Biomedical Engineering Reference
In-Depth Information
2.
MODEL
2.1. General Forms
The components of the model that will be described here are the soluble
factors, including cytokines, chemokines, their soluble receptors, and bacterial
toxins; and cells, including bacteria and phagocytes.
2.2. Cells
The number of cells of any given type is not specified in advance, and will
generally change through division, death, emigration and immigration. Each cell
has an internal state with continuum and discrete components represented by the
pair (
s
,
i
) where the
k
-dimensional vector
s
is the continuum state and
i
indexes
the discrete states. This internal state changes according to a Markov process
whose rates are functions of the cell's external environment denoted by the vec-
tor
E
(
t
). If
f
i
(
s
,
t
)
dv
is the probability of finding the cell within the infinitesimal
volume element
dv
centered at the state (
s
,
i
) at time
t
, the evolution of the prob-
ability function is given by the equation
s
S
s
S
[
]
f
(,)
s
t
=
g
(, ()) (,)
s E
t
f
s
t
+
K
(, ()) (,)
s E
t
f
s
t
,
[1]
i
ai
i
ij
i
s
t
s
s
a
=
1
j
=
1
a
which can be described by saying that with the discrete state
i
fixed, the state
s
obeys the deterministic ordinary differential equation
ds
dt
=
a
g
(, ())
sE
t
,
ai
so that all of the stochasticity comes from the jumps from one discrete state to
another (9).
All cellular types count among their internal variables their post-mitotic age
and binary variables indicating vital status and mitotic status. Beyond these, any
number of categorical or numerical degrees of freedom can be included.
In practice, the microsimulation generates sample paths of Eq. [1] using the
corresponding Ito stochastic differential equation.
2.3. Cell Motility and Chemotaxis
The motions of the cells are described by stochastic processes that I have
developed as straightforward generalizations of the Langevin equations.
Let the position of a cell be denoted
x
and its velocity
v
. Then the motion of
this cell is treated using the coupled Ito stochastic differential equations
