Biomedical Engineering Reference
In-Depth Information
d
x
=
v
dt
,
[2]
md
v
=+
(
F
GC
p
v
)
dt
+
dW
,
[3]
1
and
†
2
d
pI p
=
(
)(
H
d
W u
+
D
dt
)
H
p
,
dt
[4]
2
where
p
is a unit-vector indicating the direction of the cell's polarization and
u
is
a vector giving amplitude and direction of the bias due to the chemotactic fac-
tors. Eq. [2] simply states the usual relationship between velocity and position.
Eq. [3] describes the variation in speed from both deterministic and random
sources, given by the external forces
F
and the internal motive force G
p
, on the
one hand, and the fluctuating forces described by the three-dimensional Wiener
process,
d
W
1
. The superscript dagger indicates matrix transpose. Eq. [4] de-
scribes biased diffusion of the polarization vector on the surface of the unit
sphere. The first term gives the stochastic driving term of this process, with in-
tensity H;
d
W
2
is another three-dimensional Wiener process independent of the
first. The second term arises "automatically" and can be thought of as preserving
the length of
p
. The last term represents bias toward the fixed direction given by
the vector
u
whose magnitude gives the strength of the bias. The chemotactic
constant D gives the rate of adjustment toward the preferred direction.
The bias vector is related to the local chemokine concentration through
B
B
1
c
()
x
B
c
()
x
u
=
=
c
()
x
.
[5]
1
+
C
c
( )
x
B
2
¯
B
1
+
C
c
( )
x
¡
°
¢
±
2.4. Soluble Factors
I assign arbitrarily to each soluble factor an index represented by a roman
letter, and to each individual cell an index represented by a greek letter, for ease
of description. Designate by
c
i
(
x
,
t
) the local concentration of soluble factor
i
at
location
x
and time
t
. This concentration evolves according to reaction-diffusion
equations of the form
s
c
n
i
(,)
x
t
=
[
T
()
t
S
() (,)](
t c
x
t
E
x
x
())
t
i
N
i
N
i
N
s
t
N
=
1
£
²
¦
¦
m
¦
¦
+
D
2
R c
(,)
x
t
M
c
(,)
x
t
.
[6]
¤
»
¦
i
ij
j
i
¦
i
¦
¦
¥
¼
j
=
1
