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)).
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