Biomedical Engineering Reference
In-Depth Information
The first term within square brackets on the right-hand side represents the secre-
tion of factor
i
by individual cell N. The secretion rate of this factor by cell N is
T
i
N
, and the position of this cell is
x
N
(
t
). The positions of the cellular sources are
in turn governed by stochastic differential equations (above). Similarly, the latter
term in square brackets represents adsorption of factor
i
onto cell N. Both the
secretion and adsorption rates are time-dependent, since both depend on the state
of cell N, which itself changes in time. The first term within curly brackets
represents diffusion, with diffusion coefficient
D
i
, where /
2
is the Laplacian op-
erator. The sum in the middle of the bracketed term represents removal via com-
plex formation with other soluble factors, where
R
ij
is the effective rate of
removal of soluble factor
i
by complex formation with factor
j
. The last term
represents degradation of factor
i
or its removal by mechanisms not explicitly
treated in the model, such as complex formation with factors treated as existing
at constant concentration or adsorption onto cells likewise treated as static.
For the examples in this chapter these partial differential equations were
integrated using a semi-implicit forward scheme on a 80 + 80 + 80 cubic grid.
2.5. Specific Forms
I have used this modeling platform to simulate the induction and resolution
of microlocal inflammation during bacterial infection. This approach should be
regarded as analogous to an in vitro experiment, in the sense that the compo-
nents are simplified and highly controlled. The components of this specific study
are nonmotile bacteria and phagocytes. Each bacterial agent has three discrete
states and a continuous age, as well as position. Each bacterial agent secretes a
soluble factor chemotactic to phagocytes. Several such factors are familiar, most
notably N-formyl-methionyl-leucyl-phenylalanine (fMLP). These bacterial
agents are subject to Brownian motion but are not polarized and do not move
under their own power. This characteristic is not common to all bacteria, cer-
tainly, but many bacteria of medical significance, including
Bacillus anthracis
are nonmotile.
The phagocytic agents have three internal states: quiescent, activated, and
refractory. The transition rate from quiescent to activated is zero in the absence
of bacteria and proinflammatory cytokine, and is an increasing but saturating
function of the local concentration of proinflammatory cytokine. This transition
rate is finite and constant when in contact with bacteria; the rates from proin-
flammatory cytokine and bacterial contact are additive. While in the activated
state, the phagocyte secretes proinflammatory cytokine at a constant rate until its
transition to a refractory state. The refractory state is characterized by a loss of
responsiveness to proinflammatory cytokine and bacterial contact, as well as
cessation of PIC secretion. In addition refractory phagocytes shed a soluble re-
ceptor for PIC, which binds it and neutralizes its activity.
