Biomedical Engineering Reference
In-Depth Information
Figure 5.27
Monte-Carlo diffusion in a quasi-1D geometry. Left, from top to bottom: trajectories
of 1, 10, and 500 nanoparticles in a time interval of 50 seconds. Right, from top to bottom: end point
of the trajectories at
t
= 50 seconds.
5.4.1.2 Three Dimensional Case
In the three-dimensional case, a particle moves in a time step D
t
from the location
(
x, y, z
) to the location (
x
+ D
x
,
y
+ D
y
,
z
+ D
z
). The random walk algorithm is
D =
x
4
Ddt
cos( ) sin( )
α β
D =
y
4
Ddt
sin( ) sin( )
α β
D =
z
4
Ddt
cos( )
β
(5.49)
α
=
random
(0,2 )
π
β
=
arccos (1 2
-
random
(0,1))
The angles
a
and
b
are defined in Figure 5.28. Recall the definition of the angle
b
in (5.49). If we had taken simply
b
= random(0, 2
p
), the
z
-direction would be a
preferred direction of displacement. If we want a uniformly distributed direction
angle, we have to take a random
a
angle between 0 and 2
p
and a random
z
coordi-
nate between -1 and +1. This random
z
-coordinate is obtained by the function 1-2
random(0,1) and the angle
b
is equal to arcos(1-2 random(0,1)).