive constant and
T
CYC
is the mini-
mum cell-cycle time. The bacterial death rate depends on the presence of phago-
cytes; I neglect natural mortality. In the presence of one or more quiescent
phagocytes in contact with the bacterium, the death rate is constant and equals
1/hr. In contact with activated phagocytes, the death rate is 20/hr. Mitotic cells
are immediately replaced by a pair of age-zero daughter cells at locations adja-
cent to the location of the parent cell. Dead cells are immediately removed from
the simulation.
For the bacterial agents, Eq. [1] becomes
n
s
s
p
,
f at
(,)
=
f at
(,)
B
H a
(
T
) (,) (
f at
N
H x
(
x
R
) (,)
f at
C
C
CYC
C
i
C
s
t
s
a
i
=
1
s
s
,
f
(,)
at
=
f
(,)
at
+
B
H a
(
T
) (,)
f at
M
M
CYC
C
s
t
s
a
n
s
s
p
,
f
(,)
at
=
f
(,)
at
BN
+
H x
(
x
R
) (,)
f at
D
D
i
C
s
t
s
a
i
=
1
where
H
is the Heaviside function and B is a positive constant. The hazard func-
tion for death is zero in the absence of phagocytes, and is constant with value N
when there is a phagocyte (indexed from
i
= 1 to
n
p
) is within distance R of the
bacterium. The position in space of the bacterium is
x
and those of the phago-
cytes
x
i
.
The phagocytic agents have three internal states, but are never mitotic.
These states are quiescent, activated, refractory, and dead. The stochastic transi-
tion from quiescent to activated depends on the presence of either bacteria or the
proinflammatory cytokine (PIC). In the presence of one or more bacteria, the
rate of this transition is 0.5/min. In the presence of the optimal concentration of
PIC the rate is 2/min. As a function of the local concentration of PIC, the rate is
zero for [PIC] = 0, and increases with [PIC], saturating at large [PIC]. In the
activated state, phagocytes kill bacteria more rapidly than do quiescent phago-
cytes. Activated phagocytes go on to become refractory, in which state they are
unresponsive to stimulation. The mean time spent in the activated state is inde-
