Wound Healing: Multi-Scale Modeling - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 321

Fig. 1 The trajectory of a
bacterium originally located
at ð 0 ; 0 ; 0 Þ and moving
according to Brownian
motion with r ¼
2 : 6833 10 5 m/ p ; which

corresponds to a bacillum at
37 C

d ð x Þ¼ 0 ; for all ð x ; y ; z Þ 6¼ð 0 ; 0 ; 0 Þ;
R

ð 7 Þ

d ð x ; y ; z Þ dX ¼ 1 ;

X 3ð 0 ; 0 ; 0 Þ

3

r 2
2

; which represents the

where X is subset of R with nonzero measure. For

r 2

diffusivity, we used

2 ¼ 3 : 6 10 10 m 2 /s (bacillum at 37 C). Note that if D rep-

resents the diffusivity of a species, then r ¼

p

: Equation ( 5 ) represents a
fundamental solution to the three-dimensional diffusion equation (in an unbounded
domain), and it represents the probability density that the bacterium is localized at
position ð x ; y ; z Þ at time t. Note that Eq. ( 5 ) is very helpful in deriving the relation
between the stochastic differential equation of Langevin type with zero drift, see
Eq. ( 1 ) and the diffusion equation ( 12 ). The probability that a region X contains
the bacterium at time t is then given by
P ð t ; X Þ¼ Z

2D

f ð t ; ð x ; y ; z ÞÞ dX ; ð 8 Þ

X

and note that R
R

3 f ð t ; ð x ; y ; z ÞÞ dX ¼ 1 for t 0 : We remark that if drift is incor-
porated through l ¼ð l x ; l y ; l z Þ; then Eq. ( 1 ) becomes

dX ð t Þ¼ l x dt þ rdW ð t Þ; dY ð t Þ¼ l y dt þ rdW ð t Þ; dZ ð t Þ¼ l z dt þ rdW ð t Þ;

ð 9 Þ

for t [ 0 ; with exact solution, if l x ; l y ; l z and r are constant,

X ð t Þ¼ X 0 þ l x t þ rW ð t Þ; Y ð t Þ¼ Y 0 þ l y t þ rW ð t Þ;

ð 10 Þ

Z ð t Þ¼ Z 0 þ l z t þ rW ð t Þ;